The chemical bonds between atoms are not rigid :
Act like spring Vibration
Lattice vibrations are responsible for transport of energy in many solids
Quanta of lattice vibration are called phonons.
The 1-D Diatomic Chain
Phonon Dispersion and Scattering
harmonic oscillator
[ ] [ ] 0M K
1
mK
23
1K 2K 3K1m 3m2m
2
2
dm K
dt
0 exp( )i
K
m
Equation of motion :
Solution :
Natural frequency :
Spring-mass model
Equilibrium Position :
Deformed position :
AssumptionAll the atoms and the springs between them are the sameThe force on the n-th atom is only by its neighboring atoms
1mm 3m
1mm
mK
1n
mmK K
n 1n
a
Lattice vibrations with monatomic basis
2
1 12 ( ) ( )n n n n
dm K K
dt
2
1 12 ( 2 )n n n
dm K
dt
Solution
0 exp( )exp( )n i t inka
Substitute into equation2 [2 exp( ) exp( )] 2 (1 cos )m K ika ika K ka
1
22
(1 cos )K
kam
12 | sin |
2
Kka
m
Fre
qu
en
cy
,
Wavevector, k a
0
First Brillouin zone
2
a
a
Slope( ) = Group velocityd
dk
1st Brillouin zone : ka a
: Dispersion relation( )k
transmission velocity of a wave packet
Kv grad Kg gv d dK or
12 | sin( ) |
2
Kka
m
1cos( )
2g
Kv a ka
m
- Wave packet or Wave group
- Beats
The velocity of energy propagation in the mediumThis is zero at the edge of the zone Standing wave
Group velocity
0k - : Long wavelength or continuum limit
0g
Kv a a
m : Velocity of sound in a crystal
Speed of sound
Fre
qu
en
cy
,
Wavevector, k0
1
Edge of First Brillouin zone
a
Ka
- : Edge of first
Brillouin zone
1 0
0
exp{ ( 1) }exp( )
exp( )n
n
i n kaika
inka
Standing wave
Acoustic branch
1cos( )
2g
Kv a ka
m0 exp( )exp( ),n i t inka
0gv
Acoustic branch
22
1 2 1 2 1 22
22 1
2 2 2 2 2 12
( 2 )
( 2 )
nn n n
nn n n
dm K
dt
dm K
dt
1mm 3m2m1mK
2n 2 1n
2 1n2n 2 2n2 1n
ax
Lattice vibrations with Two atoms per primitive basis
two atoms per unit cell
2 1
2 2
exp( )exp( )
exp( )exp ( 1/ 2)
n
n
A iwt inka
A iwt i n ka
Solution
Substitute into equation
21 1 2
22 2 1
(2 ) 2 cos( / 2) 0
(2 ) 2 cos( / 2) 0
K m A K ka A
K m A K ka A
The determinant must be zero for nontrivial solutions
21 2
21 2
2 2 cos( / 2)0
2 cos( / 2) 2
K m K ka A
K ka A K m
4 2 2 21 2 1 22 ( ) 4 1 cos ( / 2) 0m m K m m K ka
Two roots for 2
Optical branch
Acoustic branch
1/ 22
2
1 2 1 2 1 2
1 1 1 1 4sin ( / 2)kaK K
m m m m m m
1/ 22
2
1 2 1 2 1 2
1 1 1 1 4sin ( / 2)kaK K
m m m m m m
Acoustic branch : The atoms move together, as in long wavelength acoustical vibration.
Optical branch : If the two atoms carry opposite charges, we may excite a motion of this type with the electric field of a light wave.
Influences optical properties of a crystal
Optical branch
Optical branch and Acoustic branch
First Brillouin zone
1 2
1 12 ( )op K
m m
0k
ka
22 /op K m
12 /ac K m
1/ 22
2
1 2 1 2 1 2
1 1 1 1 4sin ( / 2),op
kaK K
m m m m m m
1/ 2
22
1 2 1 2 1 2
1 1 1 1 4sin ( / 2)ac
kaK K
m m m m m m
1 2
1 12 ( )op K
m m
1 2
12 0ac
Kka
m m
2
1
2 /
2 /
op
ac
K m
K m
High frequency
First Brillouin zone
Characteristics of optical branch
0gv : Group velocity is negligible
12 /K m 22 /K mBetween and :no solution
s s+1 s+2 s+3s-1
us-1 us us+1 us+2 us+3
us-1 us us+1 us+2
Three modes of wave vectors for one atom per unit cell One longitudinal mode Two transverse modes
Transverse vs. Longitudinal polarization
If there are q atoms in the primitive cell, there are 3q branches to the dispersion relation
- Number of branches
3 acoustic branches : 1 longitudinal acoustic (LA)
2 transverse acoustic (TA)
3q - 3 optical branches : q - 1 longitudinal optical (LO)
2q - 2 transverse optical (TO)
Dispersion Relation for Real Crystal
- Si[100] direction - SiC
The group velocity of phonons in the optical branches is small contribute little to the thermal conductionAt low temperatures : TA are dominant contributors to the heat conduction
At high temperatures : LA are dominant contributors to the heat conduction
Frequency gap
Governs the thermal transport properties of dielectric and semiconductor
Inelastic scattering : the phonon frequency before the scattering event is different from that after the event
Normal (or N) – process : Inside the 1st brillouin zone
Phonon scattering
Phonon-phonon scattering
1 2 3 1 2 3 or
1 2 3 1 2 3k k k k k k or
: energy conservation
: crystal momentum conservation
Umklapp(or U) process : Outside the 1st brillouin zone
a
a
xk
zk
1k
2k
3k
a
a
xk
zk
1k
2k
3k
G 1 2k k
N process U process
k k G
1 2 3 1 2 3k k k G k G k k or
N - process U - process
Energy ConservationMomentum ConservationThermal conductivity
Conserved Conserved
Conserved Net momentum not conserved
DominantNot dominant
Act as a direct resistance to heat flow
Distributing the phonon energy
Thermophysicalrole
N-process vs. U-process
1/ :
, :
U
A B
Scattering rate of the U-processPositive constants
Above room temperature
Below room temperature
- Fig. 5.13 Thermal conductivity of silicon
Phonon scattering: Temperature dependence
2( )U A B T
U T 1
T
/
1 1
21Bv k Tp K
CT e
,v vC T C T
At high temperature specific heat does not change significantly
Four – phonon scattering
1
2
3
4
4
1
2
3
1
2
3
4
(Temperature range : 300 K ~ 1000 K)
: Negligible
Phonon – defect scatteringElastic scattering
Independent of temperature
Defendant on the phonon wavelength
Phonon scattering – 4 phonon & defect
2 2four T
four U
4ph-d
Dominant at high temperatureScattering by acoustic phonon is essentially elastic.
