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[email protected] http://sauvignon.mit.edu/323/ [2002-05-11 22:21:15]
Description
3.23 Electrical, Optical, Magnetic Properties of Materials (previously Physics and Chemistry of Materials) • All EOM properties derived from electrons in materials – Electronic: response of ‘free’ electrons in material – Optical: response of bound electrons and effect of electron transitions in material – Magnetic: response of electron spin and electron motion in material • Determining energy of electrons from electric and magnetic fields will determine electron response and hence properties • Real materials: effect of imperfections on these materials
©1999 E.A. Fitzgerald
323a1
Origin of Conduction Range of Resistivity
Why?
©1999 E.A. Fitzgerald
323a2
Response of material to applied potential I
I
V
V=f(I) Rectification, Non-linear, Non-Ohmic
R
Linear, Ohmic
V eV=IR
Metals show Ohmic behavior Microscopic origin?
©1999 E.A. Fitzgerald
323a3
Remove geometry of material
V I W L V=IR=IR/L R=L/(σA)
J=σE
In general,
! ~! J = σE
Isotropic material Anisotropic material
All material info
In cubic material,
©1999 E.A. Fitzgerald
J x σ xx J y = σ xy J z σ xz
E
J
E
J
σ xy σ xz E x σ yy σ yz E y σ yz σ zz E z 323a4
Microscopic Origin: Can we predict Conductivity of Metals? • Drude model: Sea of electrons – all electrons are bound to ion atom cores except valence electrons – ignore cores – electron gas J=σE=-nev, by definition of flux through a cross-section n=number of electrons per volume v=velocity of the carriers due to electric field--> drift velocity Therefore, σ=-nev/E and we define v=-µE µ is mobility, since the electric field creates a force on the electron F=-eE
σ = neµ
©1999 E.A. Fitzgerald
323a5
Current Density and Electron Density
J n=# electrons/volume A dx
©1999 E.A. Fitzgerald
J
Volume=A*dx Charge=dQ=-enAdx dx=vdt J=(1/A)(dQ/dt)=-nev define µ: v=-µE therefore J=-nev=σE if σ=neµ 323a6
Does this microscopic picture of metals give us Ohm’s Law? E F=-eE F=ma m(dv/dt)=-eE v=-(eE/m)t J=σE=-nev=ne2Et/m σ=ne2t/m
v,J,σ,I t E
Constant E gives ever-increasing J
t
No, Ohm’s law can not be only from electric force on electron! ©1999 E.A. Fitzgerald
323a7
Hydrodynamic representation of e- motion
p=momentum=mv
dp( t ) p( t ) =− + F1 ( t ) + F2 ( t ) +... dt τ Response (ma)
Drag
Driving Force
Restoring Force...
dp(t ) p( t ) ≈− − eE dt τ Add a drag term, i.e. the electrons have many collisions during drift 1/τ represents a ‘viscosity’ in mechanical terms
©1999 E.A. Fitzgerald
323a8
In steady state,
dp(t ) =0 dt −t τ
p(t ) = p∞ (1 − e ) p∞ = − eEτ
p -eEτ
τ
t
If the environment has a lot of collisions, vavg=-eEτ/m mvavg=-eEτ
Now we have Ohm’s law ©1999 E.A. Fitzgerald
ne 2τ σ= m
µ=
eτ m 323a9
Predicting conductivity using Drude ntheory from the periodic table (# valence e- and the crystal structure) ntheory=AVZρm/A, where AV is 6.023x1023 atoms/mole ρm is the density Z is the number of electrons per atom A is the atomic weight For metals, ntheory~1022 cm-3 If we assume that this is correct, we can extract τ
©1999 E.A. Fitzgerald
323a10
Extracting Typical τ for Metals
•
τ~10-14 sec for metals in Drude model
©1999 E.A. Fitzgerald
323a11
Thermal Velocity • So far we have discussed drift velocity vD and scattering time τ related to the applied electric field x • Thermal velocity vth is much greater than vD
x
x
L=(vD+vth)t
1 2 3 mvth = kT 2 2
vth =
3kT m
Thermal velocity is much greater than drift velocity
©1999 E.A. Fitzgerald
323a12
Example: Conductivity Engineering • Objective: increase strength of Cu but keep conductivity high ne 2τ σ= m " = vτ
µ=
eτ m
Scattering length connects scattering time to microstructure
Dislocation (edge)
l decreases, τ decreases, σ decreases ©1999 E.A. Fitzgerald
e-
323a13
Example: Conductivity Engineering • Can increase strength with second phase particles • As long as distance between second phase< l, conductivity marginally effected
L
L
L+S
α+L
β+L
α
S
α+β
Cu
Sn
X
Cu β
microstructure
S
Material not strengthened, conductivity decreases ©1999 E.A. Fitzgerald
β
L α
dislocation L>l
Dislocation motion inhibited by second phase; material strengthened; conductivity about the same 323a14
