Advances in Energy Transfer Processes

THE SCIENCE AND CULTURE SERIES — SPECTROSCOPY Series Editor: Antonino Zichichi

ADVANCES IN ENERGY TRANSFER PROCESSES

Editors

Baldassare Di Bartolo and Xuesheng Chen World Scientific

ADVANCES IN ENERGY TRANSFER PROCESSES

This page is intentionally left blank

•^SS^^SSS^^^SMMSiS^^MSM^MM-

ADVANCES IN ENERGY TRANSFER PROCESSES Proceedings of the 16th Course of the International School of Atomic and Molecular Spectroscopy Erice, Sicily, Italy

17 June - 1 July, 1999

Editors

Baldassare Di Bartolo Boston College, USA

Xuesheng Chen Wheaton College, USA

Series Editor

Antonino Zichichi

Vk§> World Scientific wb

Jersev London »Sint New Jersey • Singapore'Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

ADVANCES IN ENERGY TRANSFER PROCESSES Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4728-1

Printed in Singapore.

Si utilitas amicitias conglutinaret, eadem commutata dissolveret. Sed quia natura mutari non potest, idcirco verae amicitiae sempiternae sunt. (Friendships based on convenience, when circumstances change, end quickly. But, just as nature cannot change, true friendships are eternal.) Cicero, De Amicitia.

These proceedings are dedicated to the memory of Dr. Alberto Gabriele.

VI

•> - 5 , 7 1 •'

H P*

u HH

PN

W H

Drawings, by Dr. Elisabeth Kurtz (The first five drawings are scenes from Erice, the sixth drawing is a view ofFavignana, and the seventh drawing is a view of Selinunte.)

This page is intentionally left blank

fl«»M.j»g •tymsprw • i^o-jf Wivs'ls

IX

J(g

1* •«§J

XII

Ob

X

XIII

CDr. E\if3tM>

Iflufe.

IXorltfullt,

i^'>|2 m2

Summary By virtue of the so-called Addition Theorem we can express the Legendre Polynomial of the cosine of the angle between two position vectors in terms of the

19 spherical harmonics of the angular coordinates of the two vectors. This allows one to expand an electrical charge distribution in terms of monopoles, dipoles, quadrupoles, etc. In a similar way we can express the interaction Hamiltonian between two atoms in fixed positions, as they would be in a solid, in terms of multipole operators of the two atoms. The direct term of the matrix element responsible for the energy transfer is then expressed in terms of matrix elements pertaining to the individual atoms. This allows us to relate the energy transfer process to spectral parameters of the two atoms involved in the transfer.

2. Different Types of Interaction Homines dum docent discunt. Lucius Annaeus Seneca 2.1. Multipolar Electric Interactions An electric multipole of a charge distribution p(x) with x = (r,d,) is defined as follows: Dl,m = J ^ f j \dTfXx)rlYlm{0,)

(67)

If we set p(x) = YeS(x-xs)

(68)

we obtain im=J~r D,„ = lJdTYe5{x-xsyYlm(e,|2Tt|^p:|y)|2 m=-l

(71)

JL>"=-1

C ' is a dipole-quadrupole term

2

2 i>i^i«>l2T i|(«#>i 2 H i|(«'i^M T ti^iwi L

_/_6!_ Rs 3! 5!

m=-l

_L'M=-2

,m=-2

(72)

Lm=-l

10>

C* is a quadrupole-quadrupole term e4 8! fl10 5! 5!

i|(-'l^l«>|2T i||.C>i|&')| can in principle be derived from spectroscopic data. Example: dipole-dipole term In classical electrodynamics the power irradiated by an oscillating electric dipole M cos cot is given by (74)

3c3 A classical oscillator of amplitude M has two Fourier components Mcosat = -Meim

+-Me-i(0'

(75)

21 with frequency co and -co. Classically we do not distinguish frequency co from frequency -co, namely photons absorbed from photons emitted. Quantum mechanics, however, allows only one of the two components to enter the relevant matrix element M

Jessica, - »

(76)

2

\MQM\

and co

3?

-\2M\2~M

(77)

Therefore A = rate of decay = 1 — r„

energy emitted per unit time energy of a photon

4co4\M\\ , 3c3ha

8^3„,|2 r\M\ 3hc3

(78)

where xQ = radiative lifetime. Then

e2^a'\Dl\a)f =M

3hc3 1 &mo3 T„

(79)

The / number, a quantity usually derived from absorption data, is defined as follows:

f-^H

(80)

We can then write

WHWt-vt-gr

(81)

If we use Eq. 79 and Eq. 81 in Eq. 71 we obtain (

|\

AB/\

R 6

_ 1 3e2c3h6 fB R6 4mEA rA

3!3,

3hc3 1 ^^he2 , ^ > %nvy T„ 2mco"

(82)

22

where E=fua and probability of radiative decay probability of radiative decay + probability of nonradiative decay

£=-

1/T„.

(83)

1/r, XA is the measured lifetime and X0A is the radiative lifetime of atom A. The transfer rate is then given by WAB = ^Kl^ABDf \gA{E)gB{E)dE

Je'c^K -2-3n5nffBB rgA(E)g {E)gBB(E) '^ JE_ 2mR6 xA J E4 "

1* "» f y^ rA^R.

where RO = ^/B

' f c W ^ l^EYE)dE 2m

= radius at which the transfer rate is equal to the decay rate

(85)

Let w^, w^ and w% be the energy transfer rates by dipole-dipole, dipolequadrupole and quadrupole-quadrupole mechanisms, respectively. We can compare the magnitudes of w^ and w%:



^=£|KK 2>I2 13[H2 1^ f~R>a; W«

2

(aY [ R )



23 If the electric dipole transition in atom B is not allowed, then \\D/\ < a0 and it is possible that

w&>wft

(89)

Note that

V

AB dd V vAB

MV> 2

1 R4 ( D ' ) > . ) '

(90)

Summary In this section we redefine the multipoles that appear in the expansion of the matrix element of the direct term of the interaction Hamiltonian leading to energy transfer between two atoms A and B. We then deal with the expansion of the square of the matrix element considered above in terms of dipole-dipole (d-d), dipole-quadrupole (d-q), quadrupole-quadrupole (q-q), etc. For each term in the expansion we have two quantities, one related to atom A and one to atom B. These quantities are in principle derivable from the spectral data of the two atoms. An important example is provided by the dipole-dipole (d-d) term. It is shown that it is possible to use such spectral data as the lifetime of atom A and the f-number of atom B in order to calculate the relevant squared matrix element that enters the probability per unit time of the energy transfer process. An expression for such probability in terms of these spectral parameters is actually derived. A characteristic distance is introduced at which the energy transfer rate is equal to the decay rate of atom A. A quantitative comparison between the d-d and d-q transfer rate is made. It is shown that, if the electric dipole atomic transition of atom A is not allowed, then the d-q energy transfer rate may be larger than the d-d transfer rate. Finally it is found that the ratio between the q-q and the d-d energy transfer rates is approximately equal to the fourth power of the ratio of the Bohr radius to the distance between the two atoms. 2.2. Multipolar Magnetic Interactions Magnetic multipoles can be defined similarly to electric multipoles. The magnetic field due to a magnetic dipole m placed at position x, as in Figure 6, is given by

B(x) = -7^(x)

(91)

24

X Fig. 6. Positioning of a magnetic dipole.

where

\x-x'\

(92)

If we have a distribution of magnetic dipoles represented by the function M(x)= magnetic dipole per unit volume, then the potential (j>(x) is given by

J

\x~x\ (93)

where the notation IP indicates an integration by parts. Then

J

\r — r'

(94)

25 If in the two formulae above we replace the quantity [-V • M(x)j with the electric charge density p(x), then Eq. 93 and Eq. 94 give us the electric potential and the electric field of a charge distribution, respectively. We may then use for the magnetic multipoles the formulae related to the electric multipoles by replacing in them the quantity p(x) with the quantity [-V, • M(*)j which for this reason is sometimes called magnetic charge distribution. A magnetic multipole of a magnetic dipole distribution M{x) with x = (r,0,0) is then defined, in accordance with Eq. 67, as DZg" = \ ^ \

d

* P x • M(x)\ rlYlm(6,)

(95)

It is clear that £&T=J[-V-Af]rfr = 0 and

T-Jl-*-****-^*!^)* J

J

dx

dy

J

dz

=-$MzdT = Mz where Mz = z-component of total magnetic dipole. Similarly A T

=

+ ^ ( M , ± M , )

We shall now consider the static and dynamic effects of the magnetic interactions. Consider the dipole-dipole interaction between atoms in solids. The magnetic field due to a dipole UB at distance R = 2.2 x 10"8 cm is given by

""^ = lS = 1 ° 3 g a U S S On the other hand, the energy of a dipole in the internal field a ferromagnet is on the order of kTc = UB Heff, where Tc = Curie temperature ~ 1,000 K. We find for Heff\he order of magnitude

26 1Q- 16 -10 3 ,_7 ° 1 0 gaUSS in -20

„ kTc L g „)| 2 |(oL g n )| 2 - / 4

(96) (97)

where fis = Bohr's magneton. We find

l(el) w^magn)

(eaj2

-- (6.7 x l 0 4 f = 4.5 x10 s

(98)

A

The coefficient 109 makes us aware of the fact that magnetic interactions play a negligible role in energy transfer processes. Summary Magnetic multipoles can be defined in a way similar to the electric multipoles, by using the formulae of magnetostatics. In turn these formulas can be derived by similar expressions in electrostatics by simply replacing the electric charges with the so-called magnetic charges. The results of simple calculations show that the magnetic multipolar interactions are negligible in comparison to the multipolar electric interactions. We consider the possibility that magnetic interactions may be responsible for the magnetic ordering in solids. We find that the effective field necessary to set the magnetic order in ferromagnetic systems is much larger than the one provided by the dipole-dipole interactions. We know from other sources that the magnetic ordering in ferromagnets is due to exchange interactions. 2.3. Exchange Interactions We shall now examine more closely the matrix element of Eq. 58. The direct term can be written as follows:

27 2

2

{d (l)b(2)\—\a(l)V

(2)) = Ud (rjafi)—

*(r2)*tf (r2)dTidT2

(99)

and represents the Coulomb interaction between the charge distributions ea'(^)*a(^) and eb(r2)*b' (f2) at distance R from each other. The exchange term can be written e2

2

-{d (l)fe(2)|—|fc(l)a(2)) = ff a' ( r ^ (r,)— a(r2)b(?2f dr^ r

(100)

12

12

and represents the Coulomb interaction between the charge ed (r\)*b (^) and ea(r2)b(r2) at distance R from each other; these two charge distributions are very small if R is large. Therefore the exchange term is small if R is not small. The exchange term can also be written {d (l)fc(2)|—| *>' (l)a(2)) = -{d (l)b(2)\—Pn\a(W r

(2))

(101)

r

n

n

where Pn is an operator that interchanges the two electron coordinates. For many electron atoms (d bfijab')

= (exchange term) = (d b\ - £ — / ? > # )

(102)

The overlap of the electron charges makes the condition R > rAs, rat invalid. However, if the overlap is small the multipolar part can be treated as previously. As for the exchange part,

it can be replaced by an equivalent operator5: \ h

n% = - I S £ ; r ' («,*) c'm(8s, are more difficult to estimate. Magnetic order in solids is a static effect of the exchange interactions.

(5)

(6)

Summary We revisit the matrix element of the interaction Hamiltonian responsible for the transfer of energy between two atoms. It consists of a direct term and of an exchange term. Both terms can be interpreted as interactions between charge distributions. For the direct term these distributions are localized at the sites of the two atoms; for the exchange term the two distributions are strongly affected by the distance between the two atoms, and, of course, become very small for large distances. If the overlap between the wavefunctions of the two atoms is small, then the electric multipolar part can be treated as previously. If all the orbitals have the same asymptotic radial dependence, then the matrix element of the exchange term of two atoms at distance R from each other will have a dependence ~exp(-2R/ro). rois an effective Bohr radius: ro < 0.3 A for rare earths, r0 < 0.6 A for transition metal ions. In view of these small values exchange interactions are generally small and negligible.

29 2.4. Phonon-Assisted Energy Transfer The probability for energy transfer between two ions in solids is proportional to the overlap integral \gA{E)gB{E)dE =

h\gA{a)gB(a)do

(105)

where gA(a>) and gB((o) are the line shape functions for ions A and B, respectively. If we consider the case of two Lorentzian lines of width ACOA and ACOB, centered at (OA and (OB, respectively, we find a

\gA(a)gB(o))dco = -

(A©) +((O. +{(0A-aB) JT(AG))

J

(106)

where A(0= A(OA + A(OB. For sharp and well-separated lines the value of the integral in Eq. 106 and the probability for energy transfer become negligible. At low temperatures, where the lines in solids tend generally to be Gaussian, the value of the integral may be even smaller. In these circumstances the energy transfer process may be favored by the emission or absorption of a phonon whose energy compensates for the energy mismatch between the two transitions and ensures the conservation of energy in the process. The transition probability per unit time of the energy transfer process accompanied by the production of a phonon, if AE = EA - EB > 0 as in Figure 7, is given by8 2K,

\d b\HjaV)\2S[n((0)

+ \]\gA{E)gB{E - hw)dE

(107)

where S = ion-vibrations coupling parameter and h(£> = AE. If AE < 0, then the energy transfer process is accompanied by the annihilation of a phonon and In ">AB = -r-K rf b\HAB\at/fS[n(c))

+ \}\gA{E)gB{E + ha)dE

AE

A

B

Fig. 7. Two-atom system requiring phonon assistance for energy transfer.

(108)

30 If the transfer rates in both A -> B and B —> A directions are much greater than the intrinsic decay rates of A and B, phonon-assisted energy transfer processes may establish a Boltzmann distribution of populations between the excited states of A and B9. If AE » ZtOm where am is the maximum phonon frequency, energy transfer is assisted by the creation of many phonons and the probability per unit time of the process is given by1"'11: wAB(AE) = wAB(0)e^

(109)

where 0 = temperature-dependent parameter related to similar parameter for multiphonon decay rate. Summary Energy transfer can take place when there is no resonance between the energy levels of the interacting atoms. This may occur through the intervention of phonons, which may be absorbed or emitted in concomitance with the transfer process. If the gap between the two levels can be bridged by one phonon, then the probability for transfer is proportional to a atom-vibration coupling parameter and to [the number of phonons present] for an "upward" transfer, and to [the number of phonons present plus one] for a "downward" transfer. For a multiphonon downward transfer the energy transfer probability depends exponentially on the energy gap.

3. Modes of Excitation and Transfer La Republique n 'a pas besoin de savants. Head of the Tribunal that condemned Lavoisier 3.1. Setting of the Problem A typical sequence of events that includes the transfer of energy from one atom or ion called sensitizer or energy donor (D), to an atom or ion called activator or energy acceptor (A) consists of: (i) (ii) (iii)

absorption of a photon by D. energy transfer from D to A, and emission of a photon by A.

In the present treatment we shall assume that D and A are weakly interacting, so that the

31 energy level shifts due to the interaction are smaller than the width of the D and A levels. This means that the absorption bands of D and A are identifiable. Also, the present treatment will not consider the process of radiative transfer which consists of the emission of a photon by D and the absorption of the same photon by A. When such a process occurs the lifetime of D is in general not affected by the presence of A. If only ions of one type, say D, are present in the sample, and if their concentration is high, then the D -» D radiative transfer may lead to trapping of the radiation and to an increase of the measured lifetime. In such case this lifetime may depend on the size and shape of the sample. The process we will be considering consists of the nonradiative transfer from D to A (step (ii) above). Summary In this section we first follow the unfolding of the events that include the energy transfer process, starting with the excitation of the donor (also called sensitizer), the transfer of energy from the donor to the acceptor (also called activator), and finally the decay of the acceptor. We consider conditions that are not very restrictive: the two atoms involved in the energy transfer interact weakly and the energy levels of each atom are not affected by the presence of the other. We distinguish between radiative transfer and nonradiative transfer, and declare that the emphasis in what follows will be on the latter. We consider for the moment the case in which no energy transfer among donors takes place. 3.2. Pulsed Excitation Assume that we have a number No of donors and NA of acceptors and call CODA the probability of D —> D energy transfer per unit time. Assume also that O>DD the probability of D —> D energy transfer is negligible. The question we shall try to answer is the following: If we excite a number of donors with a light pulse, how will the system respond? Let the pulse of light begin at time t = -T and end at time t =0, and let T be much smaller than w ^ : T«WDA

Let Nd{0) = number of excited donors at time t=d Na{0) = number of excited acceptors at time £=0 We shall put

(HO)

32 ^(0) =0

(111)

because during the short interval of time (-T.0) no relevant D —>A transfer takes place. If Nd(0) = number of donors in the ground state at time t=0 Na(0) = number of acceptors in the ground state at time t=0 then Nd(0) = ND-Nd,(0)

(112)

Na(0) = NA

(113)

We shall call x the lifetime of the donor, in the absence of the activator: v-l=P

+ wnr

(114)

where P = probability of spontaneous emission per unit time wnr = probability of nonradiative decay per unit time We shall define pi(t) = probability that the donor at position Rt is excited at time t and p{t) = statistical average of p0) The number of excited donors at time t is given by Nd{0) p(t) and the probability of finding a donor excited at time t by p ( 0 ^ » ND

> «~>0

«

(115)

ND

The number of quanta emitted as luminescence by the donors per unit time is PNjQWt)

(116)

33 The total number of quanta emitted as luminescence by the donors is N = PNce(0)j~p(t)dt

(117)

The total number of quanta emitted by the donors, in the absence of activators, is N0 = PNj (0)j"V ( ' / r ) rff = PNd, (0)T

(118)

The quantum yield of luminescence is

iL^r-pwt

(H9)

Summary The excitation of the donor atoms is generally accomplished by sending a light beam on the sample. The exciting light can be in the form of a pulse or it may shine continuously. In the pulsed excitation mode the attention is focused on the time dependence of the donor luminescence, which gives a measure of the time evolution of the probability that a donor is excited at a certain time. The donors' decay is compared with the decay of the donors in absence of transfer. We derive an expression for the quantum yield of luminescence given by the ratio of [the number of quanta emitted by the donors in the presence of acceptors] to [the number of quanta emitted by the donors in the absence of acceptors]. 3.3. Continuous Excitation If we excite the donors with a light pulse beginning at t0 = —T and ending at t0, p(t — t0) is the response of the donor system. The response of this system to N short pulses of equal amplitude will be N _

» V->c

where V = (4/ 3)7iRy and 7fv = radius of the largest spherical volume, and

I(t) =

^je-^mR2dR

(134)