Example: Conductivity Engineering • Scaling of Si CMOS includes conductivity engineering • One example: as devices shrink… – vertical field increases – τ decreases due to increased scattering at SiO2/Si interface and increased doping in channel need for electrostatic integrity (ionized impurity scattering) – Scaling continues ‘properly’ if device shrinks fast enough to negate decrease in scattering time G S
SiO2 D
Evert Ionized impurities (dopants)
©1999 E.A. Fitzgerald
323a15
Determining n and µ: The Hall Effect Vx, Ex +++++++++++
---------
! ! ! ! F = qE + qv × B
Bz
Fy = −evD Bz
Ey
I, Jx
In steady state,
EY = vD BZ = E H , the Hall Field
Fy = −eE y
Since vD=-Jx/en, 1 E H = − J x BZ = RH J X BZ ne RH = −
©1999 E.A. Fitzgerald
1 ne
σ = neµ 323a16
A&M treatment using full equations of motion ! ! ! ! dp p ! p ! p ! xB (CGS units) = − + F = − − e E + dt mc τ τ
Separate into x & y coupled equations: ! ! ˆ ˆ p = p xi + p y j , B = Bz kˆ dp x p = − x − eE x − ω c p y τ dt dp y py 0= =− − eE y + ω c p x τ dt eB ωc = (CGS) mc
(pxxB is in +y direction, pyxB is in -x direction)
0=
px ω c p y − eτ e py ωc py Ey = − + eτ e ne 2τ σo = m Ex = −
Since ©1999 E.A. Fitzgerald
σ o E x = ω cτJ y + J x σ o E y = −ω cτJ x + J y In steady state,
Ey = −
σ o Ex = J x Jy = 0
1 ω cτ Jx = − J x B z = RH J x B z nec σo
323a17
Experimental Hall Results on Metals
• Valence=1 metals look like free-electron Drude metals • Valence=2 and 3, magnitude and sign suggest problems
©1999 E.A. Fitzgerald
323a18
Response of free e- to AC Electric Fields • Microscopic picture
EZ = EO e − iωt
e-
B=0 in conductor, dp(t ) p (t ) =− − eE0 e −iωt dt τ
try
! ! ! ! and F ( E ) >> F ( B )
p (t ) = p0 e − iωt
p0 − iωp 0 = − − eE 0 τ p0 =
eE0 iω −
1 τ
ω>>1/τ, p out of phase with E eE0 p0 = iω
ω → ∞, p → 0
ω1/τ)
p
p (ω>1, ε(ω) goes to 1 For an excellent conductor (σ0 large), ignore 1, look at case for ωτ1
ε (ω ) = 1 −
ωωp, ε is positive, k=kr, wave propagates
R
ωp
©1999 E.A. Fitzgerald
ω
323a27
Success and Failure of Free e- Picture •
•
Success – Metal conductivity – Hall effect valence=1 – Skin Depth – Wiedmann-Franz law Examples of Failure – Insulators, Semiconductors – Hall effect valence>1 – Thermoelectric effect – Colors of metals
K/σ=thermal conduct./electrical conduct.~CT 1 2 τ Κ = c v vtherm 3 3k b T 3 ∂E 2 cv = = nk b ; vtherm = m ∂T v 2 13 3 nk b2Tτ 3k bT Κ = nk b τ = 3 2 2 m m
ne 2τ σ = m 2
Therefore :
Luck: cvreal=cvclass/100; vreal2=vclass2*100 ©1999 E.A. Fitzgerald
Κ 3 kb = T σ 2 e
~C!
323a28
Wiedmann-Franz ‘Success’
Thermoelectric Effect Exposed Failure when cv and v2 are not both in property
E = Q∇T 3 − nk b cv nk = 2 =− b Thermopower Q is Q = − 3ne 3ne 2e
Thermopower is about 100 times too large! ©1999 E.A. Fitzgerald
323a29
Improvements? What are ion cores doing... • Scattering idea seems to work • any effect of crystal (periodic) lattice? • Diffraction – proves periodicity of lattice – proves electrons are waves – proves strong interaction between crystal and electrons (leads to band structures=semiconductors and insulators) – useful characterization technique • Course: bias toward crystalline materials: many applications: materials related to either end of spectrum (atomic/molecular or crystalline) extended
localized Point defects, atoms, molecules
Polymers, α Si
Bands; properties of many solids with or without extended defects
Diffraction is a useful characterization in all these materials ©1999 E.A. Fitzgerald
323a30
Diffraction • Incoming λ must be on the order of the lattice constant a or so (λ Energy Bands E
Implies a maximum in E even though k continues to increase
323e7 ©1999 E.A. Fitzgerald
k
k+G
k
Implications of Bloch Electron Waves Since E is invariant to k shifted by G (same as diffraction argument): ! 2 (k x ± mx Gx ) E= 2m
2
1-D E E(k)+E(Gy and/or Gz) E=
3-D 2 2 2 ! 2 [(k x + mx Gx ) + (k y + m y G y ) + (k z + mz Gz ) ] 2m
E(k) E(k+G)
E(k-G)
−2π/a -π/a
0 G
π/a
kx 2π/a
Note: key difference in 1-D to 3-D in graph on left is that we are looking along kx, but in 3-D the value can change due to Gy and/or Gz
323e8 ©1999 E.A. Fitzgerald
General Derivation of NFE • We know that the wavefunctions will look like plane waves • We know that lattice potential from the lattice ions have the periodicity of real space and G in k-space
Ψ (r ) = ∑ cq e iqr q
U (r ) = ∑ U G e iGr G
HΨ = E Ψ ! 2 2 iqr − !2 2 ∇ Ψ = ∑ q cq e 2m q 2m
iGr iqr UΨ = ∑ U G e ∑ cq e q G
turn the crank!
©1999 E.A. Fitzgerald
! 2q 2 − E cq + ∑ U G 'cq −G ' = 0 G' 2m
323e9
Proof of Bloch Theorem from General Derivation • For a given k, central equation shows that the coefficients cq that matter are those that are a multiple of G from k
Ψk = ∑ cq eiqr ≈ ∑ ck −G ei (k −G )r q
G
In our diffraction example, k=π/a, and cq and cq-G were 1 at the diffraction condition,
Ψ
k=
π a
=e
π i r a
+e
π −i r a
Continuing,
−iGr Ψk = e ∑ ck −G e = e ikr u (r ) G u (r ) = ∑ ck −G e −iGr Bloch waves are indeed the general solution ikr
G
323e10 ©1999 E.A. Fitzgerald
Central Equation near the Bragg Plane
E
•Evaluate at q=+ or -1/2 G •Assume since U(r) goes as 1/r, that only nearest two potentials are relevant
1/2 G
For q=1/2 G,
For q=-1/2 G,
E(q) G’=-G
G’=G ! 2 1 2 c 1 + U G c 1 = 0 G E − 2m 2 G − G 2 2
E(q-G)
! 2 1 2 − G − E c 1 + U − G c 1 = 0 2m 2 − G G 2 2
EG/2
0
π/a G
reciprocal lattice points
2π/a kx
E-G/2
(EG / 2 − E )c1/ 2G + U G c−1/ 2G = 0 U −G c1/ 2G + (E−G / 2 − E ) = 0 Also, U G = U −G 323e11
©1999 E.A. Fitzgerald
Central Equation near the Bragg Plane (EG / 2 − E ) UG
E UG
(E−G / 2 − E )
=0
(EG / 2 − E )(E−G / 2 − E ) − U G2 = 0 + EG / 2 − EG / 2 ) E (E 2 ± −G / 2 + E = −G / 2 U G 2 2 2
At the Bragg plane, EG/2=E-G/2
E = EG / 2 ± U = EG / 2 ± U G 2 G
Plugging back into one of the equations,
cG / 2 = ± c−G / 2
UG
EG/2
0
UG
π/a
ψ q = cq eiqr ; ψ G / 2 = cG / 2ei (G / 2) r ψ q −G = ±cq ei (q −G )r ; ψ −G / 2 = cG / 2 e −i (G / 2) r Ψ = ψ G / 2 ± ψ −G / 2 Ψa & Ψb
©1999 E.A. Fitzgerald
323e12
Central Equation just off the Bragg Plane E
∆q
1/2 G
E=
E(q)
E=
E(q-G) ∆E
0
π/a
2π/a 2
! 2G 2 ( ) ∆ q !2 ! 2G 2 1 2m 2 (∆q ) + ±U ± E= U 2m 16m 8 ! 2 (1 + ζ ) (∆q )2 + U o E= 2m ! 2G 2 ! 2G 2 ζ =± ; Uo = U + 16mU 16m ©1999 E.A. Fitzgerald
E=
Eq + Eq − G 2 Eq + Eq −G 2 Eq + Eq −G 2
±
(E
− E q −G )
2
q
+U 2
4
1 ∆E ± 1+ 4 U
2
∆E easier problem for S.E. •We will eventually run into the many-bodied problem --> many electrons, many atoms •We will make approximations to continue, and arrive at similar results to the NFE
Hψ atomic = Eatomicψ atomic HΨmolecule = Emolecule Ψmolecule HΨchain = Echain Ψchain
HΨsolid = (Evalence e −'s + Eions )Ψsolid ©1999 E.A. Fitzgerald
323e15
Review of H atom • •
S.E. in spherical co-ordinates Separation of variables; separation constants are basically quantum numbers; 3 for 3 dimensions Creates atom structure for periodic table
•
z r Atomic number (atomic charge)=Z
U=
e-
θ
+ φ
µ=
y
− Ze 2
− Ze 2 = 2 2 2 r x +y +z 1 1 1 + m M
Problem easier if we choose U(r)
Reduced mass keeps problem a single body problem
−! 2 ∇ ψ + U (r )ψ = Eψ 2m
x
Where the del is in spherical co-ordinates:
∂2 ∂ ∂ 1 ∂ 2 ∂ 1 1 ∇ = 2 + θ sin r + 2 2 r ∂r ∂r r sin θ ∂φ 2 r 2 sin θ ∂θ θ 2
323e16 ©1999 E.A. Fitzgerald
Review of H atom ψ = R(r )Θ(θ )Φ (φ ) Hψ = Eψ Do separation of variables; each variable gives a separation constant φ separation yields ml θ gives " r gives n
After solving, the energy E is a function of n − µZ 2 e 4 − 13.6eV E= = n2 (4πε o )2 2! 2 n 2
ml and " in Φ and Θ give Ψ the shape (i.e. orbital shape) The relationship between the separation constants (and therefore the quantum numbers are:)
n=1,2,3,… " =0,1,2,…,n-1 ml=- ", - "+1,…,0,…, "-1, (ms=+ or - 1/2)
0
U(r)
-13.6eV
323e17 ©1999 E.A. Fitzgerald
Relationship between Quantum Numbers
s
s
p
s
p
d
Origin of the periodic table 323e18 ©1999 E.A. Fitzgerald
Hydrogen Wavefunctions
323e19 ©1999 E.A. Fitzgerald
Physical Nature of Orbitals (Ψ2)
ψ 2 = ψ *ψ = P (r , l , ml ) Probability density