Note that the limit in Eq. 133 is taken for a large solid, but in such a way that the concentration of activators NAIV remains constant. In order to evaluate I(t) and then p(t) we need to know the function WDA(R)-

37 Summary We first set up the basic equation that gives us the time dependence of the average probability of excitation of a donor, following a pulsed excitation. The assumption is made that all donors are homogeneously distributed over the system. A relevant model-dependent quantity appears in the expression of the average probability of donor excitation: the probability per unit time that energy transfer takes place between donor and acceptor. 4. 2. Simple Models 1. Perrin Model. In this model13 f«>

w (

R{R0

- H o *>*,

(135)

Then

/( 0 = i* l-^WtfdR V J

= i *iL*i = i _ 4 = l - ^ 3 V 4 c0NA

(136)

where N N CA - concentration of acceptors = —— = 4— V 47iRll3 _i

AK

'K

C0 =—R0

= volume of donor's sphere of influence

— = number of acceptors in the sphere of influence of donor c o

y

(137) '

(138)

(139)

We have then

p{t) = e-{tlx)

lim [ / ( t ) f A = e-{tlx) V-»«

\NA

( l—^-\ c0NA

= e^'^e^^1^

(140)

Note that if R0 = °o, c0 = 0 and p(t) = 0 (immediate transfer). If R0 = 0, c0 = and p(t) = e~("T) (no transfer). The quantum yield of the donor luminescence is given by

38 _N_ 1 •=N„

fp(t)dt = - JV ( , / ™ c *' c^dt = e-^ '^ T o o

(141)

The transfer yield is

i__5L = i_ e -fcA/o

(142)

% 2. Stern- Volmer Model. In this model14 WDA has the form WDA (R) = w = const

(143)

Then

/(0 = i * je-^wR2dR V

= *Le-»>!L = e-» y

o

(144)

3

and p(r) = e. 1 -J*-(ifV 1 -3 c0NA \rj

y

(167)

n

Also Mm In{t) = exp iVA->°o V->oo

-^-fifvri-i c0NA VTJ

V

(168) "

and finally

i_Wi_2Yi p(0 = e - ( ' / T ) [^(0^=exp|-i-^r[l

3/n

(169)

This result was first derived by Forster15. We want to discuss the conditions under which this formula has been derived. xm » 1 means

-»1

and

Hf Take Rm = 3A, R0 = 10A and n = 6

43

f»7xl0~4r This shows that the approximation xm » 1 may be good even for very short times. xv « 1 means

Take Rv= 1cm, R0 = 10A and n = 6

i.«f^T=io« A io J

t((W42r

which is indeed always the case. Figure 9 reports a plot of the function p(t)

given in Eq. 169 for the case n = 6

(dipole-dipole interaction). The following observations can be made regarding Figure 10: (1) (2) (3) (4)

At the beginning the decay is faster, because the donors close to the acceptors decay first. After a while other donors, which are farther away from acceptors, and remained excited, start transferring energy. At very long times donors that are very far from acceptors will finally decay with their own lifetime. The greater is cAlc0 = number of acceptors in donor's sphere of influence, the longer one has to wait for the p(t) curve to become parallel to the one for cAlc„ = 0. In Figure 10 a comparison is made of the p(t) curves for the various processes: n =

6 (dipole-dipole), n = 8 (dipole-quadrupole) and n = 10 (quadrupole-quadrupole).

44

Fig. 9. The average probability of donor excitation in the case of dipole-dipole interaction.

T

1

1

1

T

Fig. 10. The average probability of donor excitation for n=6 (dipole-dipole interaction), n=8 (dipolequadrupole interaction), and n=10 (quadrupole-quadrupole interac-tion). The ratio cA/Co is equal to 5. (Reproduced from [12])

45

If n = 6 (dipole-dipole interaction) p(i) = exp

_i_WIYi r c \2)\x

1/2

ill

(170)

= exp

0

Then N iV„

1

~J

exp

7°°

•f-2?/1,

.L-EA. T

Cn

(171)

C^ J .. 1 / 2'\ 2

9°°

9

where

2q = 4n-A0

But 2 r r2 2 t r2 >rfq = 1 — = = f e _ x t/x = n = f e~* rfx

(172)

Then1:

— = \-qeq

4n(\-erfq)

(173)

We shall examine the following two limiting cases: # —> 0:

c& -> 0, er/g —> 0 iV„

g —> °o:

2 c„

c^ —> °°,

erfq —> 1

46 Summary This section is dedicated to the case when the donor-acceptor energy transfer takes place via a predominant multipolar interaction, with the parameter n being 6, or 8, or 10, It is found that the unfolding of the average probability of donor excitation is represented by a decay that, characteristically, presents deviations from the pure exponential decay as follows: (1) At the beginning the decay is faster, because the donors close to the acceptors decay first. (2) After a while other donors, further away from acceptors and still excited, start transferring energy. (3) At very long times donors that are very far from acceptors will finally decay with their own intrinsic lifetime. 4. 4. Exchange Interactions We refer back to the result in Eq. 72 and in particular to the second term, called the exchange term. The transition probability due to this term in the interaction Hamiltonian is given by

V A=

In

° T

,e2

(rf(l)a 00)

~zydy (181)

49 1

-"(In y?

J'

y

yv

*-H T yv

1 /. \3J „-zy ' :(iny) = -l-e-^(\nyvf

(ln y)3

-®dy

dy

+Ij(lny)3ze-^y

1

-z]e-V{\n yfdy-

+

J e ^ t a yfdy

(182)

z ] e - ^ ( l n y)3rfy

3 - » - - ( f]i n ) v )V>- *- -zJe-O'On y) dy 1

3

->«> ->-MH>> where g(z) = -zje" z > (lny) 3 ^

(183)

Then 3 1

_z

p(f) = lim e -(//*) 33 ( ^ , r 'Jf ^ ( h y ) -dy Y [RV) JV ->co A

V->oo

3r

= lim e

-(»/T)j

3

y 3 f*^

3

i g(z) 3

7 Uvy

V->°o

lim e - ( , / T ) V->00

(184)

ft,

r \ v*• I * ( z ) R

Taking Eq. 166 into account, we obtain \NA

p(t)= lim e~itlx) V->oo

7 c0NA

:exp

t

1

^1*(^ / T )

(185)

50 2R

u

o

where y = — - . L We note, in regard to g(z)li: (1) (2)

g(0)=0. g(z) is positive and monotonicaly increasing for z >0.

(3)

Expansion of the exponential e'zy and integration term by term gives the series g(z) = 6z1£

(4)

y>

(187)

For z >10, g(z) can be approximated by the expression *(z) = (ln z) 3 +1.7316 (In z) 2 +5.934 In z +5.445

(188)

Summary This section pertains to the case of energy transfer taking place via the exchange interaction, a case of lesser importance with respect to the previous one. The elaborate calculations lead to a time evolution of the average probability of donor excitation with an exponential decay shortened by an additional time factor, critically dependent on the quantity 2RQ/L, where RQ = donor-acceptor distance at which the transfer rate is equal to the donor's decay rate, and L = effective Bohr radius.

5. Energy Transfer with Migration of Excitation among Donors Caesar non supra grammaticos. Suetonius 5.7. Migration as Diffusion Process The case treated in the previous section 4 deals with direct transfer from donor to acceptor, with no migration among donors and is exemplified in Figure 11(a). We shall now consider the case in which WDD # 0, and the energy transfer processes can take place among donors, so that the energy of excitation may reach an acceptor after hopping resonantly among donors as in Figure 11(b). Since migration among donors can be viewed as a diffusion process, we shall present some elementary considerations regarding the phenomenon of diffusion. Consider a system consisting of N similar molecules in a volume V, and let n be the density N/V. Let ni be the density of specially labelled molecules, for example, of radio-

51

D*

*

• A

D

O

D

D

0

D

*

A^

o

•Y*

D

D

D

\

A Fig. 11(a). Direct transfer.

(b) Transfer with migration.

active molecules. Assume nj = nt(x), i.e. a non-equilibrium situation, but assume also n = const, so that no net motion of whole substance occurs during diffusion. The mean number of molecules of type 1 crossing the unit area of a plane perpendicular to the x direction in the x direction in the unit time is given by (cm 2 sec"1)

jx=-D^L

(189)

ax where D = coefficient of self diffusion [cm2 sec"1] The above Eq. 189 is valid for gases, liquids and isotropic solids19. If dn^l'0, Jx

52

A

A dx

W

H

Jx(x)

Jx(x+Ax)

Fig. 12. Crossing of plane by especially labelled molecules.

nt(x,t) i

V1' J /•* s

\\

/j

w— Fig. 13. Evolution of the density function of especially labelled molecules.

The above equation is called diffusion equation. We shall consider the initial condition of N/ molecules introduced at the time t = 0 near the x = 0: VJC.O)

= NjSM

The solution of the diffusion equation under this condition is

(192)

53 ni(x,,) =

JL-(*W)

1

=^e-(Sl*n,)

o-Jln

(193)

V4JTDI

The shape of the n,(x,t) curve is always gaussian (see Figure 13) with the standard deviation o = -JlDt

(194)

Also 1 *"

(JC 2 ) = — \x\{x,t)dx \ ' JV, •>

= o2 = 2Dt

(195)

In three dimensions the diffusion equation is 4 " i ( ? , 0 = J DV 2 n 1 (r,0 at

(196)

If the initial condition is n1(?,f) = iV15(?)

(197)

the solution is given by

3,e~ir ^^=T7^m {ATtDtf

n

°

(198>

Also (r2) = —\r\(r,

t)d3r = 6Dt

(199)

We shall now relate these notions to the case of donor to acceptor energy transfer. We define a function p(R,t) by stating that p(R,t)d3R is the probability that the donor with coordinate in (/?,/? +dR) is excited at time t. We shall now consider the following cases. 1. Diffusion Only. In this simplest case the diffusion equation gives us ^-p(R,t) = DV2p(R,t) If the excitation is initially localized at R = 0, then

(200)

54 -(R / 4 D 0

" H = : T-j4nDt S

(201)

with (fl 2 ) = 6Dt

(202)

2. Diffusion and Relaxation. In this case a term has to be added to the relevant equation -p(fi,r) = DV2p[R,t)

(203)

--p[R,t)

If we set p{R,t) = y[R,t)e

0. Sometimes, however, merely the energy balance of a subsystem is considered, e.g. the electromagnetic field may gain or loose energy by interaction with the mechanical system of the charges so that E# ^ 0 if T,E refers to the electromagnetic field alone. As the size and shape of the volume V considered in Eq. (2) is arbitrary, a local formulation of a conservation law can be derived by using Gauss' law ^ M

+ divJG(r,i) = a G ( r , t ) .

^G denotes the name as well as the quantity.

(3)

76 pG(r, t),jG(r, t) and ffG(r, t) respectively denote the density, current density, and production rate density of G. In addition, the relations of these quantities to the state variables have to be specified. It is an old question whether an energy current can be imagined as energy moving with a well-defined velocity10 yet a solution has been worked out only recently for some cases. 11 In a kinematic interpretation, one assumes a decomposition of the form j E — PEVE, like it is known to hold for the particle current of a one-component fluid, JAT = PNVN, where, pN, vN are state variables of the fluid which can be measured independently. In particular, vN is the velocity of a reference frame in which j N is locally zero. For several components the current density consists of several terms

PN =Y2pi, i

iN = Yl PiVi '

(4)

i

where z labels different particle numbers TV*. Note, j ^ = PWVJV rnay appear as selfevident, -nevertheless, it does not define a particle transport velocity v # which, in general, can be measured independently and, hence, such a velocity is of no physical relevance. This holds for the energy-current velocity, too. The situation is even more queer for the momentum balance of the electromagnetic field. For a pure electrostatic field the momentum density (=£ x %/c) is zero everywhere, whereas, the momentum-current denstity (= negative Maxwellian stress tensor) has nonzero (diagonal) components, see Landau-Lifshitz 14 (Vols. 2,8). In some special cases, however, v £ = JE/PE is identical with the group velocity of a wave puis. Furthermore, a decomposition of the form of Eq.(4) may sometimes be appropriate in terms of plane-wave modes of the system which play the part of particle numbers degree of freedom.

3.

Density, Current, and Modes of Energy Transfer

In this chapter, we will construct and list of results for the energy density, energy current density and (normal-) modes of basic systems and equations which are frequently used to describe energy transfer processes. Normal modes of a system are defined as particular solutions of the respective (homogeneous, linear) field equations supplemented by appropriate boundary conditions. (In some cases, such modes could even be chosen as complex functions even though the physical fields are real.) Their importance and utility lies in the fact, that arbitrary field configurations can be expanded in terms of these modes which themselves represent a type of stationary states. In Quantum Mechanics these modes are the eigenstates of the Hamiltonian. The following chapter gives some selected examples. For a discussion of nonlinear waves see, e.g. the review article by Bishop et al. 12

77 3.1.

Scalar Waves Propagation of scalar (complex) waves are described by the wave-equation

^£- A )* ( r ' i ) = s ( r ' f ) '

(5)

where s(r, t) describes an external source. The state of a wave-field is fixed by i]/(r,to) = $ ( r ) and 4>(r, t0) = 9 ( r ) , where $ ( r ) , 6 ( r ) are two arbitrary functions. The homogeneous part of Eq. (5) is (form-) invariant under time reversal and, thus, describes reversible processes. By standard manipulations the energy density, currentand production rate densities can be found

d*(r,t)

Pfi(r,£)

2 c2

dt

jE(v,t)

=

oE(T,t)

= pm®(^^s(T,t)^

2

1 + -/3m|grad*(r,t)|2,

-p^R^^lgrad^r,*))

,

,

(6) (7)

(8)

where pm denotes the mass-density (or another appropriate quantity) and 5R means real part, see e.g. the textbook by Barton. 13 Expressions of this form are relevant to all scalar waves, though the physical significance of the individual terms may be different. For example, for an elastic continuum \I> denotes the displacement of a volume element, c = \jEjpm is the velocity of sound, and E is the bulk modulus, see Landau-Lifshitz 14 (Vol.7). For a nonviscous compressible fluid, on the other hand, \I/ is the velocity potential, v = —grad?/> and Eqs.(6-7) can be rewritten as M r > *) = ^PmV2 + pme ,

j B ( r , t) = ( -pmv2

+ pmw J v .

(9)

Here the energy density is just the sum of the kinetic and internal energy e (per unit mass), w = e +p/pm is the enthalpy (per unit mass), and p is the pressure, see Landau-Lifshitz 14 (Vol. 6). J B consists of two terms: the term pEv represents the energy which is "transmitted convectively" whereas the term pv is traditionally described as "the work per sectional area" which is done by the fluid.11 Next we consider the modes of the (homogeneous) wave-equation Eq. (5) (s = 0) by separation of the variables r, t *K(r,i) = e-^**K(r).

(10)

K labels different mode functions which obey the (homogeneous) Helmholtz-equation

NT)'

*«(r) = 0.

(11)

78 An arbitrary solution of the (homogeneous) wave-equation can be decomposed in modes * ( r , t) = ] T [AWc-*"-* + A^e+^t]

*K(r),

(12)

K

where the two independent functions A^ are fixed by the initial conditions. For real fields, these functions are not independent. The simplest modes are plane waves, l'k(r) = exp(ikr), where K = k is the wave vector and wk = ck. Remarkably, the energy current of such a mode is just energy density pE(r,i) = pmk2 times c, j B ( r , t ) = pB(r,t)ck (see end of Chapter 2.). As an example, we consider the propagation of (longitudinal) waves in onedimension (no source term), where the general solution can be explicitely stated in terms of two arbitrary functions f^(x), which are fixed by the initial conditions. •9{x, t) = / (t) = (ct)2. Finally, we notice a property of wave propagation in two dimensions which is in strange contradiction with our daily experience. In three-dimensions, an initial pulse of finite duration propagating off the source will always create a wave packet with a leading as well as a trailing edge. In two dimensions, however, there is no trailing edge and, hence, an observer will find an infinite afterglow, see Fig. 3. Remarkably, this property holds in all space dimensions of even order d = 2,4, 6 etc. For spherical or cylindrical waves and solutions of the inhomogeneous wave equation in terms of Green-functions, see e.g. Barton's book. 13

79 1.2 1.0

f

^0.8 & 0.6 9"

*

i t = 15

I

I

in

*!

0.6

;t = 0

0.4 0.2

0

10

l\

J

-10

20

0

x 0.40 0.35

0.5

:t = 0

0.30

0.4

0.25

0.3

0.20 0.15

t = 15

0.2

t=15

0.10

0.1

0.05

10

-10

20

Figure 2: Time evolution of an initial wave-packet and its energy-density. Initial conditions: *(x,0) = *(x) (left) and *(x,0) = Q(x) (right). $(x),0(x) are Gaussians centered at x = 0. Note, the dotted curves in right upper graph display the velocity. (Dimensionless quantities.)

d=2

Y

d=3

r = ct

r^r=c,

/ dark \ ct a)

ct

I

r

b)

Figure 3: Pulse propagation in d=2 dimensions shows an infinite afterglow whereas pulses in d=3 dimensions display a leading as well as a trailing edge.

80 3.2.

Diffusion

Consider a conserved quantity with density p with a diffusive flow, j = —Z?gradp (=Fick's law) where D is the diffusion constant. By construction, p obeys the diffusion equation | | = DAp. For notational convenience, we shall write \P (which is assumed to be real) instead of p, where ^ may describe the density of particles, charge, energy, etc., and allow for an external source term s(r, t).

dt

•£>Atf(r,i) = s ( r , i ) .

(15)

The state of this system is fixed by ^ ( r , 0) = $ ( r ) . In contrast to the wave-equation, Eq. (15) is not symmetric with respect to time-reversal and, thus, describes irreversible processes. The modes of the (homogeneous) diffusion equation {s = 0) are defined as *K(r,i)=e^**K(r).

(16)

K labels different mode-functions which are again solutions of the (homogeneous) Helmholtz-Eq. (11) where (LUK/C)2 has to be replaced by \K/D. An arbitrary solution of Eq.(15) (with s(r, t) = 0) can be decomposed as

*(r,t) = X)A (t e- A - t \P (t (r),

(17)

where the real coefficients AK are determined by the initial condition. A very useful solution of Eq.(15) is the "heat pole" which belongs to ^ ( r , 0 )

(36)

Note, Eq.(36) is not a general result, in particular, it does not hold for finite damping, or in the region of anomalous dispersion.

3.5.