1s
Radial Probability Density = R*R4πr2dr Look at only probability of finding electron in a shell of thickness dr at r from the nucleus
Bohr radius
r
Ψ can be negative; cross over is 0 in Ψ2 2s
Compare to Bohr ∞
∞
r = ∫ rP(r , l , ml )dr = ∫ rR * R 4πr 2 dr 0
0
n 2 ao r= Z
rBohr =
2p
1 l (l + 1) 1 + 2 1 − n 2
2
3s 3p
2
n ao ! ; ao = 2 = 0.52 A µe Z
3d
323e20 ©1999 E.A. Fitzgerald
Physical Nature of Orbitals (Ψ2)
•
Θ and Φ functions give shape
l=3 (f shell)
323e21 ©1999 E.A. Fitzgerald
Multielectron Atoms • Recall previous wavefunctions were for single electron in different possible orbits • What about a real atom with mulitple electrons? – Screening, so E=-13.6Z2/n2 doesn’t work well always – empirically: Z=Zeff • Can solve exactly by putting other electrons in U through an electron density, ρ Pick a starting Ψ and ρ H Ψ=Ε Ψ
New cq, define new Ψ, ρ: check for minimum E ©1999 E.A. Fitzgerald
Determine cq, E 323e22
What about building up our solid: multiatoms? • Simplest case: H2+, hydrogen molecular ion (1e-, 2 protons) R
U molecule
1 1 1 = e − − + r1 r2 R 2
+
+ r1
r2
•ions repel •electrons attract ions together when in between ions (bond)
e-
•Born-Oppenheimer Approximation !protons fixed (R const.): get energy of electron !determine proton-proton energy separately and superimpose 1 1 U el = e − − r1 r2 2
Solve HΨmol=EΨmol
Emol
e2 = Eel (R ) + R
323e23 ©1999 E.A. Fitzgerald
Multiatoms:H2+ • Assume that the molecular wavefunction for the electron is a linear combination of the atomic wavefunctions (LCAO)
Ψ = c1φ1 + c2φ 2 1s : φ1 =
− r1 ao
1
πa
3 o
1
e ; φ2 =
πa
3 o
e
− r2 ao
HΨ = EΨ Ψ * HΨ = EΨ *Ψ E=
* Ψ ∫ HΨdV * Ψ ∫ ΨdV
=
Ψ* H Ψ Ψ* Ψ
ˆ (c φ + c φ )dV ( ) c c H φ + φ ∫ = ∫ (c φ + c φ )(c φ + c φ )dV * 1 1
* 1 1
* 2 2
* 2 2
1 1
1 1
2 2
2 2
323e24 ©1999 E.A. Fitzgerald
Multiatoms:H2+ We will get three kinds of integrals that we will abbreviate:
Coulomb Integrals
H11 = ∫ φ1 Hˆ φ1dV ; H 22 = ∫ φ 2 Hˆ φ 2 dV
Bond Integrals
H12 = ∫ φ1 Hˆ φ 2 dV
Overlap Integrals
S12 = ∫ φ1φ 2 dV
* * Assume φ1 = φ1 ; φ 2 = φ 2 (i.e. real functions)
and
* φ ∫ i φi dV = 1 (they are normalized)
c12 H11 + c22 H 22 + 2c1c2 H12 E= c12 + c22 + 2c1c2 S12
Apply variational method: pick Ψ’s to get the lowest E; minimize E with respect to c1 and c2 323e25
©1999 E.A. Fitzgerald
Multiatoms:H2+ ∂E = 0 = c1 (H11 − E ) + c2 (H12 − ES12 ) ∂c1 ∂E = 0 = c1 (H12 − ES12 ) + c2 (H 22 − E ) ∂c2 For identical atoms, H = H11 = H 22 ≡ atomic integrals c1c2 (H − E ) − c1c2 (H12 − ES12 ) = 0 2
EA = H +
2
EA
H12 − HS12 1 + S12
H − HS12 E B = H − 12 1 − S12
anitbonding
H
H EB
bonding
323e26 ©1999 E.A. Fitzgerald
Multiatoms:H2+ Use E’s in original equations:
c1 ± c2
ΨB = c1 (φ1 + φ 2 ) ΨA = c1 (φ1 − φ 2 )
Note: 2 atoms, 2 levels Bonding
Anti-Bonding
Ψ2
323e27 ©1999 E.A. Fitzgerald
Multiatoms:H2+ E
Bonding
E
Nuclear repulsion
Anti-Bonding Nuclear repulsion
2
e R
e2 R
Ro R K (R − Ro )2 2 F = − K (R − Ro )
Electron energy
E=
R
Electron energy
Atoms connected by ‘springs’ 323e28 ©1999 E.A. Fitzgerald
LCAO: Adding more atoms • Assume H2 solution similar to H2+ • Let’s use Li as the atom to build our solid Molecular orbital notation
Ψ1σ * = c(φ1 − φ 2 ) Ψ1σ = c(φ1 + φ 2 )
Representation of Ψ2 1σ*
anitbonding
+
Sign of wavefunction
1s
1s 1σ
bonding
+
323f1 ©1999 E.A. Fitzgerald
LCAO: Adding more atoms • Li: 1s22s1 • Look at Li2
+
2σ*
-
E 2s
2s
+
2σ
Li2
1σ* 1s
1σ
1s 1s are ‘buried’ inside above pictures and more atomic-like (experience less delocalization since they are not valence electrons)
323f2 ©1999 E.A. Fitzgerald
LCAO: Adding more atoms • Conservation of states • more nodes for higher energy molecular wavefunctions nodes
Ψ4 = c41φ1 − c42φ 2 + c43φ3 − c44φ 4
Li4
Ψ3 = c31φ1 − c32φ 2 − c33φ3 + c34φ 4 Ψ2 = c21φ1 + c22φ 2 − c23φ3 − c24φ 4 Ψ1 = c11φ1 + c12φ 2 + c13φ3 + c14φ 4
3 2 1 0
+ - + +
-
+ -
+ +
j=4 j=3
2s
j=2
2s
4 states, 4 valence e- in extended orbital
j=1
323f3 ©1999 E.A. Fitzgerald
LCAO: Adding more atoms • Generalize to N Li atoms N
Ψ j = ∑ c jiφi ; c ji = i =1
2 πji jπ sin ; E j = α + 2 β cos N +1 N +1 N +1 bond integral
atom index
orbital index
known as the Debye-Huckel model examples: N=2 α-β
N=large
N=4 j=4
α+β j=1
α-1.6β α-0.6β α+0.6β α+1.6β
α-2β EF α+2β 323f4
©1999 E.A. Fitzgerald
Generaliztion of Debye-Huckel for large N • Same as “tight-binding” model N a
Index i=Z/a
N+1~N=L/a
Z
L=Na
cj ≈
E
2 πjz sin N L
j ⇒ N, k ⇒
π L
πj 2 sin (kz ) ⇒c~ L N E = α + 2 β cos(ka )
k=
α
2β 2β
-π/a
π/a
k
vg =
1 ∂E 2 aβ =− sin ka ! ∂k ! 323f5
©1999 E.A. Fitzgerald
Bonding of other elements across periodic table • LiN : all spherical shells • across periodic table (1s2 2s2 2px), encounter p-orbitals • leads to π-bonds as well as σ-bonds
+
-
-
+
2pz
+
+
+
2py
+ More wiggles, higher energy
3σ
-
=
+
+
+
+-+-
=
+ -
=
+-
-
-
3σ*
πy
-
+ -
+
+ ©1999 E.A. Fitzgerald
-
+
2pz
-
2py
=
+-
+
πy* 323f6
Bonding of other elements across periodic table • Example: Oxygen (O2) •From Hund’s rules: put 1e- in each π orbital first (maximize spin) •O2 is paramagnetic
3σ∗ πx∗, πy∗ 2p
2p
πx, πy 3σ
•For O2 and F2, the 3σ is lower than the π-orbitals •For Li2 to N2, the πorbitals are actually lower than the 3σ
2σ* 2s
2s 2σ 1σ* 1s
1σ
1s
•O has eight electrons, thus O2 distributes 16 electrons over 10 molecular levels 323f7
©1999 E.A. Fitzgerald
Visualization of π orbitals Example: 4 Carbon chain, unsaturated (C4H6)
H C
H C H E
Nodes 3 2 1 0
E=α+2βcoska ©1999 E.A. Fitzgerald
+-+-+-+ + - + - + + - + + -
β may change from σ to π, but principle the same
σ bonds π bonds
H C H C H
π bonds extend over entire chain
Ψ4=c1φ1-c2φ2+c3φ3-c4φ4
α-1.6β
Ψ3=c1φ1-c2φ2-c3φ3+c4φ4
α-0.6β
Ψ2=c1φ1+c2φ2-c3φ3-c4φ4
α+0.6β
Ψ1=c1φ1+c2φ2+c3φ3+c4φ4
α+1.6β
π electrons: each carbon contributes 1 e- to the extended π
323f8
Bonding of atoms with different levels/orbitals • Example: LiF
5.4 eV
F px
σ*
18.6 eV
Li 2s
Li 2s
0
+
+
+
Zero overlap
F 2p
σ
+ +
+
-
=
-
Ψ = c1φ Li + c2φ F
σ bond
Ψσ * ⇒ c1 >> c2
Li
+F ionic
Ψσ ⇒ c2 >> c1 323f9
©1999 E.A. Fitzgerald
Hybridization • Two ways to look at it – solutions to S.E. that minimize E are Ψ’s that look like combinations of s and p orbitals – orbital ‘distorts’ to reach out and bond Examples
Hybrid
H Be H
Linear, sp
H
+
+
+
Be
H
- -
+ +
H
H sp hybridization
+
Tetrahedral, sp3
B
-
H H H C H H
Planar,
H
-
H B
sp2
+ 1s2s
+
+
-
+ 323f10 ©1999 E.A. Fitzgerald
Bonding and Hybridization • • •
Energy level spacing decreases as atoms are added Energy is lowered as bonding distance decreases All levels have E vs. R curves: as bonding distance decreases, ion core repulsion eventually increases E
E
Debye-Huckel
p s
R NFE picture, semiconductors
hybridization 323f11
©1999 E.A. Fitzgerald
Properties of non-free e• Electrons near the diffraction condition are not approximated as free • Their properties can still be viewed as free e- if an ‘effective mass’ m* is used !2k 2 E= 2mec*
! 2k 2 E= 2m
!2 m = 2 ∂ E ∂k 2 * ec
! 2k 2 E= 2mev* !2 m = 2 ∂ E ∂k 2 * ev
−π/a
Note: These electrons have negative mass!