Energy Transport Velocity and "Superluminal" Pulses

Presently there is a remarkable interest concerning the velocity of energy transport and the interpretation of "superluminal" pulses in dispersive media and wave guides, see e.g. a special issue of the "Annalen der Physik". 20 Although such phenomena are not new a great fuss arose. In particular, it was well known that the propagation of a puis with a Gaussian envelop 23-25 is essentially different compared to a semi-infinite envelop, which was initially studied by Sommerfeld and Brillouin. 21 Pulse propagation experiments usually measure the cross-correlation function between the pulse travelled through matter along distance L and a reference pulse, see Fig.6. From that a delay/advance distance As = cAt a pulse velocity in matter c/(l + As/L) is deduced. It is well known that the group velocity vgr{k) =

dcu(k) dk

d[n(ui)uj] dw

n(u) + um'(u>) '

(37)

describes the propagation in a linear dispersive, nonabsorbing medium outside the stop band, where u(k) = ck/n. (If the refractive index is complex, one extrapolates Eq.(37) replacing n(u) by its real part.) In the region of normal dispersion {n'(ui) > 0) the group velocity is always smaller than c. However, in regions of strong anomalous dispersion (n'(uj) < 0), vgr can exceed c or even become negative. The common belief is that the meaning of Eq.(37) breaks down and the behaviour of the pulse becomes much more complicated. This is indeed true for the semi-infinite sinussoidal puis, 22 for

85

GOPtt SAMPLE (17K)

<S2•

&

%

^O^n

Figure 6: Left: Schematic of a Gaussian pulse propagating in a dispersive medium and of the experimental arrangement to measure the cross-correlation function. Right: a sample of the crosscorrelation data as the laser is tuned through the exciton line of the GaAs : N sample, N = 1.5 x 1017cm~3. According to Chu and Wong.23 a Gaussian shaped pulse, however, the situation is different. As discussed by Garrett and McCumber, 24 and in more detail by Oughstun and Balictsis, 25 the leading edge of the pulse is less attenuated than the trailing edge so that the pulse maximum speedsup at the expense of the pulse height, see Fig.6. For a Fourier transform limited puis whose spectral width is much less than the width of the absorption line, the pulse propagates with the group velocity as given by Eq.(37) and even the shape and width of the pulse can remain almost intact after it emerges from the sample. Moreover, for undamped, harmonically bound electrons with a single resonance frequency the group velocity is identical with JE/PE', compare Eq.(36) with Eq.(37). The physics of puis propagation in wave guides below the cut-off frequency is similar, yet the pulse reshaping and attenuation is due to reflection rather than by absorption. 19 The propagation of an electron wave-packet through a potential barrier was studied by Krenzlin et al., see Ref.20 (p. 732). Note, there is no indication of advance propagation of the leading pulse wing which, trivially, can never be surpassed. In addition, the time separation of two pulses is not affected so that there is no superluminal transmission of information, even if vgr > c. If a physically motivated definition of an energy transport velocity is desired, it ought to be based on a (local) Lorentz-transformation to a moving frame (denoted by a prime) of vanishing energy current

£

• e

B

•Bf

7 (£ + V x B), V x£ 1\B

(38) (39)

86 where 7

= 1 - (V/c)2.

Equating £' x B' = 0, we obtain in the limit V < c ~ 2 ^ ~ e 0 / i 0 £ 2 + £2'

(40)

Remarkably, this velocity is numerically only, but one half of the "self evident" form.

3.6.

Quantum

Mechanics

In quantum physics, expressions for particle probability density and its currentdensity are well known from text books, yet expressions for energy-density and energy-current-density are less familiar. To construct these quantities, we first consider a single particle in one dimension in a time-dependent potential V(x,t). The wave function ip(x,t) obeys the Schrodinger-equation

tt»*M = iMM,

£=4 S

+

^ -

(4i)

Under a time-reversal operation Tip = ijj*, so that all expectation values remain unchanged. Hence, Eq. (41) describes reversible processes, see Landau-Lifshitz 14 (Vol.3, §7). First, we find the energy-density starting from the expectation value of the Hamiltonian H =

J ip*(x, t)Hijj(x, t)dx=

f pE(x, t) dx.

(42)

By partial integration we rewrite the integrand such that it is intrinsically positive, hence

pE{x t)=

2

'L

Q_,./„ ,> dip{x,t) dx

2

+ V(x,t)\i;(x,t)f .

(43)

Next, we study the variation of < H > with respect to time. Elimination of V^by Eq.(41) and performing a partial integration we obtain for the integrand of ^ < H > I =

i— 4m2

(i',)r

(44)

Now, the task is to rewrite the integral over I to extract the current-density via —§^3E{X) + 0E(x,t). For the first term of Eq.(44) we get dx •> -1

_^LA/^1^_ C C \ 4m 2 dx \ dx dx2

J

(45)

87 By partial integration, we find that the second term compensates the third one. Finally, the desired results for the energy-current and energy-production rate densities are 3E{x,t)

3

f h3 dip* d2tp \ I QITTL dx

dx

I

2

aE(x,t)

=

^ W i , f ) |

2

.

(47)

For time-independent potentials, aE{x,t) = 0, as the energy is conserved. The generalization to three-dimensions is obvious by replacing Jj by grad . As an example, we consider a free particle in a plane wave state tp(x, t) = ,7-exp(ikr), where Vo is the normalization volume. pB(r,t)

= ^P-,

jj5(r,t)=pB(r,t)—,

VQ

(48)

Til

where -E(k) = h2k2/2m is the energy of the particle. The Schrodinger-equation of a free particle bears some analogy to the (homogeneous) diffusion-equation in "imaginary time", to —• its- Eq.(17) can be viewed as the result of the application of the time-evolution operator Uo(t) on the initial state * ( r , i ) = £z>(t)*(r,0),

UD{t) = e-Bt,

(49)

where H = —DA is the "Hamiltonian". However, there is a profound difference between the diffusion equation and the Schrodinger-equation. Eq.(41) describes reversible processes and the solution can also be propagated backwards in time whereas it cannot for the (irreversible) diffusion-equation, because there are arbitrary large positive eigenvalues A^. In mathematical terms, the Schrodinger time evolution operator, Us{t) = exp(—itH/h), generates a group of unitary transformations, whereas Un(t), Eq.(19) is not unitary and generates only a semigroup which has no inverse element.

3.7.

Collective Excitations in Solids

Condensed matter is a strongly interacting many body system which, in chemical terms, forms macromolecules with unsaturated bonds which allows for unlimited aggregation of particles. Due to the strong interaction between the (bare) particles (electrons, nuclei) no exact treatment is possible. However, in many cases a number of relevant features of the ground state as well as for the low-lying excitations have

88 been recognized. For a review on basic notions in solid state physics see Anderson's book. 29 For our purposes, the most important properties are: Ground state properties: • broken symmetry: crystal, ferromagnet, superconductor . . . • rigidity: mechanical, magnetic, gauge . . . Low energy excitations: • behave like a gas of weakly interacting (quasi-) particles, • single particle excitations resemble the bare particles, at least at not too strong interactions (=adiabaticity), • collective excitations are dynamically equivalent to the creation/absorption of bosons. Broken symmetry means that the ground state of the system has a lower symmetry than the Hamiltonian. For example, the Hamilton of a solid is invariant under translations and rotations whereas the crystal state is not. (This is different for fewbody systems, e.g. the hydrogen atom where the ground state has the full symmetry of the Hamiltonian.) Rigidity refers to external perturbations. For example, condensed matter (fluids, solids) is almost incompressible, in addition, solids are rigid with respect to shear deformations. As an example of gauge-rigidity we consider the London-equations of a (type I) superconductor | j ( r , t ) = — £{T,t), Ov

curlj(r,t) = - ^ B ( r 1 t ) ,

Tft

(50)

if li

where n is the density, m the mass, and —e the charge of the electrons. Remarkably, these equations remain unchanged when replacing the electrons by Cooper-pairs with density n/2, charge - 2 e , and mass 2m. We compare Eqs. (50) with the quantum mechanical result for the particle current of an electron in a magnetic field of vector potential A, B = curl A: JN{r,t) = ~ ZiTThZ

(*Vad *-cc)-

— \V(r,t)\2A(r,t).

(51)

Tit

For JV particles, * = \P(ri, r 2 , . . . rjv) summation and integration of Eq.(51)on JV - 1 coordinates has to be included. If the many electron wave function is not affected by the magnetic field Eq.(51) (times —e) reduces to just the second London-equation (in the Coulomb-gauge, div A = 0).

89 The concept of quasi-particles was originally developed by Landau who realized that there is a continuous mapping of the low energy excitation spectrum with the strength of the interparticle interactions, see Landau-Lifshitz, 14 (Vol. 5). Amazingly, this description holds even in relatively strong interacting systems like in metals or in liquid helium 3He, 4He. As an example of single-particle excitations we consider the transformation of free electrons to Bloch electrons in crystals, see Fig.7. In terms of quasi-particles, the excited states are described by occupation numbers na = 0,1,2..., where a labels the one-particle states (or "modes") with energy eQ. Hence, the excitation energy of the total system becomes

E({na}) -E0 = Y^ tana + Eint.

(52)

a

E0 is the ground state energy and Eint({na}) contains nonlinear terms in na which describe interactions between quasi-particles. For a translationally invariant system or a crystal, a is identical with the wave number k, in addition, there may be several branches of wave-like excitations which will be labelled by an additional index (= v). Collective modes are equivalent to a set of (uncoupled) harmonic oscillators with frequencies ua. Quantization of these oscillators directly leads to Eq.(52) which is dynamically equivalent to a system of bosons with energies hu>a, see Fig.8. In contrast to massive bosons, however, (e.g. .ffe-atoms in liquid Helium) these quasi-particles can be easily created and destructed. An appropriate description of such processes is not possible within the "ordinary" wave-function formulation of quantum mechanics, and "second quantization"* is needed.30 In the extended zone scheme quasi-momentum ?ik plays almost the same role as real momentum. The two most important properties of particles and quasi-particles are • The transport velocity of energy and momentum is given by vT = — — . dp

(53)

• The overall (quasi-) momentum and energy of particles and quasi-particles is conserved, see Fig.9. Unfortunately, there is no general rule under which conditions such a scenario exists, and how to find the collective variables. Some examples will be given in the next sections. A survey on the dynamics and spectroscopy of collective excitations in solids, 32 and a collection of common and different properties of particles and quasiparticles 28 can be found in the Proceedings of two previous Erice-Schools. *The name "second quantization" is misleading. Besides h no second quantum constant arises. A better name is "occupation number representation", yet it is used in a different sense as position or momentum representation.

90

-KK-K-KK O M K

'AY.

Figure 7: Left: Energy (quasi-)momentum relation of free electrons in an empty lattice (thin line) and in a weak periodic potential (solid line), where K = 2im/a, n = 0 , ± 1 , ± 2 . . . are reciprocal lattice vectors. Right: Energy bandstructure of GaAs (in the reduced zone scheme). According to Cohen and Chelikowski.31

energy bosons

oscillators

0--

o -

Figure 8: Equivalence of a system of N (noninteracting) bosons with single—particle energies e 0 and occupation numbers na and an infinite (uncoupled) set of harmonic oscillators with frequencies ua = £a/Ti- Note that the zero-point energies of the oscillators are omitted. Dots symbolize particles, crosses excited states, respectively. N = 6. According to Ref.28

impurity

neutron

b)

c)

Figure 9: Examples of interactions between particles and quasi-particles. (a) Excitation of a phonon by neutron scattering, (b) scattering of a phonon by impurities, and (b) decay of a phonon due to anharmonic interactions. Energy and momentum are conserved at each particle/quasi-particle vertex whereas the impurity takes momentum but no energy.

91

D haq n m ') ^•ftAO Wilt C * w A w O - W W W U O ^

1 2 H a I-

3 I

4 I

N-l I

N I

H-q

Figure 10: (a) Linear monoatomic chain with equal masses m and nearest neighbour springs D and periodic boundary conditions, (b) frequency spectrum for N = 10 "atoms". Note, there is no vibrational q = 0 mode (at u> = 0); this degree of freedom is taken over by the common translational motion. According to Ref.28 A.

Phonons

Phonons are quantized vibrations of a crystal-lattice. For notational simplicity, we consider first a linear chain of equal masses m and nearest neighbour springs, see Fig.10. Starting from the Hamiltonian in terms of canonical momenta pj and displacements Uj of the masses at sites j = 1 , . . . N, {pj, Uf} 5jf N

j=i

1

2m^

i. +

(54)

2D{u^

we first find the collective momenta Pk and coordinates Qk (55) {a, b} denotes the Poisson-bracket symbol, k = |j^/s, K = 0, ± 1 , ± 2 . . . ± 7V/2 is the wave-vector which is restricted to the first Brillouin-zone, and a is the lattice constant. Eq. (55) is canonical, {Pk,Qk'} = &kk' and transforms Eq. (54) to a set of uncoupled oscillators with frequencies u>k H(P,Q)

E ^ + ! '-QkQl

k>

W

* =

\

S

(56)

In d dimensions, a vibrating lattice has d acoustic (uk = cs|fc| for k —¥ 0) and d(s — 1) optical branches (u>k ^ 0, k —• 0), where s denotes the number of (inequivalent) atoms in the (primitive) unit cell, v = 1,1...d • s, see Fig.11. The quantization of uncoupled oscillators, Eq. (56), is almost trivial. The state

92

(1,0,0)

L

\

9?

\

F 0.2 0.4 0.6 0.8

L

f

\L T \

Jd

0

(1,1,0) £.

(i.i.i)

1.0 0.8 0.6 0.4 0.2

ifTi' l//\ 0

0.2 0.4

b)

a)

Figure 11: Structure and Brillouin-zone (a), and phonon dispersion curves (b) for potassium (bcc, one atom per primitive unit cell). Along the horizontal axis we plot q, q/\/2, and q/\/Z for the (1,0,0), (1,1,0), and (1,1,1) directions, respectively. According to Cowley et al.33 of each oscillator (labelled by k) is fixed by quantum numbers nk = 0,1, 2..., hence E({nk}) = Y, *"* (n* + \)

•

(57)

This is already of the form of Eq. (52), and we may identify e(k) = htok with the energy of the quasiparticles which are called phonons. To describe arbitrary phonon states or interaction processes between ghonons it is convenient to use creation and anihilation operators ak,a[ rather than Pk,Qko-k =

mukQk + iPk /n

fe

f^ ^ t \ak,ak,

5kk' •

(58)

These are also called ladder operators as their repeated application on \n > creates the "ladder" of stationary states n = 0 , 1 , . . . , where a+ \ n >= %/n + 1 | n + 1 >, a\n>= s/n\n-l> "climb" up/down the ladder by one "rung".

E H

=

K e'k>a(alk + ak), 2NmuJk

^^ tiu}kakak .

(59) (60)

The zero point energy of the oscillators has been omitted as it is not relevant for the dynamics of the system.

93 Note, there are three different types of particles in the game: • N coupled "bare" particles of mass m described by Pj,Uj, • N uncoupled harmonically bound particles of unit mass described by

Pk,Qk,

• phonons of unlimited number described by 1ik,a[. In addition, we can consider phonon states which are not even eigenstates of the phonon number operator (see example below).

(61)

N = J^^k

For example, we consider two particular phonon states: One-phonon state:

| <j>(k) > = ^ 0 ( f c ) | l f c > = ^(fc)aJ,|0 >. k

k

For notational simplicity, |0 > = \{nk} > with nk = 0 for all k. a£|0 > = |l/t > represents a steady state of the oscillator #fe in excited state nk = 1, all other oscillators being in the ground state. By construction, \(k) > is an eigenstate of the phonon number operator with eigenvalue 1: N\(j>(k) > = l\(j>(k) >. The time-evolution of this state is e~'Ukt\lk >, hence (f>(k) —> 4>{k,t) = <j)(k)e~lh}kt which may be interpreted as wave function of a phonon (-wave packet) in momentum spaced For such a state, the expectation value of the position operator of mass # j is < \v,j\<j> > = 0 for all times so that it cannot be the quantum analog of a classical wave, it is "quantum noise". 00

a-state:

n

\a >= V"* e - ? ' 0 ' —y=\n >. 71—0

Here k is fixed, n = nk, and will be suppressed for notational simplicity, and a = |a|e , , p denotes a complex number. Again, the time evolution of | a > is easy to find just by replacing a by a(t) = a • e~'w*. Remarkably, these states describe almost classical (wave-like) motion of the masses in the chain < a(t)\uj\a(t)

>oc |a| cos(kja — uikt — (p).

(62)

Hence, \a\ and ip fix the amplitude and phase of the wave, a-states are not eigenstates of the phonon number operator and the expectation value and uncertainty are =

\a\2,

AN = \a\.

(63)

§In contrast to massive quantum particles it is not possible, however, to define a wave function in position space. The reason is that for phonons (or photons) u> <x \k\, which does not permit a Fouriertransformation from k to r space, see Landau-Lifshitz14 (Vol. 4).

94 For large amplitudes AN/ < N >-> 0. In addition, AujApj = §, withy timeindependent uncertainties. Thus, in real space, \a > describes a Gaussian wavepacket. Although these states have been already introduced by Schrodinger they are nowadays called "coherent" states, Glauber-states, or just a-states. In quantum optics they play a fundamental role for the description of laser radiation and coherence phenomena. 34

-77777 77777 Figure 12: Interacting two-level systems (a) and their pseudo-spin analogon (b). Left: ground state, right: a localized excited state.

B.

Excitons

Excitons are propagating electronic exitations in matter. Traditionally, one considers two limiting cases: • Frenkel-excitons are almost localized atomic or molecular excitations in molecular crystals with little coupling to neighbouring units. • Wannier-excitons, on the other hand, are loosely bound hydrogen-like electronhole pairs in semiconductors. These pairs are delocalized over many lattice sites so that a continuum description is possible. For simplicity, we consider only Frenkel-excitons and approximate the spectrum of molecular excitations by two-level systems, see Fig.l2a. As the Coulomb interaction between the molecules is isotropic, the coupled molecules are described by a Heisenberg model for (pseudo-) spin |

H = J2 ^e0ajz - Y^ Ji&di > i

( 64 )

v

where d{dj denotes the scalar product of the spin vector operator with Pauli-matrices (ax,dy,az) at site j . The broken symmetry of the ground states is evident from Fig. 12b. Due to the interaction, the excitation of an isolated molecule (as described by a spin flip at site j) may spread off to neighbouring sites. To bring Eq.(64) to the quasiparticle form Eq. (52), we first transform to exciton creation/annihilation operators b^bj by the Holstein-Primakoff transformation 30 (omitting the site index for notational simplicity)

dM=as±iZv,

a^ = (l-W2X

?(-> = (? (+) ) f •

(65)

95 2.00

E(k)

1.75

,.