π/a 323f12
©1999 E.A. Fitzgerald
1-D Crystal Metals and Insulators • • • • •
How do band gaps affect properties of materials? Only electrons near EF participate in properties If EF is in the middle of the band, free e- and metallic behavior If EF is near the band gap, changes in materials properties may occur Need to find out where EF is! Where kF=π/a if we 2k L 2 L want to see how many N= F = a π electrons are in first band Note: L/a is the number of unit cells in the 1-D crystal; therefore, the number of electrons per unit cell, which depends on valence and the crystal structure, determine where EF is with respect to the band gap
EF with 2e- per unit cell EF with 1e- per unit cell
−π/a ©1999 E.A. Fitzgerald
π/a
323f13
1-D Crystal Metals and Insulators • 2e- per unit cell: EF at band edge: 2 possibilities – Band gap >> kT: electrons at band max can not accept energy from electric fields; no conduction, insulating behavior – Band gap near kT: some thermal fluctuations large enough to allow population of second band; carriers are there, but less than for free e-, semimetal • 1e- per unit cell: EF in middle of band: free e-, metallic Note: crystal structure (number of atoms per primitive cell) and valence (number of conduction electrons per atom), combined with band gap size, determine the electronic properties 323f14 ©1999 E.A. Fitzgerald
Higher Dimensions (2 and 3-D) • 1-D: E(kx); 2-D: E(kx,ky); 3-D: E(kx,ky,kz)
‘Fermi Surface’ E Zone center
ky First zone
π/a First zone
0
kx −π/a
−π/a
0
π/a
π/a
3-D
−π/a
π/a 2-D
1-D Zone center ©1999 E.A. Fitzgerald
−π/a 0
323f15
Higher Dimensions: Visualizing the Fermi Sphere and Bragg Plane Intersections • Use Ge band diagram to demonstrate how to show EF intersecting two important directions
EF 323f16 ©1999 E.A. Fitzgerald
Bands and Zones • Examine 2-D simple cubic crystal, or a slice through a 3-D cubic crystal • Bragg planes exist at bisection of every pair of reciprocal lattice points • As kf increases, Ef increases and crosses Bragg planes; when crossing, going to next band
323f17 ©1999 E.A. Fitzgerald
Bands and Zones 1st band Electron surface
Hole surface
G
2nd band
Reduced zone
Extended zone
323f18 ©1999 E.A. Fitzgerald
Carrier Dynamics at Fermi Surfaces ∂E
• Recall velocity is proportional to vg ∝ ∂k • Carriers have mass inversely proportional to curvature • Bulk electronic properties average over all surfaces
∂2E m* ∝ 2 ∂k
[010] •In magnetic field, carriers move along the Fermi Surface •Can measure the Fermi Surface this way
vg m l*
m t* v total since F~vxB [100] vg from E-field
Example: Si 323f19 ©1999 E.A. Fitzgerald
Metals and Insulators • Covalent bonds, weak U seen by e-, with EF being in mid-band area: free e-, metallic • Covalent or slightly ionic bonds, weak U to medium U, with EF near band edge – EF in or near kT of band edge: semimetal – EF in gap: semiconductor
• More ionic bonds, large U, EF in very large gap, insulator 323f20 ©1999 E.A. Fitzgerald
Insulators • Very large band gaps=no conduction electrons at reasonable temperatures • All electrons are bound • Optical properties of insulators are derived from the electric field being able to temporarily move electrons: polarization • We will return to the interaction of E-field with bound electrons in Dielectrics Section 323f21 ©1999 E.A. Fitzgerald
Semiconductors • Band gaps not large enough to prevent all carriers from getting into next band: three mechanisms – photon absorption – thermal – impurity (i.e. doping)
• Carriers that make it to the next band are like free carriers with new mass, m* 323f22 ©1999 E.A. Fitzgerald
Semiconductors: Photon Absorption • When Elight=hν>Eg, an electron can be promoted from the valence band to the conduction band E Ec near band gap E=hν k Creates a ‘hole’ in the valence band
Ev near band gap 323f23 Note: Most absorption near the band gap since the density of states is highest there ©1999 E.A. Fitzgerald
Holes and Electrons • • •
Instead of tracking electrons in valence band, more convenient to track vacancies of electrons, or ‘holes’ Also removes problem with negative electron mass: since hole energy increases as holes ‘sink’, the mass of the hole is positive as long as it has a positive charge Both carriers at the band edge can be thought of as classical free carriers like the Drude model had, as we shall see
Decreasing electron energy
Decreasing hole energy Decreasing electron energy 323f24 ©1999 E.A. Fitzgerald
Conductivity of Semiconductors • Need to include both electrons and holes in the conductivity expression
ne 2τ e pe 2τ h + σ = neµ e + peµ h = * me mh*
p is analogous to n for holes, and so are τ and m*
Note that in both photon stimulated promotion as well as thermal promotion, an equal number of holes and electrons are produced, i.e. n=p
323f25 ©1999 E.A. Fitzgerald
Thermal Promotion of Carriers •
EF
E
•
We have already developed how electrons are promoted in energy with T: Fermi-Dirac distribution Just need to fold this into picture with a band-gap
gc(E)~E1/2 in 3-D
Eg
f(E)
1
gv(E)
©1999 E.A. Fitzgerald
Despite gap, at non-zero temperatures, there is some possibility of carriers getting into the conduction band (and creating holes in the valence band)
g(E)
323f26
Density of Thermally Promoted of Carriers Number of electrons per volume in conduction band
n=
∞
∫ f ( E ) g ( E )dE
Ec
Density of electron states per volume per dE Fraction of states occupied at a particular temperature 1
f (E) = e 1 g c (E) = 2π 2
2m 2 ! ∞
* e
≈e
(E − E F )
3 2
−
(E − EF ) k bT
when ( E − EF ) >> kbT
+1
k bT
1 2m n = 2 2 2π !