^

1.50

\

/" '--

1.25

\

1.00

\

0.75

0.25 -7t

-7C/2

0

JC/2

Jt

ka

Figure 13: Energy (quasi-)momentum relation of Frenkel-excitons. (Dimensionless quantities, eo : 1,/i! = 0.25.) Operators at different sites commute. From [oj- , Sj-, '] = 4azj5jj> we obtain [6j, 6|,] 5if, hence "3>"j bj,b], describe bosonic excitations. (66)

H(b\b) = 5 > 6 & + ! > « • ' * & • ' j=i

Higher order terms in bj,bj describe interactions between excitons which can be omitted at low exciton densities. Due to the long-range structure of the Coulombinteraction many contributions to the coupling elements hjji have to be taken into account in a realistic description. 35 In crystals hif = hj^ji discrete Fourier-transformation (i.e. transformation to Bloch states) b* =

E4VN e

ikja

ak,

\ak

>«t] ='

(67)

diagonalizes the exciton Hamiltonian, Eq. (66) H@,a)

= ^2 ^. Note, this definition does not necessarily imply that < G >^ is "sharp", i.e. has vanishing uncertainty AG = 0, where AG = V < G 2 > — < G > 2 . There are two distinct classes of states: • pure (or ideal) states which have zero entropy • mixed (or statistical) states of nonzero entropy. Irreversibility and loss of coherence are intimately connected with increase of entropy. Hence, the transformation of pure states into mixed states plays a fundamental role in the description of energy transfer processes in macroscopic systems. According to the dogma of classical physics there are (infinitely many pure) states in which all observables have a sharp value. Hence, AG ^ 0 is always caused by errors, either in the state preparation or in the measurement of G. In quantum physics the situation is different. There are particular states in which a specified observable G has sharp values, AG = 0. These states are the eigenstates of G: G\ip > = g\tp >. But there are no states in which AG n = 0 for all Gn. Therefore, AG„ ^ 0 is not necessarily caused by errors! In classical mechanics pure states are specified by the values of coordinates and velocities (or momenta) of all particles which defines a point in phase space: i1 = (Pi?) = (Pi>QV,p2, [or wave functions tp(x,t)], observables are described by hermitian operators G, and the (expectation) value of G is defined by < G >,/,=< 1>\G\ip >=

fip*{x,t)Gil){x,t)dx.

(74)

The dynamics of pure states is determined by and the Schrodinger equation ih-\i>(t)

>= HW)

(75)

>•

A mixed state, as in classical physics, is a statistical mixture of pure states, where the phase space density is replaced by a density operator

p[t) = ^2pn\n>0.

(76)

| n > denote an arbitrary set of state-vectors (which might not even be orthogonal!) with (real) positive weights pn, and the expectation value Eq.(74) is replaced by =tr

(pG) = Y^ < a\pG\a > = ^

pn < n\G\n > .

(77)

where a labels an arbitrary base. The last part of Eq. (77) can be interpreted as the "usual" quantum average of G with respect to the pure states \n > with a classical average with weights p n put on top. Eventually, we mention that the density operator

98 a) E(r 0 ,t 0 )

f^- mirror 2 "E, |

V ir

t screen

E

i

I

— ** mirror 1

J

10

15

20 cm At

Figure 15: Schematic of the Michelson interferometer (a) and visibility curve of the red Cadmium line (b). Al is the difference in path length of mirrors 1,2. At = Al/c is the coherence time. According to Born and Wolf.44 of a pure state \ip > is the projector p = \ip > < tp\. The dynamics of a quantum mixture follows the v. Neumann-equation^ (78) Mixed states have nonzero entropy S = —kBtr (plrip) .

(79)

In the classical case the trace is replaced by phase-space integration. Noticeable, both the Liouville- and the v. Neumann equations describe reversible processes, —gp- = 0! We are able to describe states of nonzero entropy, but not irreversibility!

4-2.

Coherence and Correlation

The notion of coherence originates from optics where it describes space-time correlations of the electromagnetic field. Familiar consequences are the appearance of interference fringes in a double-slit experiment or other oscillatory phenomena. A prominent instrument to study coherence phenomena is the Michelsoninterferrometer, see Fig. 15. Field on screen E:

£ = £i + £2

C [£in(r0, t0 + 7i) + £in(r0, t0 + r 2 )]

Intensity:

I = \£\2

C 2 [G(1,1) + G(2 ) 2) + 25RG(1,2)]

Correlator:

G(r2,i2;ri,ti

(t) = 1. In general, peq refers to a local rather than to global equilibrium in order not to violate particle number conservation. The relaxation time approximation is expected to be good for weak coupling and not too small times. A nice application of this approximation has been given by Mermin 43 who included damping in the Lindhard dielectric function e(q, w). b. Master equation (incoherent limit). This description considers only the diagonal elements of the statistical operator (with respect to the eigenstates of the isolated system) Pn(t) =< n\p\n >; nondiagonal elements are considered to be zero. -pm(i) = ^[rmnpn(i)-rnmpm(i)]

(90)

n

The transition rates Tmn can be obtained from the "Golden Rule" and obey the symmetry relation Tmnexp(—Em/kBT) = Tnmexp(—En/kBT), where T is the temperature of the bath. 57 c. Stochastic Liouville/v. Neumann equations. The interaction of an electronic system (like excitons) and the phonons in a crystal is approximately treated as a heat bath pushing the electrons or excitons in a stochastic manner. This description is justified if the temperature is not too low, otherwise polaron states are formed. However, it is not possible to take the reaction of the electron or exciton on the phonons into account. There is action but not reaction! For example, for Frenkel-excitons the interaction part of the Hamiltonian Eq. (66) is replaced by

Hint = Yifmn(t)blbn,

(91)

mn

where fmn(t) is assumed to descibe a Gaussian stochastic process with zero mean value. The diagonal element fnn(t) describes fluctuation of the exciton energy e at lattice site n, whereas the non-diagonal elements represent the stochastic variations of the (coherent) interaction matrix element hjj> in Eq.(66). For details we refer to the book by Kenkre and Reineker7 (see p. 120 ff). d. Kinetic (Boltzmann) equation We consider a gas of (quasi-) particles with weak short range interactions. Instead of the (classical) phase space distribution function p ( r i , p i . . . rjv,Piv) the one-particle distribution function / ( r , p, t) = / / p(i, p; r 2 , p 2 ; . . . rN, pN; t) dr2 d p 2 . . . rfrjy dpN

(92)

103 is used which obeys the kinetic equation ^

^

+v ( p ) ^ M

+

F ( p , r ) ^ ^ = C;

(93)

where v is the velocity, Eq.(53), and F an (external) force acting on the particles. Boltzmann had the ingenious idea to approximate the interactions between the particles, as summarized by C, by the concept of collisions, i.e. short range, instantaneous gain-loss processes with no memory to previous states (=Markov-process). Collision term:

C = Nin(r, p, t) - Nout(r, p,t).

Nin/aut count the number of particles scattered in/out dp at p . It is almost a miracle how this equation has withstood all criticsms and how it could be adapted to quantum mechanics, [ f(r,p,t) becomes the Wigner-function, in thermal equilibrium, / is the Fermi/Bose function.] In contrast to the Liouville equation, Eq.(93) is no longer symmetric under time-reversal: the Ihs changes sign whereas the rhs does not. Therefore the Boltzmann equation describes irreversible processes. Cohen and Thirring 52 give a historical survey and discussions of the celebrated Boltzmann equation, whereas Landau-Lifshitz 14 (Vol. 10) deal with physical kinetics in the wide sense of the microscopic theory of nonequilibrium processes. For example, for elastic scattering of electrons by impurities in a metal or semiconductor the collision term becomes** C(f) = j M p ' -»• p ) / ( r , p , t) - w(p -> p ' ) / ( r , p , t)] dp'

(94)

with the intrinsic scattering rate (in Born approximation) ™(p -> p') = Nimp^-1
-¥ (CT)2 we deduce a (dimensionless) mean propagation velocity c = l / v ^ , where c2 is just the mean square of the group velocity. B.

Two-level-system We study a system with two base states represented by 1>=

1\ 0

,„ { 0 12 >= 1

(97)

and a coupling parameterized by a real, positive constant e. The Hamiltonian of this system and its eigenstates \I >, \II > and energies Ef = -e, En = e are given by H =

0 -t

-e 0

" - ; * ( !

'" > - ; * ( - .

(98)

105 A realization of such a system is a spin 1/2 in a magnetic field in x-direction, H = —HBB -a ( a = (jyx,oy,az) denote the Pauli-matrices) or a particle in a double-well potential. Further examples can be found in the Feynman-Lectures 16 (Vol. III). (a) Schrodinger dynamics The Schrodinger equation for \ip(t) > |V>(i)>=Cl(t)|l>+c2(t)|2>

(99)

yields two coupled differential equations for the amplitudes ihdi(t) = -ec2{t),

ihc2{t) = -ec^t)

(100)

which can be easily solved by insertion. For \ip(0) >= |l > we obtain Ci(*)=cos(^t),

c2(i)=lsin(^t)J

(101)

where LO0 = 2e/h is the transition frequency between levels 1,11. As a result, the probability finding the system in (base) states |1 >, |2 > and expectation values of the spin vector are P1{t)

=

| C l (t)| 2 = c o s 2 ( ^ t ) ,

(102)

P2(t)

=

|c2(t)|2=sin2(^t),

(103)

< ax > = 2$tc*1{t)c2{t) = 0 ,

(104)

< av > = 29ct(t)c 2 (t) = sin(w 0 t),

(105)



=

2

2

| C l ( t ) | - | c 2 ( t ) | = cos(o;ot).

(106)

Note, that < ax > is time independent as ax commutes with H and, thus, is conserved. (b) Relaxation dynamics It is convenient to expand the density operator with respect to the eigenstates |7 >, \II > of H because peg is diagonal in this representation

»=(-.'!)• *.-iCo A ) Z = exp(/3e) + exp(—/3e) is the partition function and /? = l/(kBT) temperature. The general form of the density operator is

Pit) =(B*

IBA)-

(-> is the inverse

(108)

106 A is real, 0 < A < 1, whereas B may be complex. Initially, p(0) = |1 > < 1|, hence 4(0) = 5(0) = 1/2. Writing Eq. (89) in a matrix form, we obtain for the (1,1) and (1,2) components of p

Mi) + yA(t)

=

,/e

§
=

.

(c) Master equation As states |1 >, |2 > have equal energy the transition rates must be equal, I \ 2 = T 2 i = F and so that the master equations Eqs.(90) become P1(t)=T(P2-P1),

P2(t) = T(P1-P2).

(120)

By conservation of probability, Pi + P2 = 1 these equations can be easily solved, thus Pi(t) = \[l

+ e-™],

P2(t) = \ [ l - e ^ } .

(121)

Results of coherent, fully incoherent, and relaxation dynamics are displayed in Fig. 18.

107 1.4 1.2

1

-° V

£ 0.8 \ a. 0.6 0.4 0.2 0.5

1.0

1.5

2.0

2.5

3.0

T

Figure 18: Dynamics of the two-level-system. Coherent (solid line), relaxing (dashed line), and fully incoherent motion (dashed-dotted line). C. Exciton chain (a) Schrodinger dynamics Instead of expanding the wave function into the stationary states of Eq.(68), we try to solve the (time-dependent) Schrodinger equation for a single exciton directly. In site representation, the amplitudes cm(t) of the state vector

M*)

> =

I] C mW|lm > ,

Cm(0) = 1, P m (i) -> l/(-\/47iT t), which resolves the diffusive character of the process, < (ma)2 > = 2Dt, D = Ya2 is the diffusion constant. Problems: 1.) Calculate the entropy, Eq.(79), of the damped two-level system as a function of time. [Hint: Evaluate the trace in eigenbasis of /?(£)]• 2.) Show that the coherent and fully incoherent dynamics of a two-level system can be described in terms of a generalized master equation with a memory kernel Y(t — t') dPi(t) dt

Jr(t-tl)[P2(t')-P1(t')}dt'.

(133)

Which T(t — t1) corresponds to the relaxing system? 3.) An interesting, yet pathological case (of Ca,b), is a cluster of molecules with equal coupling strength between all TV sites. [Hint: use ^ncn =const, ^2nPn = 1]-

109 0.14 0.12

T=10

0.10

\ 0.08 0.06 0.04 0.02

1 , .

0.14 0.12

T = 10

0.10

i °08 0.06 0.04 0.02

III,Figure 19: Spatio-temporal evolution of the energy per site in the excitonic chain Pm(r) (= probability finding an exciton at m). (Top) coherent, (bottom) incoherent motion, according to Eqs.(127,132).

D. Gas of (quasi-) particles (a) Approach to equilibrium We consider a homogeneous, dilute gas of hard disks in two-dimensions. At t = 0 all particles are located on a square grid and have the same magnitude but random directions of momentum (velocity). By numerical integration of the Newton-equations, 53 / dp is obtained by counting the number of particles in dp. As time proceeds, the particle collisions tend to realize a Maxwellian distribution /o(p) = /oexp

2mkBT

(134)

which is accompanied by a monotonous increase of entropy. The time interval for reaching equilibrium roughly corresponds to 200 collisions. However, this state is highly correlated and apparently only describes "molecular chaos": Reversing all particle velocities after 50 or 100 collisions reproduces the initial state, see Fig. 20. Therefore,

-fcs / /

f(p,r,t)£nf{p,T,t)drdp

(135)

obviously does not give the correct (change) of entropy. (Note the difference between

110

r*

-1

- o o

X o o

0 eq.

0

O m
0, there is no damping of the plasma wave in the Landau-Vlassov approximation. For wave vectors q « kp, however, plasmons can decay into electron-hole pairs (Landau-damping), although Eq.(142) is invariant under time-reversal! For a survey on plasmons, see, e.g., Ref.55

5. 5.1.

M e m o r y Functions and non-Markovian Behaviour Memory function concept

The dynamics of closed systems as studied in the previous sections is determined by a set of first order differential equations with respect to time. Given the state at time t0 these equations determine the state at all later times, see e.g. the Hamiltonian equations Eq.(70). In addition, in classical mechanics damping may often be phenomenologically included by introducing a friction-force —7oi. For instance, for a particle with mass M in a potential V(x) the Newton-equation reads Mx(t) = -MJox

- ^ ^

+ /(i)

,

(146)

where f(t) is an external force. Actually, the friction-force has to be supplemented by an additional fluctuating force f(£), (with zero mean) which represents e.g. the collisions of molecules impinging on the (heavy) particle. This is the Langevin formulation which not only describes the damped average motion but also the fluctuations around it, see van Kampen. 57 In a more detailed description of open systems the heat bath is often modelled by a system of harmonic oscillators and the bath degrees of motion are eliminated to construct an equation of motion for the system (particle) alone. This will be illustrated by two examples in the following sections. In such a description the reaction of the bath on the system particle is also taken into account which leads to an equation of the form Mx + f

M 7 ( i - t') x{t') dt' + ^ ^

= f(t).

(147)

This equation is nonlocal in time and the memory function y(t — t') determines how far the "history"of x(t') (t' < t according to causality) influences the present x(t). With "/(t - t') = 7 0 5(i - t') we are back at Eq.(146). Note, x(t0) and x(t0) do no longer determine the state of the particle, at least not in the same sense as it has been used in Chapters 3. and 4. Additional assumptions for t —>• — oo are needed. Equation (147) describes a non-Markovian process which will be discussed in the next section. As an example, we consider a harmonic oscillator with frequency UJ0 under the

113 action of a monochromatic force, f(t) = /ocos(wt). As a result, the solution is x(t)

= »[xMfoe-**]

X(UJ)

=

±--2

T{u)

=

/

,

(148)

l

,

(149)

dt = r 1 (w) + ir 2 (a;).

(150)

2

/>oo

7 (t)e

iwt

Jo Note, memory effects do not only lead to a frequency dependent scattering rate (= I \ ) but to a shift in the resonance frequency (= r 2 ) , too. Memory effects have an interesting consequence on the frequency dependence of conduction electrons (mass m, density n, w0 = 0). Expressing the electron current as j(t) = —enx(t), f(t) = —e£(t), we obtain for the conductivity aM = — „ , * . , (151) m 1 (u) — IUI which is a generalization of the Drude result Eq.(141), see Fig.22. Deviations the measured conductivity of conduction electrons from the Drude result are always indications of memory effects. For experimental evidence, see, e.g., measurements by Dressel et al. 56 on some organic conductors. In quantum physics, the inclusion of dissipation requires more care because quantum systems are described by a Hamiltonian which, in the absence of time dependent external potentials, ensure the conservation of energy. On the other hand, dissipative forces cannot be included in the Hamilton itself. A successful and rather general approach is by the concept of an (infinite) reservoir and elimination of the reservoir degrees of freedom. For details and applications we recommend the book by Dittrich et al. 50 which gives an excellent introduction and overview on the relation of quantum transport and dissipation. Master and Boltzmann equations with memory kernels have been respectively studied by Kenkre 5 ' 42 and Hauge. 54 A. Caldeira-Leggett model As an example how to eliminate the bath variables we consider a classical particle of mass M and coordinate q, which is bilinearly coupled to a set of harmonic oscillators ("bath"), see Ingold's article in Ref.50 (p. 213). H H

-

=

Hs + Hbath + Hint,

=

V & 1M + (?)>

(152) (153)

H

»^ = Y,^: + lm^xl

(154)

i

Hmt

=

-qJ^CiXi i

+q

2

^ — ^ ' i

*

(155)

114

with suitable constants for CJ,WJ, rrij. This model has been used by several authors and is nowadays known as the Caldeira-Leggett model. The equations of motion of the coupled system are:

Mq + V'(q)+qJ2~-2=J2C'Xi' i

miU)

i

(156)

i

set + ujfxi = —q(t).

(157)

nil

As the bath represents a system of uncoupled oscillators their equation of motion can be easily solved in terms of the (unknown) driving term ~ q(t). Xi(t) = Xi(0) cos(uiit) + ^ - ^ sin(wji) + / rrii

~^— sin[ui(t - t')]q(t')dt'.

(158)

Jo rriiU){

Inserting this solution into the Newton-equation of the particle, we obtain Mq(t) + [ My{t - t')q(t') dt' + V'(q) = £(t), Jo

(159)

where 7(f — t') is the damping kernel (memory function) and f (t) is a fluctuating force (which depends on the initial conditions of the bath variables and is not stated here) 7 (t)

=

-

r

^ -

7T JO

'(«)

cos(tut)dcj,

(160)

U

= -E^^"^).

(161)

i

For a finite number of bath oscillators the total system will always return to its initial state after a finite (Poincare) recurrence time or may come arbitrarily close to it. For N —> oo, however, the Poincare time becomes infinite simulating dissipative behavior (see also Chapter 4.3.). We therefore first take the limit N —> oo and consider the spectral density of bath modes as a continous function. Frictional damping, y(t) ~ joS(t), is obtained for J(ui) ~ 7oW- A more realistic behavior would be the "Drude" form J( W ) = 7 o w - ^ x h r ,

7(*)=7o7ue- 7 D *,

(162)

which behaves as in the friction case for small frequencies but goes smoothly to zero for uj > 7£), see Fig.21. B. Rubin model A rather nontrivial yet exactly solvable model is obtained by a linear chain (see Chapters 3.7.A and 4.4.A), with one mass replaced by a particle of (arbitrary) mass

115

Figure 21: Spectral density of the bath oscillators (left) and memory functions (right). Solid lines: Caldeira-Leggett model with a Drude form, dashed dotted lines: Rubin model. (Dimensionless quantities, M0/M = 1).

2.0

b

£ 1.0 0.5

Figure 22: Real and imaginary parts of the electrical conductivity for the Caldeira-Leggett and Rubin models. Notation as in Fig.21. Mo, see Fig. 10. The left and right semi-infinite wings of the chain serve as a bath to which the central particle is coupled. As a result the damping kernels are

7(*) =

M0

Ji{ujLt)

(163) 2

I »

=

M

I —u V>i*}\sgn(u))

+ iuj ,

Ul\=o.

(169)

V 27T Y which is a function of time Yx(t)=f(X,t).

(170)

Such a quantity is also called a random function. On inserting for X one of its possible values x, an ordinary function of time results, which is called a sample function or a realization of the process, see Fig. 23. Averages are defined by

= =

JYx(x,t)Px{x)dx, fYx{x,t1)Yx{x1t2)Px(x)dx.

(171) (172)

117

A/*V^^^^'

1

t'

#3

Figure 23: Schematic of sample functions. The latter is [apart from a redefinition of Yx{t) —> Yx(t) — < Yx{t) >] the autocorrelation function of Yx(t). A stochastic process can also be specified by a hierarchy of distribution functions Pn{yu *i; 2/2, h;...; yn, tn), n = 1,2,.... Px{y, t) is the probability density for Yx(t) to take the value y at time t, P2 is the joint probability that Yx(t) has the value y\ at t\ and y2 at t2, etc. Functions Pn are symmetric in all arguments yi,tt but ij 7^ tk is implied. The conditional probability Pi\i(y2, *2|l/i, *i) i s the probability density for >x(£) to take the value y2 at t 2 , given that its value at tx is j / x . More generally one may fix the values at k different previous times t\,t2,.. .tk and ask for the joint probability of Yx at t other later times tt+i, • • • tk+i- This leads to the general definition of the conditional probability P£|fc(fc + 1 ; . . . k + e\l, 2,...k)

=

p (l 2

k)

'

^

where "i" is a shorthand for yi,U. Like Pi, Pn, Pt\k is non-negative and normalized. (Notice the difference in sequence of the arguments in joint and conditional probabilities). A process is called a Gaussian process if all its P„ are (multivariate) Gaussian distributions. For a stationary process all P n depend on time-differences alone and Pi(y,t) is time-independent. A Markov-process is defined as a stochastic process for any set of successive times ti IP exceeds the ionization potential IP of a single electron. Then the interaction between the two excited electrons can lead to energy transfer, where one electron is deexcited and the other is ionized (Fig. lb). Measurements of energies and lifetimes of these doubly excited states gives detailed information on the correlation between two electrons in different quantum states of the atom. The correlation describes, how the wavefunction and the energy of one electron is changed by the presence of the other electron. Such 'planetary atoms' have been therefore extensively studied over the past 4 5

years ' . 3

Vibrational Energy Redistribution in Molecules

In their lowest vibrational levels polyatomic molecules can be well described by weakly coupled harmonic oscillators, because the potential is nearly harmonic at these low energies and the total vibrational energy is to a good

125 Autoionization

g I P 2 9dn'd-

IP1 Ba+

••UM--"

6S

6snd-

6s*

Ba

6s*

Figure 1. Autoionization by inner shell excitation (a) or by simultaneous excitation of two valence electrons (b). The atomic orbital model is only schematic.

approximation the sum of the energies of the participating normal vibrational modes. A normal mode qi is a vibration, where all nuclei of the molecule oscillate at the same frequency and pass through the equilibrium configuration in Fig. 2a) at the same time. The situation changes with increasing excitation energy, where the potential becomes more and more anharmonic. The potential V(qi, ...,73^-6) in which the N nuclei of a molecule oscillate can be expanded into a Taylor series of the (3./V-6) normal coordinates of a nonlinear molecule, which gives

126

AE

^c

v-, a)

v2

v3

b)

Figure 2. a) Comparison of harmonic and real potential for a diatomic molecule, b) Level scheme of normal vibrations for a nonlinear triatomic molecule.

" = * + ?(5i) 0 « + 5?(ss;)„« f + dV qiqjqk + ••• -T

(1)

3

where VQ = 0 represents the potential minimum at the equilibrium configuration qi = 0 and therefore the first derivative is (dV/dqi)o = 0. In a harmonic potential the third sum in (1) vanishes and there is no coupling among the normal vibrations. In an anharmonic potential, however, the cubic term is non zero and mixes different normal modes, since the potential energy and the restoring force depends on the products of different normal mode amplitudes. This anharmonicity leads therefore to an increasing coupling between different vibrational modes and the total vibrational energy is no longer equal to the sum of the normal mode energies (Fig. 2b). In fact the model of vibrational normal modes can no longer be applied and the motion of the nuclei may become very complicated, in some cases with strong nonlinear

127 o..08 o..07 0,.06 C 0..05 o .04 'B. 0 .03 o o .02 \ j) which measures the absorption a(t) = Ni{t)-Oij at variable time delay At either directly or via the probe laser-induced fluorescence intensity which is proportional to the population Ni(t). If the probe laser is tuned to a transition starting from a neighboring level | J " 4- A J ) , the time dependent absorption of molecules in this level gives the individual contribution of the collision induced net transfer rate | J!' + A J ) -•! Jf) to the level | i) depleted by the pump.

142 (V k \ Jk')

( V i , Ji')

pump Laser

(Vf, Ji")

i l pump

Llfl.

\

I 1

1 / 1 / T

/

rel.

Figure 15. Measurements of collisional population transfer in electronic ground states by time-resolved optical-optical double resonance techniques.

The most detailed information about collisional transfer in ground state molecules can be gained from crossed beam experiments 20 . Molecules M\ in a collimated beam collide with atoms or molecules Mi in another beam, which crosses the first beam perpendicularly (Fig. 16). The molecules M\ scattered under an angle •& are detected state-selective by laser-induced fluorescence with a probe laser tuned to a transition | _;') —»| m). The intensity of the

143

probe laser

pump laser •&\

molecules

I m> —

—

J

pump

depletion of I i >

detection of molecules in I j >

Figure 16. State selective measurements of differential cross sections in crossed molecular beams.

fluorescence is proportional to the population Nj of scattered molecules Mi in level \ j). If a pump laser depletes a selected level | i) of the molecules Mi before they enter the collision region, this level cannot contribute to the population Nj after the collision. Therefore, the difference ANj of the population Nj with the pump laser on and off yields the individual collisional transfer rate from level | i) to level | j). Since the scattering angle defines the impact parameter, such a 'complete scattering experiment', where initial and final states and the scattering angle are measured, gives the most detailed information on the dependence of an individual collision-induced transition | i) —»| j) (e.g. a

144

rotational transition J ->• J 4- A J, or a vibrational transfer v -> v + Av) on the impact parameter, the collision partners and the relative velocity. With other words it tells us, at which distance between the collision partners the individual transitions have maximum probability. 9

Conclusion

The examples, given in the preceding sections illustrate, that laser spectroscopy is a powerful tool for the detailed investigation of the various energy transfer channels in atoms or molecules. In intra-molecular transitions the total energy of a molecule and its total angular momentum is preserved. The transitions cause a redistribution of energy and angular momentum among the different degrees of freedom, for example between the kinetic energy of the nuclei and the energy of the electron cloud. In case of collisional energy transfer either the kinetic energy of the relative motion of the collision partners is transferred into excitation energy of one or both of the partners or the internal energy of one partner is transferred to the other (inter-molecular energy transfer). These collisional energy transfer processes form the molecular basis of all chemical reactions and together with energy transfer by absorption or emission of photons form the basis of biological life. It is, therefore, worthwhile to study them thoroughly.

References 1. K.K. Lehmann, G. Scoles, Intramolecular Dynamics From EigenstateResolved Infrared Spectra, Ann. Rev. Phys. Chem. 45, 241 (1994) A. Beil, D. Luckhaus, M. Quack, J. Stohner, Ber. Bunsenges. Phys. Chem. 101, 311 (1997) 2. St.R. Leone, State-Resolved Molecular Reaction Dynamics, Ann. Rev. Phys. Chem. 35, 109 (1984) 3. see for instance the Proceedings of the Int. Conf. On Spectral Line Shapes, Vol. I-X, Library of Congress Catalogue 1979-99 4. I.C. Percival, Planetary Atoms, Proc. Roy. Soc. London A353, 289 (1977) 5. J. Boulmer, P. Camus, P. Pillet, J. Opt. Soc. Am. B4, 805 (1987) 6. A. Delon, R. Jost, J. Chem. Phys. 95, 5701 and 5687 (1991) 7. A. Holle, J. Main, G. Wiebusch, H. Rottke, K.H. Welge in: Atomic Spectra and Collisions, ed. By K.T. Taylor, M.H. Nayfeh, C.W. Clark (Plenum Press, New York 1988)

145

8. H. Wenz, R. GroBklofi, W. Demtroder, Laser und Optoelektronik 28, Febr. 1996, p. 58 9. R. Grofiklofi, P. Kersten, W. Demtroder, Appl. Phys. B58, 137 (1994) 10. P.C.D. Hobbs, Appl. Opt. 36, 803 (1997) 11. R.E. Miller, Infrared Laser Spectroscopy in: G. Scoles, ed.: Atomic and Molecular Beam Methods, p. 192 (Oxford Univ. Press, 1992) 12. T. Platz, W. Demtroder, Chem. Phys. Lett. 294, 397 (1998) 13. V. May, 0 . Kuhn, Transfer Phenomena in Molecular Systems (Wiley VCH, Weinheim 1999) 14. F.P. Schafer, ed.: Dye Lasers (Springer, Berlin, Heidelberg 1977) 15. H. von Busch, Vas Dev, H.-A. Eckel, S. Kasahara, J. Wang, W. Demtroder, P. Sebald, and W. Meyer, Phys. Rev. Lett. 8 1 , 4584 (1998) 16. H.-G. Kramer, M. Keil, C.B. Suarez, W. Demtroder, W. Meyer, Chem. Phys. Lett. 299, 212 (1999) 17. LB. Bersuker, The Jahn-Teller Effect, Plenum Press, New York 1984 18. M. Berry, The geometric phase, Scientific Am. 12, 26 (1988) 19. K. Bergmann, W. Demtroder, J. Phys. B5, 1386 and 2098 (1972) 20. K. Bergmann, U. Hefter, J. Witt, J. Chem. Phys. 7 1 , 2726 (1979); 72, 4277 (1980)

This page is intentionally left blank

ADVANCES IN THE TECHNIQUES FOR THE STUDY OF ENERGY TRANSFER

Daniele Hulin Laboratoire d'Optique Appliquee ENSTA, Ecole polytechnique, Palaiseau, France

1. Introduction When an external source of energy excites a material, there is a transfer of energy from the source to the sample. This newly excited state is usually not an equilibrium state and the acquired energy is transferred to the surrounding world. This mechanism can be studied either through the relaxation of the initially excited state or through the apparition of newly excited states. Monitoring these phenomena after an excitation gives an insight on the fundamental mechanisms that govern the material. In this chapter, the excitation will be an optical excitation, although many features could be generalized to other sources of excitation. Recent progresses in the techniques of observation have opened new fields of investigation and the purpose of this chapter is to provide a (non-exhaustive) overview of the most-frequently-used recent techniques. How can we monitor an excited state and its evolution after excitation? In a simplified manner, one can say that there is two different ways to perform this study. A change of the properties of the material, mainly but not exclusively the optical properties, can be analyzed or an emitted signal (spontaneous, stimulated emission) can be recorded. The separation between the two methods is not as abrupt as it is presented here. This chapter will first present different recent techniques used to monitor the changes in the optical properties. Then it will discuss techniques allowing to study emitted signals. Finally, phase-related techniques will also be described.

2. Pump-probe spectroscopy : absorption 2.1 Experimental methods The optical excitation is often referred as "the pump". In order to monitor the changes of the optical properties resulting from the excitation, light again has to be used. This light is referred as "the probe". In this paragraph, we will be concerned by the changes in absorption.

147

148 In order to follow the energy transfer, the absorption changes have to be time-resolved. This is usually done in two different ways:

pump

•4

cw probe

H

Time-gated detector

(a)

pump

Timeintegrating detector

(b)

After a pulsed excitation, the temporal evolution of the sample absorption can be recorded through the change of the transmitted probe beam in function of time by a timeresolving detector (part (a) of the figure). This can be realized through a photodiode, a streak camera or an optical gate. In the second case (part (b) of the figure), the detector is time-integrating (long response time), but the temporal resolution is provided by the fact that the probe beam is also a light pulse. If the probe pulse reaches the sample before the excitation (situation 1), it sees an unexcited sample; if it arrives at a delay x after the excitation (situation 3), it records the sample situation that has already evolved after excitation. This sequence can be repeated and the delay x can be varied on purpose. This method is analogous to the stroboscopic technique. The choice between the two methods depends on the need in time resolution. For times longer than few tens of picoseconds, the first solution may be easier to handle. However, if one wants to take advantage of the recent advances in ultrashort-pulse lasers, only the second method can be used. Lasers with ultrashort pulses are mainly based on oscillators using Kerr-lens modelocking and Titanium-doped Sapphire crystal. They deliver currently pulses as short as 15 femtosecondes (15xlO"15 s) with a repetition rate of 15 MHz. Because of the short duration, the pulses have a quite large spectral width (40 nm) and this allows already to do some spectral measurements. However they are centered around 800 nm and this restricts the range of study.

149 In order to overcome this limitation and also to obtain a larger intensity per pulse, amplification chains are used to boost the energy from the picojoule level up to the kilojoule level but with only few shots per day! For most of the studies this extreme performance is not necessary so that the repetition rate after amplification can remain large enough to accumulate the signal over a reasonable time. Following the desired energy per pulse, it varies from 10 Hz to more than 100 kHz (Rulliere, 1998). In the pump-probe experiment, the time resolution depends on the pulse duration and on the precise determination of the delay x. This last requirement is fulfilled by splitting the initial pulse in order to avoid any jitter between pump and probe pulses. The two pulses have different optical paths before reaching the sample. These paths can be varied on a controlled manner. The zero delay (coincidence between pump and probe pulses) is determined through the help of physical phenomena occurring only when the two pulses overlap. The signal is recorded in function of the delay x, providing the evolution of the absorption after excitation.

In the above scheme, pump and probe have the same wavelength since they originate from the same laser pulse through the beam splitter (BS). It may happen quite often that a different wavelength is needed for the pump or for the probe. In order to change the wavelength of the laser pulse, different methods can be used. Three examples are given below: •

Raman effect may be efficient enough to shift the laser toward longer wavelengths:

150 G^laser

•

^

tOiaser " tOphonon

Parametric generation is more and more widely used (Nisoli, 1994; Yakovlev, 1994; Seifert, 1994). Parametric effects are based on the use of non-linear quadratic crystals having large second-order susceptibility (%(k)

(2b)

has a finite value. For this topic see also

Fig. 1. The model of the Lorentz oscillator, unexcited (a) a collective excitation with X => oo (b) or with the minimal physically meaningful wavelength X^a (c); the excitation of a wave paket (d) and an improved version used to describe spatial dispersion (e). According to 3 ' 5 .

168

* E=Uco

* E=1ico

0>n

2 71

7C

a

•W:

Fig. 2: The dispersion relation for the ensemble of uncoupled oscillators of Fig. la - d (a) and for the coupled ones of Fig. le (b). According to 3 ' 5 .

The vanishing of the group velocity is immediately clear for the wavepaket sketched in Fig. Id. If a more realistic model is required, which includes a k-dependence of the eigenfrequency, the so-called spatial dispersion (flo(k), it can be modeled by a coupling between the Lorentz oscillators. See Figs, le and 2b. The coupling between the oscillators results in finite vg and a non-local response of the medium 9. In order to couple the oscillators to the electric field of a light wave, we give every mass a small electric charge e and place for neutrality reasons the opposite charge in the equilibrium position of the oscillator. The elongation of the oscillators x(t) is then connected with a dipole moment p(t) given by P(t) = ex(t)

(3)

In principle we could also introduced a magnetic dipole moment, but since the magnetic coupling is usually much weaker than the electric one, we neglect the magnetic dipole-, electric and magnetic quadropole and higher transitions in this article, if not explicitely stated otherwise. We assume now that an electromagnetic wave is falling on the (threedimensional) ensemble of model oscillators, which propagates in z-direction and has an electric field polarized in x-direction i.e. E = E0 exp [i(kzz - cot)], E0 = ( E0, 0, 0)

(4)

169

Then the equation of motion for each oscillator is given by mx(t) + 2ymx(t) + (3x(t) = eE0 exp (- icot)

(5)

We introduce here a phenomenological damping term y, which is justified for quasistationary conditions, on which we concentrate here. For the regime in which this approximation is valid see e.g. two contributions to this book ' . Eq. (5) is nothing but the well-known equation for a damped, driven, harmonic oscillator. The solution is for the case of weak damping the sum of a transient feature (actually the general solution of the homogeneous differential equation) plus the driven oscillation X(t) = XtransientO) + X P exp(-ifflt)

(6)

Since we are not interested in the transient features here (for this topic see e.g. the contributions 8 " 12 ) we concentrate on the second term on the r.h.s. of (6) and obtain the well know resonance term for xP(co). We define the polarizability a of every oscillator by d =^ E„

(7)

and obtain e2

I m „^ « =— —— (») co0-co -lmy The polarization of the medium P(co) i.e. the dipole moment per unit volume is then given by P = Nexp

(9)

A comparison of the material equation in its general form D = e0E + P

(10a)

where D is the electric displacement (see e.g. 5) and its linear approximation used in the following if not stated otherwise D = s0s(co)E yields with Eqs. (8, 9) for the dielectric function e(co)

(10b)

170

, . , Ne 2 /me„ e(©) = l + — j—^(D0-a) -icoy

„_ (11)

The numerator of the r.h.s. of Eq. (11) is usually called oscillator strength f. If we have several eigenfrequencies Oo; in the system, their contributions simply add in linear optics, resulting in e(co) = l + S —

^—

i C0oi-G)

(12)

-1C0Y,

For very close lying resonances, the sum in Eq. (12) may be replaced by an integral. We note that every term in the sum of Eq. (12) tends to zero for co » co0i and to a constant value f; Ia20{ for co «co 0 1 . If we consider the simple case of a single resonance, spectrally well separated from the others we can rewrite Eq. (12) as e(co)=e b [l+ 2 f / f b , j V. co0-co -icoyy

(13)

where the so-called background dielectric constant 8b includes the „1" of Eq. (12) and the constant contributions of all higher resonances, while the contributions of all lower laying resonances are neglected, following the argument given above. In principle several corrections have to be included like local fields, which alter f and coo, but the overall shape of Eq. (11) to (13) is not changed 5. Optical anisotropy can be included by a directional dependence of the parameters 8b, f, co0 and y which leads to a tensor character of s(co)5. Spatial dispersion results in a dependence of s on two independent variables co and k. We do not go into details of these topics here but refer the reader for details to 5 and references given therein. In Fig. 3a and b we show the real and imaginary parts of e(co) according to Eq. (13) for vanishing and weak damping y. For co => 0 si(co) starts with the static dielectric constant es = eb + {/a>I, goes through the resonance and approaches 8b for high frequencies from below. For the highest resonance in a given system 8b is 1. The imaginary part s2 (co) has a Lorentzian shape and develops towards a 8-function centered at coo for vanishing damping. The resonancefrequencycoo is also called transverse eigenfrequency, while a longitudinal wave may exist at the frequency co = COL for which e(co = COL) vanishes. This becomes clear when inserting a plane wave ansatz E = E0 exp [i(kr - cot)] in the Maxwell equation divD = p = 0

(14)

171 for vanishing free space charge density p. With Eq. (10b) and executing the Nabla operator we obtain 80ikE(co)E = 0

(15)

Eq. (15) is fulfilled for k 1 E. This is the usual way to show that electromagnetic radiation in vacuum is a transverse wave with (16)

E l k l B l E

where B is the magnetic flux density. In vacuum this is the only solution since e = 1. In matter there is for co = coL also the solution S((D) = 0. This means, that at this frequency a longitudinal wave may exist. A closer inspection of Maxwell's equations shows, that this is a wave with longitudinal E and P fields which are antiparallel to each other according to Eq. (10a) and vanishing magnetic fields.