(E − E g )
* e
1 2
π Since ∫ x e dx = , then 2 0 1 2 −x
©1999 E.A. Fitzgerald
e
1 2
EF ∞ k bT
∫ (E − E ) e g
−E k bT
dE
Eg
− Eg
m k T kbFT kbT e e n = 2 2 ! 2π * e b
EF − Eg
n = NCe
3 2
3 2
E
NC
k bT 323f27
Density of Thermally Promoted of Carriers • A similar derivation can be done for holes, except the density of states for holes is used • Even though we know that n=p, we will derive a separate expression anyway since it will be useful in deriving other expressions 1 g v (E) = 2π 2
2m 2 !
* h
3 2
1 (− E )2
0
p=
∫f
h
( E ) g v ( E )dE , where f h = 1 − f ( E )
−∞ 3 2
−E
m k T kbTF e p = 2 2 π ! * h b 2
p = Nve
− EF k bT
323f28 ©1999 E.A. Fitzgerald
Thermal Promotion • Because electron-hole pairs are generated, the Fermi level is approximately in the middle of the band gap • The law of mass action describes the electron and hole populations, since the total number of electron states is fixed in the system mh* 3 + kbT ln * n = p gives E F = 2 4 me Eg
Since me* and mh* are close and in the ln term, the Fermi level sits about in the center of the band gap 3 2
(
kT p or n = ni = 2 b 2 me*m 2π!
)e
3 * 4 v
− Eg 2 k bT
323f29 ©1999 E.A. Fitzgerald
Law of Mass Action for Carrier Promotion 3
(
kT ni2 = np = 4 b 2 me*m 2π!
)e
3 * 2 h
− Eg kbT
− Eg
;
ni2 = N C NV e kbT
•Note that re-arranging the right equation leads to an expression similar to a chemical reaction, where Eg is the barrier •NCNV is the density of the reactants, and n and p are the products E [N C NV ] → [n]+ [p ] g
[n][p ]
[N C NV ]
− Eg
=e
kbT
=
[ni ]2
[N C NV ]
•Thus, a method of changing the electron or hole population without increasing the population of the other carrier will lead to a dominant carrier type in the material •Photon absorption and thermal excitation produce only pairs of carriers: intrinsic semiconductor •Increasing one carrier concentration without the other can only be achieved with impurities, also called doping: extrinsic semiconductors 323f30 ©1999 E.A. Fitzgerald
Intrinsic Semiconductors • •
Conductivity at any temperature is determined mostly by the size of the band gap All intrinsic semiconductors are insulating at very low temperatures
Recall:
ne 2τ e pe 2τ h σ = neµ e + peµ h = + * me mh*
σ int = ni e(µ e + µ h ) ∝ e
− Eg 2 k bT
This can be a measurement for Eg
For Si, Eg=1.1eV, and let µe and µh be approximately equal at 1000cm2/V-sec (very good Si!) σ~1010cm-3*1.602x10-19*1000cm2/V-sec=1.6x10-6 S/m, or a resistivity ρ of about 106 ohm-m max
•One important note: No matter how pure Si is, the material will always be a poor insulator at room T •As more analog wireless applications are brought on Si, this is a major issue for system-on-chip applications –E-M waves lose strength since e- are responding to wave: loss and low Q resonant circuits 323f31 ©1999 E.A. Fitzgerald
Extrinsic Semiconductors •
•
Adding ‘correct’ impurities can lead to controlled domination of one carrier type – n-type is dominated by electrons – p-type if dominated by holes Adding other impurities can degrade electrical properties Impurities with close electronic structure to host isoelectronic
hydrogenic
x x
Ge
Impurities with very different electronic structure to host
x x
x
x
P
+
deep level
-
x x
x
x
x
Si
Si
Ec Ev ©1999 E.A. Fitzgerald
Au x
Ec ED
Ec
Ev
Ev
EDEEP 323f32
Hydrogenic Model • •
For hydrogenic donors or acceptors, we can think of the electron or hole, respectively, as an orbiting electron around a net fixed charge We can estimate the energy to free the carrier into the conduction band or valence band by using a modified expression for the energy of an electron in the H atom me 4 13.6 En = 2 2 2 = − 2 n 8ε o h n 4
(in eV)
e2 2 =e εr
me m *e 4 1 13.6 m* 1 En = 2 2 2 → 2 2 2 2 = − 2 n m ε2 8ε o h n 8ε o h n ε r
•Thus, for the ground state n=1, we can see already that since ε is on the order of 10, the binding energy of the carrier to the center is Energy
if
J positive, E~-S1S2--> Energy
if
Fe, Ni, Co ---> J positive! Other elements J is negative Rule of Thumb: r interatomic distance ≡ > 1 .5 2ra 2(atomic radius)
J is a function of distance! 323j16 ©1999 E.A. Fitzgerald
Ferromagnetism M
M (T ) ∝ (TC − T )
β ≈ 0.33 - 0.37
χ (T ) ∝ (TC − T )
γ ≈ 1.3 - 1.4
β
−γ
B=H+4πM Br, Ms
Domain rotation
Irreversible boundary displacement reversible boundary displacement
‘normal’ paramagnet
TC
Hc
H
T Easy induction, “softer”
Magentic anisotropy hardness of loop dependent on crystal direction comes from spin interacting with bonding 323j17 ©1999 E.A. Fitzgerald
Domains in Ferromagnetic Materials B N
S N S N
S
N S N S
M
Magnetic domain
Magnetic energy 1 = ∫ B 2 dV 8
Domain wall or boundary
Flux closure No external field
323j18 ©1999 E.A. Fitzgerald
3.23 Electrical, Optical, Magnetic Properties of Materials Prof. Eugene A. Fitzgerald Rm 13-4053, x8-7461,
[email protected] Prof. Yoel Fink, x3-4407,
[email protected] Grader: ? Class: Tuesday/Thursday 8:30-10, 4-237 Recitation: to be announced (3 recitations) Problem Sets: one a week, 1-3 problems, Due outside Prof. Fink’s office Exams: 3 exams, NO final Grading: problem sets 10%, each exam 30% Outline {first numbers refer to chapters to be read in A&M, second reference after comma refers to text and chapters for background reading in the topic area} Dates are approximate Conductivity and Bands Sept. 7 Origin of Ohms Law {1, O 4} Hydrodynamic representation of electrons in solids {1, O 4} Sept. 12 Hall effect {1, O 4} Sept. 14 AC response of electron particles; The plasma frequency; E-M waves in materials {1 O 4} Sept. 19,21 Electron waves and diffraction {4 5 6 , C 2 3 4} Origin of structure factor, Diffraction {4 5 6, C 2 3 4} Sept. 26 Wave-particle duality: DeBroglie, Compton, Planck, Einstein {-, E&R 1 2 3} Sept. 28, Oct. 3 Electron waves in solids {2, O 5} Quantized electron energy {2, O 5} Density of electron states {2, O 5} Boltzmann and Fermi-Dirac distributions {2, O 5} Heat capacity of electron waves {2, O 5} Oct. 5 Nearly free electrons in solids
Origin of band gaps and band structure {9, O 5} Schroedinger equation and application to electron waves in solids {8, E&R 5 O 5} Oct. 12 Bloch waves and general solution from S.E. Oct. 17 First Exam Oct. 19 H atom {-, E&R 7} Chemistry approach: H2+, tight-binding model {-, O 5} Bonding: connection to mechanical properties Oct. 24,26,31 Buliding materials atom by atom: Debye-Huckel model Electronic structure of polymer chains Hybridization Metals and insulators Bands and zones Carriers in bands Effective mass {-,K 8} Nov. 2 Intrinsic/Extrinsic Semiconductors {28, P&N vol. 1} Electrical activity of defects Hydrogenic model of extrinsic semiconductors {28, P&N vol. 1} Nov. 7 Carrier Scattering, Recombination, and Generation in semiconductors Defects: Traps/R-G centers Nov. 9,14 Drift, diffusion, and the continuity equations {-, P&N vol. 1} The ideal diode: depletion region, built-in voltage, and operation {-, P&N vol. 2} Band offsets Electrons in V(x): step, well, infinite well {-, S&W 3} Quantum wells Resonant tunnel diode operation 2DEG in Si/SiGe and transistor applications Nov. 16 Second Exam Dielectric and Optical Properties of Materials: The Capacitor {27, O 8} Nov. 21 Application of Maxwell’s equations to capacitance Dielectric constant and polarizability Dielectric response at optical frequencies Nov. 28,30 Hydrodynamic representation of polarizability Local field and Clausius-Mossetti Orientational, electronic, and ionic polarizability
Pyroelectrics and ferroelectrics Defects and dielectric loss Dispersion, attenuation in optical fibers Dec. 5 ‘Photonic Band Gaps’ Magnetic Properties of Materials: The Inductor {31 32 33, O 9} Dec. 7 Application of Maxwell’s equations to inductance Magnetization: paramagnetism, diamagnetism, ferromagnetism {31} Microscopic origin of magnetization {31} Central QM equation for magnetization and Hund’s rules {31} Pauli paramagnetism {31} Exchange and ferromagnetism {32 33} Mean-field theory {32 33} Applications: bubbles, magneto-optic storage Defects and domain pinning Dec. 12 Third Exam Texts Main text: Ashcroft and Mermin, Solid State Physics {A&M} Highly recommended: Omar, Elementary Solid State Physics {O} Cullity, Elements of X-ray Diffraction {C} Eisberg and Resnick, Quantum Physics {E&R} Kittel, Introduction to Solid State Physics {K} Pierret and Neudeck, Modular Series on Solid State Devices {P&N} Solymar and Walsh, Lectures on the Electrical Properties of Materials {S&W} Pankove, Optical Processes in Semiconductors {P} Livingston, Electronic Properties of Engineering Materials