£i

(b)

Y=0 Y = 0.2 A LT

ji

1 1 1 1

Y = 0.2A LT

1 1 I

/

\ \\\\ ). This quantity is given Fresnel's formulare as (n(g>)-l)2+K2(cp) R±(o>) =

(20)

(n(co) + l) 2 +K 2 (o)

The stop-band or reststrahlbande between co0 and COL is clearly seen. The minimum of Ri(co) occurs where n(co) = 1.

>-A LT

Fig. 5: Reflection spectrum in the vicinity of a resonance for normal incidence. From 5

Increasing damping y tends to wash out all spectral features in e(co), n(a>) and R(co). It should be mentioned that a finite oscillator strength f is for small damping necessarily connected with a finite splitting between coo and coL according to (O2L-(O20=f/Eb

(21)

and the so-called Lyddane-Sachs-Teller relation (22)

174

For a discussion of Eq. (22) see also 6. A frequency dependent, complex electrical conductivity o(co) can be included by extending the model of Lorentz oscillators to the Drude-Lorentz model. Formally this can be done by considering a term like Eq. (5) with a vanishing restoring force Px(t). Such a term then corresponds to the relaxation approximation to desribe electric conductivity. In Eq. (11) to (13) this results in a term with vanishing eo0 and an oscillator strength given by the plasma frequency
) are different from one. From the internal photoeffect we know, that this mixed state is also quantized into integer multiples of energy to accompanied by a quasi-momentum h k. For a discussion of the similarities and differences of momentum and quasi-, crystal- or pseudomementum of the quanta of elementary excitations in solids see e.g.8'15. We call the quanta of the mixed state of electro-magnetic and polarization wave „polaritons". The dispersion relation of the polariton in Fig. 7 starts on the lower polariton branch (LPB) with photon-like behaviour. The term photon-like is used, because the dispersion relation is linear like for photons but the slope is different and given by ftc(es)"2, where the factor (es ) " 2 is resulting from the accompaning polarization wave.

photon UPB

tico

energy laco

AM

1

long. lacor taco - -

LPB

wave vector k Fig. 7: The dispersion relation of light in vacuum (---) and in matter (—) for vanishing damping. From

177 Then the dispersion relation bends over to the behaviour of the resoncance. Depending on what this resonance is, one speaks e.g. of the phonon-like, exction-like, etc. part of the LPB. There is a finite longitudinal transverse splitting ALT (see Eq. (21)), eventually a longitudinal eigenmode and an upper polariton branch starting at ha L and bending over to a photon-like behaviour again, however with a steeper slope c(£b)"1/2 (see Eq. (22)). It should be mentioned that in the quantum-mechanical description the oscillator strength f contains the transition matrix element squared. The quantum-mechanical description of the polaritons proceeds via the diagonalization of the Hamiltonian containing the number operators for the photon-field B k (co)B k (co), of the Lorenztoscillator field Ck(co)Ck(co) and their interaction according to H = z M k ) B 1 > ) B k ( f f l ) + lE(k)C + k (a>)C 1 >) + k

k

Zgk,kC+k,((o)Bk(co)+c.c.

(27)

kk'

The gkk are the coupling coefficients. More details on this topic can be found e.g. in 16. The „user-friendly" aspect is, that the dispersion relation of light in matter remains the same inedpendent if we use the classical description in terms of an ensemble of Lorentz oscillators or the quantum-mechanical one in terms of polaritons. The difference is „only" the prefactor h to both co and k, more prescisely to Re{k} which describes the propagation. If not stated otherwise, we mean with k in the following the real part Actually one uses often even a mixture i.e. E = ha as a function of k. Sometimes one can hear the statement, that the polariton concept works only for ordered or crystalline matter. Indeed this concept has been introduced e.g. in 3 for a periodic arrangement of oscillators. But, as we will show in the following section 2, it can be easily extended to non-crystalline matter like amorphous solids, alloys, liquids and gases, as long as inequality Eq. (1) is fulfilled. For the opposite regime one faces rather a scattering problem. The only difference is, that in ordered matter the quasi momentum h k is conserved only modulo integer multiples of unit vectors in reciprocal or k space G = hibi + h 2 b 2 + h 3 b 3

(28a)

where the bj are connected with the primitive translation vectors aj, in real space by b,=27r

'— V(ajxak)

andc.p.

(28b)

which is one of the reasons why h k is called quasi-momentum. Eq. (28a) allows to restrict all dispersion relations to the first Brillouin zone. See e.g. every textbook on solid state physics like 17. In disordered matter, k is a good quantum number, as long as Eq. (1)

178

holds, i.e. an effective medium description is adequate. For increasing k the concept of k gets more and more vague when one leaves the effective medium regime. However, this is not a serious problem in our case. In the IR and VIS range X is much longer than the interparticle distances in solids and liquids and also for not too diluted gases. In the X- and y-ray region the refractive tends to the value in vaccum n = 1 indicating the transition from polaritons to photons, except for the spectral vicinity of well defined resonances as shown in section 2. This above statement is nicely illustrated by the following facts. In the IR, VIS and UV the refraction of light is equally well described by Snellius's law, which is based on the conservation of the momentum parallel to the interface h ky according to ENoether's theorem 5, for crystalline quartz SiC>2 and fused, amorphous glass or silica. However, in the X-ray regime where Eq. (1) does not longer hold the crystal gives still sharp reflection beams according to Ewald's construction i.e. kout

=

ki„ + Or,

| kout | =

| kj n |

(29)

in elastic scattering, while the pattern is blurred for the amorphous material, especially for higher diffraction orders, indicating that the concept of the conservation of the quasimomentum h k in matter starts to break down. 2.

Application of the Polariton Concept to Three-Dimensional Matter In this section we apply the concept of model oscillators to three-dimensional or bulk materials, starting with crystals and proceeding to disordered matter. The Lorentz oscillator will be replaced by the quanta of the elementary excitations in solids like phonons, excitons, plasmons or magnons, by (two-level) transitions in atoms or by transitions in the nuclei, which couple to the light field. The justification to replace the harmonic Lorentz oscillators in the above mentioned way follows from the close relation between the excited states of a harmonic oscillator and the occupation numbers of a Boson field as detailed e.g. in 8 ' 9 . There are, however, also some deviations between a two-level system and the harmonic Lorentz oscillator. An ideal harmonic oscillator can be driven into higher and higher quantum states, which still remain equally spaced in energy, while a two level system will be saturated by the compensation of absorption and stimulated emission, when ground and excited states are equally populated. If we restrict ourselves to the low density regime, also an ensemble of two or more level systems will be perfectly modeled. Presently there is a trend to describe the optical properties of matter by the optical - and the semiconductor Bloch equations. See e.g. 8"10,16. This theory is necessary for the understanding or modeling of transient features especially in the regime of (ultra-)short timeresolved spectroscopy as shown e.g. in 8" or 18. This theory is also capable to reproduce the linear optical properties discussed in this contribution and the elementary excitations in solids like the excitons. However, it is a rather complicated theory which allows in many cases numerical solutions only. Since it is in addition also not a „complete" theory, e.g. since it treats generally the electric- and

179

polarization fields on a classical (Maxwellian) basis and the electronic excitations on a quantum mechanical level, we recommend to use the didactically simple and intuitive model developped above, where it is applicable, and this is a wide range of linear and partly even of non-linear optics, and to restrict the use of the much more complex semiconductor Bloch equations to the situations where there are really necessary. 2.1. Polaritons in Crystalline Solids In this section we present polaritons in crystalline solids which result from the coupling and mixing of the electromagnetic radiation with the quanta of the elementary excitations like excitons, phonons, plasmons or magnons. We start with exciton-polaritons and treat them in same detail to introduce various techniques to measure the polariton dispersion, which will be used later on. Excitons are the quanta of the elementary excitation in the electronic system of semiconductors and insulators. In simple approximation they can be considered as a hydrogen- or positronium atom like system of a negatively charged crystal electron in the conduction band and a positively charged hole in the valence band, bound together by Coulomb interaction. The excitonic Ryelberg energy is modified by the reduced mass of electron and hole and by the dielectric function compard to the value of Hydrogen. Instead of 13.6eV these corrections result in values between 5 and 200meV for typical semiconductors. The onset of the ionization continum reached for main quantum numbers nB => oo coincides with the energy of the band gap. We present here data only for direct gap semiconductors with conduction and valence band extrema at k = 0, the so-called Ypoint. As an example we show in Fig. 8 the bandstructure, the reflection and transmission spectra and the polariton dispersion of excitons in CdS. There are three exciton series with holes in the A, B and C valence bands (Fig. 8a). The first two series overlap energetically and are shown in Fig. 8b and c in reflection and transmission. There is some dichroism and birefringence since the A exciton couples strongly to the radiation field only for light polarized with E 1 c where c is the crystallographic axis of the hexagonal CdS crystals. The dispersion of the m = 1 AT5 exciton polariton in Fig. 8d corresponds to the dispersion relation of Fig. 7 with the inclusion of a k-dependence of the eigenfrequency co0(k) which represents just the kinetic energy of the center of mass motion of excitons. The ne = 1 BTs resoncance shows a complication, namely the appearance of an intermediate polariton branch, which results from a mixing of the dipole allowed singlet T5 states and the dipole forbidden Y2 triplet states by the k-linear term shown schematically for the T7 valences bands in Fig. 8a. This intermediate polariton branch is the reason for the small dip in the reflection spectrum of the B exciton indicated in Fig. 8b by an arrow. The dip coincides with the energy at which the refractive index of the intermediate polariton branch has the value one. The experimental points in Fig. 8d result from reflection and hyper-Raman spectroscopy. For details see 22.

180 band structure

(b) —1—'

1 1 ' 1

l r i

i — i — r * ~ r "1

IT

1 1 1 1 1

BnB=l

AnB= 1

CdS B n B = 2;n B = 3 A n B == 2;n B = 3

"it

^A

,

i

,

,

,

flic*

A/

,

1

2.55

2.56

i

Elc

i

J—i-_l

L

2.57

\ i

,

,

,

,

2.58

X

2.59

(ev;

10-'

.p A

10"2

10"J

104 2.54

2.55

156

2.57

2J8

photon energy (eV)

159

2.60

5

10

15

20

k (10 5 cm"1)

Fig. 8: The bandstructure of CdS in the vicinity of the T-point (a) the reflection (b) and transmission spectra (c) in the vicinity of the A- and B-exciton resonances showing various members of the hydrogen like series up to n B = 4 and the dispersion of the n B = 1 A and B exciton polaritons (d). From 1 9 , 2 0

181

Fig. 9 shows the results of a four wave mixing experiment (FWM) in reflection from21. In FWM one excites a polariton by afirstpulse (ki, ha,) and probes its decay by the interference with a second delayed pulse (k2, ha 2 ). Generally one observes the diffracted order in the direction 2k2 - ki in transmission and reflection. For details of this We choose here the reflection geometry since transmission expetechnique see e.g. riments in thicker samples involve propagation and absorption effects, which makes their interpretation rather complicated. See e.g. 21 and examples below. The lOOfs pulses used in the experiment are spectrally so broad that they excite simultaneously the nB = 1 Aand B-exciton resonances. Consequently a beating is observed with a period corresponding to the splitting between the nB = 1 A- and B-excitons of about 15meV. In additon there is a slow modulation with a period of about 4.4ps corresponding to an energy splitting of about lmeV. This is just the splitting between the lower and the intermediate polariton branches of the ne = 1 B-exciton resonance in the region where both dispersion curves are rather flat and almost parallel, i.e. where the (combined) density of states is high. 1 LI i r r r n i rp"' i i i i i i 11 i f i i i I T ri [ i i i"i i i i i i pTT'rri r i i | i n n n i'T| I I I n r r r r

• 0.1

^ c p

0.01

4K In

42 c ^ •e ra t——i

CdS

: . • •

, . 0.001

• I . . I l l }I . I l l 1

0.0001 -

3

-

I

2

1 1 • • I

-

1

1I

0

1

'

t

» • • ' • ' •

2

'

3

'

I »

11

'

4

Timedelay x [ps] Fig. 9: The FWM signal in reflection in the spectral region of nB = 1 A and B excitons in CdS. From 21.

The overall decay of the signal results from damping. The value deduced here (< lmeV) coincides with data deduced from a quantitative fit of the reflection spectra 5'22.

182 A first experimental technique to measure directly the dispersion relation of polaritons is discussed in Fig. 10. If one measures the refraction of a light beam as a function of Aco at a prism with know angle a (see inset) it is possible to deduce the spectrum of the refractive index n(Aco ). Data are shown in Fig. 10 for the first three dipole-allowed exction resonances in CdS. It should be noted, that in the lowest resonance n(ha>) could be followed up to values of 25! The dispersion relation is closely related to n(fao ) via Eq. (18). Especially the lowest resonance reproduces thus the exciton polariton resonance including the k dependence of a>o, if one just interchanges the x and y axes.

400

n 100 25

2.55

2.56

2.57

2.58

irco, (eV) Fig. 10: The spectrum of the refractive index of exciton polariton resonances in CdS deduced from the refraction at a thin prism. From 23.

Now we present three examples of propagation effects in the vicinity of the exciton polariton resonances.

183 In CuCl the ALT of the lowest exction resonance is about 5meV. With a Fourierlimited pulse of about lps duration and lmeV width it is therefore possible to scan the group velocity over the resonance. Results shown in Fig. 11. The right hand side gives the polariton dispersion, the left the group velocity deduced from the time of flight through a sample of a few urn thickness.

3.22 II i i i i i i i i i \/

i i i mn|

1 i i IIIIII

1 i I MINI—I

I I IIIII

3.21 -

3.19 ••

(a)3.18

i

i

i

•

i

i

•

•

i

io io-k (x 106 cm"1)

Fig. 11: The exciton polariton dispersion in CuCl (a) and the measured (•) and calculated (—) group velocity (b). From 24.

The calculated spectrum of the group velocity is just the slope of the polariton dispersion relation Eq. (2a). There is excellent agreement between experiment and theory and it should be noted that vg can be as low as 5 • 10"5c in the resonance region. The semiconductor Cu 2 0 has also a direct band gap at k = 0. Since both bands have the same parity only excitons with P-type envelope function have a small oscillator strength in dipole approximation. The na = 1 singlet- (or ortho-) exciton couples to the radiationfield only in quadropol approximation with an extremely small oscillatorstrength, which is two to three orders of magnitude smaller than e.g. in the examples CdS and CuCl above 25. Consequently one may think, that the strong coupling or polariton-picture is not necessary to describe the interaction of theses resonance with light and one can limit oneself to the perturbative or weak coupling case. For many applications this statemtent is true, but there are also experimental data, which can be described only in the polariton picture 26. We discuss these so-called propagation quantum-beats with Fig. 12. A ps pulse

184

is send on a Cu20 sample with a thickness in the mm range. Its spectral width AE is in contrast to the above example such that it covers the whole energetic region around the resonance i.e. AE » ALT-

I

'

•

'

10 r

'

I

•

'

Cu 2 0 T=2K

i\i

J

, sj \ i \

experiment

"' \ i \

10

theory

Laser

10

I

w a v e

vector

0%

f

\

.$%:

10 _l

I

I

I

500

I

I

L_

1000 time (ps)

1500

2000

2500

•

Fig. 12: The temporal evolution of a ps pulse travelling through a sample of Cu 2 0 in the spectral region of the nB = 1 ortho exciton resonance. From le.

After transmission through the sample, the pulse has a length exceeding a ns and shows a temporal modulation, the period of which increases with time. The explanation in the polariton picture is straight forward. As shown on the l.h.s. of Fig. 12, the pulse excites the upper and lower polariton branches simultaneously. There are, as indicated, always pairs of states on the lower and upper polariton branches which propagate with the same group velocity. During the propagation the interference between these two states oscillates periodically between constructive and destructive, and they arrive with a certain relative phase at the other side of the sample and radiate a photon field into the vacuum. An inspection of the dispersion relation shows immediately, that the pairs with propagate fast have a short oscillation- or beatperiod and the ones which are slow have a

185

long period. This explains quantitatively the experimentally observed behaviour The overall decay of the curve follows from the damping. A theoretical fit to the data gave ALT « 5ueV and Tij « 0.9ueV. Similar propagation quantum beats have been observed also with free and bound exciton resonances in InSe or CdS 27. The third example of propagation phenomena is observed in a platelet type CdSi. xSex crystal involving the lower polariton branch only. Alloy semicondcutors like CdSi. xSe* show exciton localization effects at their band edge due to disorder induced by spatial fluctuation of the composition x. For a recent review see28 and references therein The spectrally resolved FWM signal (see Fig. 13a) shows for increasing delay a periodic modulation of the signal intensity. The period increases very rapidly with increasing photon energy. The spectrum of the laser-pulse covered the tail of the localized exciton states in the alloy, i.e. only states on the lower polariton branch of this strongly inhomogenously broadened resonance have been excited. The plan-parallel surfaces of the platelet-type sample formed a Fabry-Perot resonator, exhibiting a clearly modulated reflection spectrum (Fig. 13b). The interpretation of the data in Fig. 13a in first approximation is as follows. The first pulse creates a polariton wave-paket which propagates through the sample. If the second pulse arrives without time delay there is optimal overlap between the wave-pakets and a strong diffracted signal e.g. in direction 2k2 - ki. If the second pulse comes with some delay, the spatio-temporal overlap between the two pulses decreases and the signal drops. However, with further increasing delay, signal maxima occur for delays which correspond to integer multiples of round trip times of the first paket in the Fabry-Perot resonator, since then optional overlap between the two pulses is recovered. The long dephasing times of localized excitons in CdSi.xSex and the good reflectivity of the sample surfaces favour this effect, which has been addressed shortly in a theoretical work 30. If the above model is correct, one can deduce the group velocity from Fig. 13a See the open circles in Fig. 13c. The group velocity can be deduced independently from the data of Fig. 13b since the Fabry-Perot modes are equally spaced in k by Ak = 7t/d where d is the geometrical thickness of the sample. These data are shown by closed squares in Fig. 13c. The good agreement gives strong support for the above interpretation. The data are nothing but the derivative of the dispersion curve of the lower polariton branch. The upper one is not accessible by this method because it falls in the absorbing regime of the sample. A further argument comes from the fact, that the modulation disappears, when the temporal pulse length in the sample is made longer than the round trip time by spectral filtering 29. Similar propagation effects have been observed in the stimulated emission of CdSi.xSex samples31. However, there must be additional effects in this experiment, which are not yet fully understood and which involve e.g. excitation induced modifications of the optical properties of the samples, since we observe e.g. that the modulation period at one and the same energy changes slightly with the spectral width and the center frequency of the laser pulse29. Now we mention shortly the polaritons formed with other resonances in crystalline solds.

186

2.085

2.090

2.095

2.100 \

energy (eV)

2.105

\ d o m a i n e of extracted time cuts

1.70

1.75

1.80

1.85

1.90

1.95

2.00 2.05 2.10 2.15

energy (eV)

i

•

•I

i

c

T

i

-

O)

2.0x10

, 7

-

I

•

8

-

I i

"

^

o p/VM-signal

• 2.04

Fabry-Perot modes ( E i c) 2.05

IT

2.06




k„

a barrier

QW

Jr \

J\

ypL. A sketch of a surface plasmon polariton is given in Fig. 6c. Examples for the dispersion of exciton surface polaritons can be found e.g. in 56. An alternative way to couple to surface polaritons is to produce a grating on the surface of the material under investigation with a grating period A, which modifies the conservation law for k to k|| + m27t/Awithm = 0, +1,+2, ...

(30)

and allows for a suitable combination of to , kj m and A a coupling to surface polariton modes.

196 Evidently the concept of polaritons is also well established for the restricted, quasi-twodimensional geometry of surface- (or interface-)polaritons.

- HooL iKQr

licor

1.5

2.0

2.5

wave vector k (k^)

/

.,..—

1

/ / •~0

OU>0

0 V m+1

Fig. 21: Two plan-parallel mirrors forming a Fabry-Perot resonator (a), the spectra of transmission and reflection as a function of the phase shift per round trip 8 for R « 1 (b) the geometry of oblique incidence (c), the dispersion relation of photons in vacuum (d) and in a Fabry-Perot resonator (e).

198

To approach the problem, we introduce first the eigenmodes of a Fabry-Perot resonator or an etalon, assuming for the moment that the resonator contains nothing (i.e. vacuum or air). A Fabry-Perot resonator is in simplest case an arrangement of two plan-parallel, lossless mirrors with a reflectivity R < 1 of each at a distance d. See Fig. 21a. An incident plane, monochromatic wave will be partly reflected and partly transmitted at each of the two mirrors. In most cases the partial waves in the resonator will interfere destructively resulting in a reflectivity of the whole structure R,ot « 1 a transmission T,„t « Rtot -1 » 0 and a very low field amplitude in the resonator. See Fig. 21b. There are, however, special situations namely when an integer number of half-waves fits into the resonator, i.e. if m - = d or kj_ =m— 2 d

m = l,2,3,...

(31)

Then all partial waves in the resonator interfere constructively, the field amplitude in the resonator can exceed considerably the incident one, and the transmission of the whole arrangement is close to unity (Fig. 21b). This situation is shown schematically in Fig. 21a. We can reformulate this fact by saying that the eigenmodes of the Fabry-Perot resonator occur for discrete values of the normal component of the wave vector of the incident light beam k.

=m-

m = l,2,...

(32)

For oblique incidence shown in Fig. 21c the wave-vector of the incident light beam k can be decomposed in components parallel and normal to the plane of the resonator k = k i + k||

(33)

The normal component of k has to fulfil Eq. (32) for the eigenmodes of the resonator while a conservation law holds for ka due to the translational invariance of the problem in the two-dimensional plane of the resonator. This situation is shown in Fig. 21d for light in vacuum, i.e. for photons. The dispersion relation to (k) is a cone with slope h c. The cross section of this cone with a Fabry-Perot mode with a fixed ki is a parabola as shown in Fig. 21e. This figure gives the dispersion of light in an empty Fabry-Perot resonator as a function of kn for a given m. If one tilts a Fabry-Perot resonator in a parallel light beam away from normal incidence i.e. from kp = 0, the photon energy of the transmitted light shifts to higher values, i.e. to shorter wave-length. This effect has been demonstrated during the lecture. If a semiconductor physicist (and the author belongs to this community) sees a parabolic dispersion relation, he thinks automatically about the concept of effective masses, beeing defined as 5 '"

199 -i *-2 5 2 to(k) "U=ft gk2

,,.. ( 34 )

An inspection of this concept gives for this situation the physically reasonable result, that the „effective mass" of photons in a Fabry-Perot resonator is given by the energy equivalent mass of the photons in the minimum of the parabolic dispersion to 0 i.e. by m e f f c 2 =to 0

(35)

In reality the mirrors used in Fabry-Perot resonators are generally not simple surfaces but dielectric- or Bragg-mirrors. The way how they work will be outlined only in chapter 4. In a next step, we bring some matter into the resonator, which has an eigenmode at a certain frequency to M . Since such a resonance is generally connected with absorption, and since absorption deteriorates thefinesseof a Fabry-Perot resonator 5, only a thin layer of matter is recommended. Therefore one places usually one, or a few quantum wells in the resonator at the position of an antinode of the Fabry-Perot mode and with an eigenfrequency of the lowest free exciton resonance close to the one of the Fabry-Perot mode. In Fig. 22a we show schematically the dispersion relations of the Fabry-Perot resonator mode to pp (ky) and of the matter to M (ky). The curvature of the exciton resonance to M is negligible compared to the one of to ^ . If both eigenmodes do not couple, their dispersion relation cross. If there is a finite coupling, we observe again the anti-crossing behaviour, and the dispersion describes mixed states of the Fabry-Perot or cavity mode and of the exction resonances, the so-called cavity polaritons. In Fig. 22b we show an example in which the luminescence has been observed as a function of the angle relative to the normal of the cavity. A variation of this angle is equivalent to a variation of ky. The fact, that both branches of the cavity polariton show up in luminescence under non resonant band-to-band excitation proofs that they are really the quanta of a mixed state. A Fabry-Perot resonator mode alone would not luminesce at to „, if light with a different photon energy is sent on the resonator. The system of Fig. 21b consists of the hh exciton resonance in Ini.yGayAs quantum wells in a 3 A/2 cavity with dielectric Bragg-mirrors on both sides 57b, but there are many other cavity-polariton systems known in literature. A small selection from III-V and II-VI compounds is given in" To conclude, a necessary prerequisite for the appearance of cavity polaritons should be mentioned. The dephasing time of the excitation in matter T2 must be much longer than the round-trip time of light in the resonator. Otherwise the coupled modes cannot develop, because the excitation in matter loses too quickly the phase coherence to the Fabry-Perot mode. The lowest exciton resonances of quantum wells fulfill usually the above condition at low temperature and density with T2 values of a few and up to some ten picoseconds 43. For increasing excitation, T2 becomes shorter, e.g. by excitation induced dephasing by exciton-exciton colloissions or the transition to an electron-hole plasma and the characteristic cavity polariton mode structure dissappears 58.

200 theory without coupling -•

•

•—

-10° 1.35-

experiment and theory with coupl. 0

external angle 10° 30°

L

hco

1.32

-

Fig. 22: Schematic drawing of the dispersion relations of a Fabry-Peropt resonator mode h(£> ^ (k|) and of a resonance in matter h(Q M (kj) for vanishing and finite coupling (a) and experimental luminescence data for the exction cavity-polariton mode from57b (b).

4.

Photonic Crystals

After having established the polariton concept for bulk matter and for structures of reduced dimensionality, we investigate the consequences of a spatially periodic modulation of the optical properties. This will lead us to the concept of photonic (more precisely polaritonic) crystals. In prinicple it is however not so terribly new but only a repetiton of what is known as Ewald-Bloch theorem in X-ray diffraction or in the bandstructure of crystalline matter.

4.1. Introduction of the Basic Concept To introduce the concept of photonic crystals with photonic bandstructures we consider a dielectric- or Bragg-mirror and the Fabry-Perot resonator formed from two dielectric mirrors with a cavity in between. It is long known, that a high reflectivity close to unity can be obtained over a certain frequency interval, if one produces a stack of layers of different refractive indices ni and nn, where each layer has a thickness d; of a quater wavelength, i.e.

201 d

1

AL=_5L

4

i=i,n

(36)

2n,co

The partial waves reflected at every interface interfere under this condition in a way, that high reflectivity occurs over a certain intervall around co. The spectral width of this high reflectivity band increases with increasing difference of the n; and the reflectivity converges to unity with increasing number of pairs of layers, provided, the layers are lossless. If a cavity is placed between two of these dielectric mirrors, with a thickness given by an integer number of XJ2 or X

CTt

r d c a v l t y =m^ =m m^—=

2

(37)

n co

one obtains a Fabry-Perot resonator with a spectrally very narrow dip and peak in the reflection- and transmission-spectra respectively. These things are known since many years and recipies can be found in textbooks like 59. Now we approach the problem from another point of view, assuming that the reader is familiar with the properties of an electron in a periodic potential as outlined e.g. in 5, 17

We write down Schrodinger's equation and the wave equation of light deduced from Maxwell's equations in linear approximation here for the electricfieldstrength E - ^ A < K r ) + V(r)cKr) = E o.a

i

.

O.S

,-v'

s«

y

/

0.1

0.7

02

0.3

0.4

0.5

06

0.7

C0a/2rrc

COa / 2nc

energy

Fig. 24: The calculated photonic band structure of a close packed fee lattices of air spheres in Si (a), the resulting density of states (b) and the density of states for a close packed fee lattice of Si0 2 spheres in air corresponding to natural opal (c). The quantities a and c stand for the lattice constant and the vacuum speed of light, respectively. From 61.

206

Another, application oriented use would be to influence the bandstructure by external fields e.g. by cladding the air spheres with a liquid crystal, which changes its refractive index under the influence of an external electric field, allowing to manipulate the bandstructure so ' 61 and thus the refelction, transmission or luminescence spectra. Similar phenomena can be predicted with ferro electric photonic structures. In Fig. 25 we present another system of a two-dimenisonal photonic structure 60' 62 63 ' namely a hexagonal array of tubes in silicon. This structure has a true twodimensional bandgap around coa/27tc « 0.4 which is narrower for the electric field polarized parallel to tubes than for orthogonal polarization. A chain of defects, here missing holes, acts for frequencies in the gap as a wave guide. The waveguide structure in Fig. 25a has much narrower curvatures than classical waveguides or fibers can have. A realization of a beam splitter is shown in Fig. 25b and of a resonator in Fig. 25c. The central defect „localizes" the lightfield,which is contacted by the two waveguides.

Fig. 25: A hexagonal periodic array of holes in silicon with a wave-guide (a) a beam splitter (b) and a resonator (c). From 62. In a similar way a ridge waveguide of Si on SiC>2 has been converted in a FabryPerot resonator by a periodic arrangement of holes, which act as Bragg mirrors and a central part as cavity60'62'6i. We present here as a last example the building of a one-dimensional photonic structure from photonic atoms64. The concept of a photonic atoms is explained with Fig. 26. A quantum well is grown in a cavity consisting of two stacks of dielectric Bragg mirrors as explained already in section 3.3. The luminescence is shown in the lowest trace of Fig. 26b and we see the by now well known effect, that the emission maximum is shifted away from the exction resonance of the quantum well without cavity. In a next step a three dimensional resonator is formed by etching a small, square mesa of widht w. See Fig. 26a. The natural reflectivity of the side walls resulting from refractive indices n around three is sufficient to form a resonator with three-dimensional confinement for light as seen from the increasing spacing of the emission peaks with decreasing w in Fig. 26b.

207

I i

i

i

i

i

t

i

i

i

[

PL = 200 W/cm2

T = 2K

i

i

i

i' i

i

i

i

excjton

J

1.412

1.400

1.410

1.420

energy (eV) b

1.400

1

2

3

4

5

lateral size (/im)

Fig. 26: A photonic atom consisting of a square mesa prepared from a structure containing a quantum. well in a cavity with stacks of Bragg-mirrors on both sides (a) the luminescence of the mesa with w => oo and for finite w (b) and the coincidence of the measured and calculated emission peaks as a function of w (c). From6".

208

The spectral position of the emitted quanta is given by

to=^(k2+k*+kJ)1/J

(40a)

where ko is the wavevector denned by the Bragg mirror cavity while kK and ky are given by k i =(m i +l)—, w

mi =0,1,2,...; i = x,y

(40b)

Fig. 26c shows the excellent agreement between the experimental data and the results of Eqs. (40a, b) for various modes64. The next step is to couple two of the photonic atoms to a photonic molecule by a bridge between them shown in Fig. 27a ". The two lowest modes have even and odd parity in close analogy to the binding and antibinding states in a H2 molecule. Increasing coupling realized in Fig. 27c by an increase of the width of the bridge at a constant length of lum leads to a splitting of the otherwise degenerate levels without and with one node line in each dot. The natural extension is of course to couple more and more photonic atoms in a linear array. Actually one observes a splitting into more and more discrete levels as expected for coupled harmonic oscillators66. These levels develop with increasing number of coupled dots N > 10 into a one-dimensional bandstructure with small gaps at the borders of the Brillouin zones 66 while an unmodulateed photonic wire shows a smooth dispersion relation 61. More information on photonic structures and their possible applications can be found e.g. in the reviews60,68 and references given therein. 5.

Conclusion and Outlook

The concept of polaritons as the quanta of light in matter has been developped for three-dimensional ordered and disordered matter, for structures of reduced dimensionality like surfaces, quantum wells or cavities and for the spectral range from the IR through the visible to the soft y-ray regime. A recent „offspring" of thisfieldare structures with an artificially created periodic modulation of the refractive index in one, two or three dimensions, so-called photonic or polaritonic crystals. These structures bring phenomena, which are usually only known from X-ray diffraction 69 at natural crystals, into the cm-, um-wave or IR and visible range of the electromagnetic spectrum and make these phenomena much more pronounced because larger variations of the refractive index can be realized in the latter spectral ranges. Exciting new experimental findings, theoretical insights and applications can be expected from thisfieldof research in the future.

209

0++

0- +

1++

1- +

1- +

1MT

.

1.4M

IMS

' 1 channel length: 1pm 1 " " " ^ a

\M*

• *

*

m -» .

-«~<J~-»

a

• o-

1.441

channel width [ym|

Fig. 27: A photonic „molecule" made by coupling two „atoms" with a bridge (a) calculations of the field distribution of the lowest six coupled states (b) and their observation as a function of the bridge width (c). From 65 .

210 Acknowledgements: Stimulating discussions with my academic teachers, with many colleagues and with my coworkers about the topics treated here are gratefully acknowledged. Without trying to be complete I should like to mention Profs. Drs. K. Hummer, M. Wegener and R. v.Baltz (Karslruhe), Prof. Dr. H. Stolz (Rostock), Prof. Dr. E. Mollwo (f) and Prof. Dr. R. Helbig (Erlangen), Prof. Dr. A. Forchel (Wurzburg) or Prof. Dr. D. Frohlich (Dortmund). I should likte to express my special thanks to Prof. Dr. A. Forchel (Wurzburg), Dr. U. van Biirck (Miinchen) and last but not least Dr. K. Busch (Karlsruhe) for making available to me their excellent and valuable results prior to publication. References 1. 2. 3. 4. 5. 6.

7.

8. 9. 10. 11. 12. 13. 14. 15.

Energy Transfer Processes in Condensed Matter, in NATO ASI Series B 114, ed. B. Di Bartolo (1984). C. Klingshirn, in Ref. 1, p. 285. J.J. Hopfield, Phys. Rev. 112 (1958) 1555. J.J. Hopfield and D.G. Thomas, Phys. Rev. 132 (1963) 563. E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. S. John, Phys. Rev. Lett. 58 (1987) 2486. C. Klingshirn, in Semiconductor Optics, 2nd printing, (Springer, Berlin, 1997) F. Comas, C. Trallero-Giner and M. Cardona, Phys. Rev. B 56 (1997) 4115. A. J. Sievers, NATO ASI Series B 356 (1997) 227. U. Woggon, in Springer Tracts in Modern Physics 136 (Springer, Berlin, 1997). H. Stolz, in Springer Tracts in Modern Physics 130 (Springer, Berlin, 1994). H. Wang, J. Shah, T.C. Damen and L.N. Pfeiffer, Phys. Rev. Lett. 74 (1995) 3066. S. Haacke, R.A. Taylor, R. Zimmermann, I. Bar-Joseph and B. Devaud, ibid 78 (1997) 2228. D. Birkedal and J. Shah, ibid. 81 (1998) 2373. W. Langbein and J.M. Hvam, ibid. 82 (1999) 3206. J.M. Hvam, contribution to this volume. R. v. Baltz, contribution to this volume. R. v.Baltz, NATO ASI series B 372 (1998) 323. M. Wegener, contribution to this volume. D. Hulin, contribution to this volume. J.M. Hvam, contribution to this volume. C. Klingshirn, NATO ASI series B 372 (1998) 143. A. Jolk, C. Klingshirn and R. v.Baltz, NATO ASI series B 372 (1998) 397. R. v. Baltz and C. Klingshirn, NATO ASI series B 372 (1998) 381.

211 16.

H. Haug and S.W. Koch, in Quantum Theory of the Optical and Electronic Properties of Semiconductors, 2nd ed, (World Scientific, Singapore, 1993). N. Peyghambarian, S.W. Koch and A. Mysyrowicz, in Introduction to Semicondcutor Optics (Prentice Hall, Engle wood Cliffs, New Jersey, USA, 1993). 17. Ch. Kittel, in Introduction to Solid State Physics, 6th ed., (J. Wiley and Sons, New York, 1986). O. Madelung, Introduction to Solid State Theory, in Springer Series in Solid State Sciences 2, 2nd, printing (Springer, Berlin, 1981). H. Ibach and H. Liith, in Festkorperphysik, 4th ed. (Springer, Berlin, 1995). 18. J. Shah, in Springer Series in Solid State Sciences 115, (Springer, Berlin, 1996). H. Haug and A.-P. Jauho, ibid. 123 (1996). 19. G Blattner, G. Kurtze, G. Schmieder and C. Klingshirn, Phys. Rev. B 25 (1982) 7413. 20. J. Voigt, F. Spiegelberg and M. Senoner, phys. stat. sol. b 91 (1979) 189. 21. I D . Weber, W. Petri, U. Woggon, C. Klingshirn, S. Shevel and E.O. Gobel, Proc. 23. Intern. Conf. Phys. Semicond. Berlin (1996), eds. M. Scheffler and R. Zimmermann, (World Scientific, Singapore, 1996) 713. D. Weber, W. Petri, U. Woggon, C. Klingshirn, S. Shevel and E.O. Gobel, Phys. Rev. B 55 (1997) 12848. 22. B. Honerlage, R. Levy, J.B. Grun, C. Klingshirn, K. Bohnert, Physics Reports 124, (1985) 161. 23. M.V. Lebedev, M.I. Strashnikova, V.B. Timofeev and V.V., Chernii, JETP Lett. 39 (1984) 366. 24. Y. Masumoto, Y. Unuma, Y. Tanaka and S. Shionoya, J. Phys. Soc. Japan 47 (1979) 1844. 25. Ch. Uihlein, D. Frohlich and R. Kenklies, Phys. Rev. B 23 (1981) 2731. 26. D. Frohlich, A. Kulik, B. Uebbing, A. Mysyrowicz, V. Langer, H. Stolz and W. von der Osten, Phys. Rev. Lett. 67 (1991) 2343. H. Stolz, phys. stat. sol. b 173 (1992) 99. D. Fohlich, A. Kulik, B. Uebbing, V. Langer, H. Stolz and W. von der Osten, ibid, p. 31. 27. M. Jutte, H. Stolz and W. von der Osten, phys. stat. sol. b 188 (1995) 327 and JOSA B 13 (1996) 1205. S. Niisse, P. Haring Bolivar, H. Kurz,. F. Levy, A. Chevy and O. Lang, Phys. Rev. B 55 (1997) 4620. 28. A. Reznitsky, S. Permogorov, T. Breitkopf, M. Grun, M. Hetterich, C. Klingshirn, V. Lyssenko, W. Langbein, J.M. Hvam, Phys. Rev. B 59 (1998) 12947. 29. R. Ell, Diploma Thesis, Kalsruhe, to be published (1999). 30. P. Schillak and I. Balslev, Phys. Rev. B 48 (1993) 9426. 31. T. Breitkopf, A. Osterlow, V. Lysenko, A. Reznitsky and C. Klingshirn, Proc. 23. Intern. Conf. Phys. Semicond. Berlin (1996), eds. M. Scheffler and R. Zimmermann (World Scientific, Singapore, 1996) 721.

212 32. 33.

Ch. Henry and J.J. Hopfield, Phys. Rev. Lett. 15 (1965) 964. G Borstel, H.J. Falge and A. Otto, Springer Tracts in Modern Physics 74 (1974) 107. 34. H.J. Bakker, S. Hunsche and H. Kurz, Review of Modern Physics 70 (1998) 523. 35. K.A. Nelson, contribution to this volume. 36. G.W. Spitzer and H.Y. Fan, Phys. Rev. 106 (1957) 882. 37. M. Goppert, F. Gehbauer, M. Hetterich, J. Miinzel, D. Queck and C. Klingshirn, Intern. Conf. Lumin. Prag (1996), J. of Luminesc. 12-1 A, (1997)430-431. C. Klingshirn, phys. stat. sol. (b) 202 (1997) 857. 38. M.R.F. Jensen, S.A. Feiven, T.J. Parker and R E . Camley, Phys. Rev. B 55 (1997) 2745. 39. Muller-Pouilets Lehrbuch der Physik und Meteorologie, 2. Band, 3. Buch (1909)227. 40a. I E . Rothenberg, D. Grischkovsky, A C . Balant, Phys. Rev. Lett. 53 (1984) 552. 40b. L.V. Hau, S.E. Harris, Z. Dutton and CH. Behroozi, Nature 397 (1999) 594. 41. R.W. Pohl, in Optik undAtomphysik, 13th ed. (Springer, Berlin, 1976). 42. U. van Biirck, Hyperfine Interaction, in press (1999). Yu. Y. Shvyd'ko and U. van Biirck, ibid. U. van Biirck, in Nuclear Resonant Scattering of Synchrotron Radiation, eds. E. Gerdau and H. de Waard (Baltzer, Oxford, 1999), in press. 43. Landolt-Bornstein, New Series, Group III, Vol. 34C, in press (00) 44. D.S. Kim, J. Shah, D.A.B. Miller, T.C. Damen, A. Vinattieri, W. Schafer and L.N. Pfeiffer, Phys. Rev. B 50 (1994) 18240. T. Stroucken, A. Knorr, A. Anthony, A. Schulze, P. Thomas, S.W. Koch, M. Koch, S T . Cundiff, J. Feldmann and E.O. Gobel, Phys. Rev. Lett. 74 (1995)2391. D. Weber, J. Feldmann, E.O. Gobel, D.S. Citrin and K. Kohler, // Nouvo Cimento 17D (1995) 1769. 45. L.C. Andreani and F. Bassani, Phys. Rev. B 41 (1990) 7536. F. Tassone, F. Bassani and L.C. Andreani, ibid. 45 (1992) 6023. D.S Citrin, ibid. 50 (1994) 5497. 46. S. Jorda, U. Rossler and D. Broido, Phys. Rev. B 48 (1993) 1669. 47. D. Oberhauser, K.-H. Pantke, W. Langbein, V.G. Lyssenko, H. Kalt, J.M. Hvam, G. Weimann und C. Klingshirn,/?^. stat. sol. b 173, (1992) 53. 48. J. Feldmann, G. Peter, E.O. Gobel, P. Dawson, K. Moore, C. Foxon and R.J. Elliot, Phys. Rev. Lett. 59 (1987) 2337. L.C. Andreani, Solid State Commun. 11 (1991) 641. J. Martinez-Pastor, V. Vinattieri, L. Carraresi, M. Colocci, Ph. Roussignol and G. Weimann, Phys. Rev. B 47 (1993) 10456. 49. K. Ogawa, T. Katsuyama and H. Nakamura, Phys. Rev. Lett. 64 (1990) 796.

213 50.

51. 52. 53. 54. 55. 56. 57a. 57b. 57c. 57d. 57e. 58.

59. 60. 61. 62. 63.

64. 65.

66.

67.

F. Tassone and F. Bassani, IlNuovo Cimento 14D (1992) 1241. D.S. Citrin, Phys. Rev. B 48 (1993) 2535. S. Jorda, Solid State Commun. 87 (1993) 439. M. Kohl, D. Heitmann, P. Grambow and K. Ploog, Phys. Rev. B 37 (1988) 10927. G. Benedek, NATO ASI Series B 88 (1983) 523 and ibid. B 372 (1998) 295. Surface Polaritons, eds. V.M. Agranovich, D.L. Mills, Mod. in Prob. Condens. Mat. Sci. Vol. 1 (North Holland, Amsterdam, 1982). N. Marshall and B. Fischer, Phys. Rev. Lett. 28 (1972) 811. N. Marshall, B. Fischer and H.-J. Queisser, Phys. Rev. Lett. 27 (1971) 95. J. Lagois and B. Fischer, Festkorperprobleme /Adv. Solid State Phys. 18 (1978) 197. C. Weisbuch, M. Nishioka, A. Ishikawa and Y. Arakawa, Phys. Rev. Lett. 69(1992)3314. R. Houdre, C. Weisbuch, R.P. Stanley, U. Oesterle, P. Pellandini and M. Ilegems, Phys. Rev. Lett. 73 (1994) 2043. V. Savona, J. Cryst. Growth 184/185 (1998) 737. F. Quochi, G.R. Hayes, R. Andre, G. Bongiovanni, A. Mura, J.L. Staehli and Le Si Dang, ibid. p. 754. P. Kelkar, V. Kozlov, A.V. Nurmikko, C.-C. Chu, D C . Grillo, J. Han, C.G. Hua, R.L. Gunshor, Phys. Rev. B 52 (1995) R5491. P. Kelkar, A.V. Nurmikko, C.-C. Chu, J. Han, W.-L. Chen and R.L. Gunshor, J. Cryst. Growth 184/185 (1998) 745. F. Jahnke, M. Ruopp, M. Kira and S.W. Koch, Festkorperprobleme /Adv. Solid State Phys. 37 (1998) 191. H.M. Gibbs et al., ibid. p. 227. A. Thelen, in Design of optical interference coatings (Mc Graw-Hill, New York, 1989). A. Birner, K. Busch and F. Muller, Physikal. Blatter 55, Heft 4 (1999) 27. S. John and K. Busch, IEEEJQE, in press (1999). K. Busch and S. John, Phys. Rev. E 58 (1998) 3896. J.S. Faresi, P R . Villeneuve, J. Ferrera, E.R. Thoen, G. Steinmeyer, S. Fan, J.D. Joannopoulos, L.C. Kimerling, H.I. Smeth and E.P. Ippen, Nature 390 (1997) 143. J.P. Reithmaier, M. Rohner, H. Zull, F. Schafer, A. Forchel, P.A. Knipp, T.L. Reinecke, Phys. Rev. Lett. 78 (1997) 378. M. Bayer, T. Gutbrod, J.P. Reithmaier, A. Forchel, T.L. Reinecke, P.A. Knipp, A.A. Dremin and V.D. Kulakovskii, Phys. Rev. Lett. 81 (1998) 2582. A. Forchel, private communication A.I. Tartakovskii, V.D. Kulakovskii, P S . Dorozhkin, A. Forchel and J.P. Reithmaier, Phys. Rev. B 59 (1999) 10251. A. Kuther, M. Bayer, T. Gutbrod, A. Forchel, P.A. Knipp, T.L. Reinecke and R. Werner, Phys. Rev. B 58 (1998) 15744.

214

68.

69.

in Photonic Band Gap Materials, ed. CM. Soukoulis (Kluver Academic Publishers, Dordrecht, 1996). in Phtonic Crystals, eds. J.D. Joannopoulos, R.D. Meade and J.N. Winn (Princeton Academic Press, New Jersey, 1995). M. v.Laue, in Die Interferenz von Rontgenstrahlen (Akad. VerlagsGes. Leipzig, 1923).

NON-MARKOVIAN SCATTERING PROCESSES IN SEMICONDUCTORS

MARTIN WEGENER Institut fur Angewandte Physik, Universitat Karlsruhe, Kaiserstrafie 12 D-76128 Karlsruhe, Germany

ABSTRACT Often we think about scattering processes in solids in terms of classical point-like particles that bounce against each other in instantaneous collisions. In particular the only time scale that appears in the corresponding mathematical description, e.g. in the Boltzmann-equation or in Fermi's golden rule, is the mean free time between collisions. This picture is obviously in contradiction with the quantum mechanical, i.e. the wave-like nature of the collective excitations, which obey the time-dependent Schrodinger equation. Quantum mechanics tells us that any interaction between collective excitations is nothing but an interference phenomenon, including obviously the scattering processes that are responsible for decoherence and 'dissipation'. Intuitively, we can think about the dynamics of such many-body systems in terms of a finite duration of scattering processes which can be described by non-Markovian relaxation. Here we give an introduction into this growing field of quantum kinetics in semiconductors. 1.

Introduction

In the first semester of any physics course we learn that the energy transfer from a harmonic oscillator to its surrounding can be well described by a phenomenological Stokes damping parameter 73. Stokes damping means that the corresponding force acting on the oscillator at time t is proportional to its velocity at the same time t the dynamics is local in time. The resulting damped harmonic oscillator equation for the displacement P is one of the paradigmas of physics: P(t) + 7sP(t) + n2P(t)

= 0.

(1)

fi is the eigenfrequency of the undamped oscillator. An example of such a system is shown in Pig. 1. While the phenomenological Stokes damping can properly describe many aspects of the damped oscillator, it completely fails in other situations. Let us analyze the case where the oscillator motion is started by an impulse and then is stopped at the origin, P = 0, for a short but finite time, thus P = 0. For these two initial conditions the second order differential equation, Eq. 1, determines that P = 0 from thereon, which is in sharp contradiction to the experimental observation. It is clear that the water

215

216

Figure 1: Scheme of a simple mechanical system showing non-Markovian relaxation. in the tank, which is still in motion, acts on the oscillator and sets it in motion as well. If we wish to write down an appropriate equation of motion for the oscillator only, a more general form has to be used, j(t - t') P(t') dt1 + Q2P(t) = 0 .

Pit) + f

(2)

J — oo

This equation is non-Markovian, it is nonlocal in time and the memory function •y(t - £') determines how far the history (causality => t' < t) of the system influences its present state. For j(t - t') = 2^sS(t - 1 ' ) , i.e. immediate loss of the memory, Eq. 1 is recovered. For the simple example 7(t - H) oc e- ( '-*' )/T -'

(3)

we have the characteristic memory time constant r m for the decay of the memory. In the limit rm —> 0, the exponential decay of P is recovered. On the other hand, if we had written down Newton's law for the oscillator (without phenomenological damping) coupled to the Navier-Stokes equation for the fluid in the tank, we clearly would have obtained an equation of motion which is local in time. This is the same in quantum mechanics. Consider that we prepare a quantum mechanical system, e.g. a semiconductor, in a known state at time t = 0. It is clear that from thereon its wavefunction follows the time-dependent Schrodinger equation. Thus the evolution of its wavefunction at time t is completely determined by the state of the crystal at the same time t - the dynamics is again local in time. The semiconductor, however, is such a complicated many-body system that its Schrodinger equation is not solvable, and it is necessary to restrict oneself to treat only certain degrees of freedom of the problem, for example the electrons in a conduction and a valence band. With this approach, we have to consider that this subsystem interacts with other degrees of freedom, e.g. vibrations of the nuclei or other electrons, which are modeled as a thermal bath. Then the equation of motion of the subsystem becomes nonlocal in time, i.e. the evolution at time t not only depends on the state of the system at time t but also on its history. This is the regime of quantum kinetics 1 . 2> 3 . While the classical oscillator model suggests that memory effects arise due to nonequilibrium effects within the bath, we will see later that this is not necessary in quantum mechanics.

217

Figure 2: Scheme of an optical excitation process in which a photon with energy hCl takes an electron from the valence to the conduction band. This leads to an occupation / at state k in the conduction band. Before we come back to non-Markovian relaxation in solids, we first have to discuss briefly the Markovian relaxation, i.e. the Stokes-like energy transfer from the system to a bath in a solid. Consider an electron in the state with wavevector k in the conduction band of a two-band model semiconductor with parabolic energy dispersion depicted in Fig. 2. As this electron is clearly not in the lowest energy state, it eventually relaxes down to the bottom of the conduction band parabola. For the dynamics of the occupation function / at state k (the probability that state k is occupied, a number between 0 and 1 for fermions), it is often argued along the lines of the radioactive decay: The change of / , df, is proportional to the occupation itself, proportional to the discussed time intervall dt and proportional to some specific energy relaxation rate 71 and we obtain / + 7i / = 0.

(4)

As a result, the occupation / decays exponentially with rate 71 or equivalently with time constant 7\ = 7-f1. If we actually want to prepare such a nonequilibrium initial condition, we can employ optical excitation with a short laser pulse shown in Fig. 2. The optical transition at frequency fi means that we take an electron from the fully occupied valence band and transfer it to the conduction band. Because of the small photon momentum (large speed of light) these transitions look almost vertical in the Brillouin zone. It is intuitively clear (and will also be shown later) that the corresponding polarization follows a harmonic oscillator equation with eigenfrequency fi. The phenomenological damping rate of this coherent oscillation is called the phase relaxation rate 72 or decoherence rate. T2 = 72-1 is the dephasing time. The free oscillation P + Q2P = 0. (5) can be rewritten to p + iQp = 0 (6) where P is simply the real part of the complex quantity p which we want to call the optical transition amplitude.

218 Back to non-Markovian relaxation. Actually, how valuable is the concept of such decay rates ji and 72 ? Later we will discuss quantum mechanical models for the interaction of crystal electrons with lattice vibrations - a problem which is far from being easily solved. To get a first feeling at this point already, we consider a simple physical approximation: the fluctuation model. 1.1.

The Fluctuation

Model

Lattice vibration means that the local lattice constant fluctuates as a function of time. As a result, the local band structure also fluctuates, in other words: The eigenfrequency fl(£) becomes time-dependent. We can write it in the form

n(i) = n0 + sn(t)

(7)

and choose Q0 such that the average fluctuations SCl are zero, i.e. (S^t(t)} = 0. At this point, the average (...) is a purely classical one and can be interpreted as an ensemble average over many individual experiments. Let us see what the influence of these fluctuations is on the transition amplitude p. Formal integration of Eq. 6 with Eq. 7 immediately leads us to p{t) = p(0) (exp ( - i j * [Oo + Sn(t')} dt'^j) = p(0) e- ifio( fl - i f (5Q(t')) dt' --[*[* L

Jo

»

'

(Sn(t')Sn(t11))

2 Jo JO •

„

dt'dt" + ... 1 . '

(8)

J

= 0 In the second line we have employed a Taylor expansion of the exponential function. In this line we face a correlation function, (5£l(t')6Cl(t")), for the first time. If we consider these frequency fluctuations as a random process, the correlation function is similar to a combined probability in an experiment where we throw two dice. Let us consider three cases i)-iii). i) If the two events of throwing dice are independent, the probability of throwing, let us say first a six and then a one, V(6,1), is simply the product of the probabilities throwing a six, V(6), times the probability of throwing a one, V(l), and thus given by ^(6,1) = •p(6) , P(l) = i • | . Thus for actually different fluctuation events, i.e. t' ^ t", we have (