Scattering by optical phonon is inelastic : Polar scattering
negligible
Facilitates heat transfer between optical phonon and electron (Joule heating)
(i : initial state f : final state)
Phonon scattering- phonon-electron
ac eE E
f i phE E : energy conservation
f i phk G k k
: momentum conservation+ : phonon absorption- : phonon emission
Phonon and photon inelastic scattering
called Raman scattering, X-ray scattering, neutronscattering, and Brillouin scattering
i: incident photon
s: scattered photon
ph: phonon
Phonon scattering- Raman scattering
s i ph
s i ph
Stokes shift (phonon emission)
anti-Stokes shift (phonon absorption)
Dependence on temperature
used for surface temperature measurements
2
i ph phanti-Stokes
Stokes i ph B
expI
I k T
Photoelectric Effect:
electromagnetic wave
metal plate
Heinrich Hertz observed the photoemission in 1887J. J. Thomson discovered electron as a subatomic particle
Albert Einstein explained the photoelectric effect in 1905(Nobel Prize in 1921)
Photoemission
Electron Emission and Tunneling
h e
e
e e
Measuring the Ejected Photoelectrons
electrode electrode
A
loadvacuum
incident photon
Frequency, of incident radiation is not high
enough
no electric current
threshold frequency for photoemission in given material
e
e
h
Work Function
work function: energy needed to remove
electron from metalAg, Al, Au, Cu, Fe: 4 ~ 5 eV (ultraviolet region)Na, K, Cs, Ca: 2 ~ 3 eV (visible region)
2e e,max
1
2h m v
maximum kinetic energy of ejected electron
2e e,max
1
2m v
h
A photon can interact only one electron at a
time.
electron right at the Fermi
level
Application of Photoemission
XPS (x-ray photoelectron spectroscopy)
measurement of the electron binding energy, Ebd
sample
chemical composition of the substance near the
surface
electron energy analyzer
2bd e e
1
2E h m v e
h 2bd e e
1
2h E m v
Thermionic Emission
hot cold
AJ
Loadvacuum
Similarity to photoemission → Work function
current density B2RD 1 k TJ A r T e
e
e
x
EF
Fermi-Dirac distribution at T = 0, E < EF, all
states are filled by electron and E > EF all
states are emptywhen T > 0,Some electron having more than EF + ,Small fraction of electron must occupy energy levels exceeding EF + .
Current Density
Particle flux
Current density
ˆNJ fv nd
velocity space: x y zd dv dv dv
ˆx y zfv ndv dv dv
e e, ,1N x N xJ eJ J e r J
e, 1x x x y zJ e r v fdv dv dv
: fraction of electron reflected
r
ˆˆxv n v i v
number of electrons per unit volume
B
3 2e
/
1 1( ) 8
1k T
mdN dg dN vf v
V dv V dv dg h e
Recall
24 x y zv dv dv dv dv
F B
3
e/
( ) 21
x y z
E E k T
dv dv dvmf v
h e
F ,E E 2 2 2e / 2x y zE m v v v
current density in the x direction
e, 1x x x y zJ e r v fdv dv dv 2
e ,0,0 F2
xx x
m vv v E
ejected velocity (vx) of electron > binding Ebd
FE E B/ 4k T when
F B( ) /FD( ) E E k Tf v e
less than 2% error
2 2 2
eF
B B
,0
( )3
2ee, 1 2
x y z
x
m v v vE
k T k Tx x x y zv
mJ e r v e e dv dv dv
h
F
B
,0
3 2e e
e,B
2 1 exp2x
E
k T xx x xv
m m vJ e r e v dv
h k T
2 2e e
B B
exp2 2
y zy z
m v m vdv dv
k T k T
e, 1 ,x x x y zJ e r v fdv dv dv F B
3
e/
( ) 21
x y z
E E k T
dv dv dvmf v
h e
Let2
e B
B e2x x
x
m v dv k Tt
k T dt m v
,xv t
F
B
F
B
B B
e e
E
k TtE
k T
k T k Te dt e
m m
2e ,0
F2xm v
E
2e ,0 Fe F
B B e B
2
2 2xm v Em E
tk T k T m k T
,0 F
B
2e B
B e
exp2x E
k T
txx x xv
x
m v k Tv dv v e dt
k T m v
Richardson-Dushman eq.
Richardson constant ARD = 1.202×106 A/(m2K2)
F F
B B B
3
e B Be,
e e
22 1
E E
k T k T k Tx
m k T k TJ e r e e e
h m m
B
22e B
e, 3
41 k T
x
m ekJ r T e
h
B2RD 1 k TJ A r T e
2 2e e
B B
exp2 2
y zy z
m v m vdv dv
k T k T
2 2e e
B B2 2 B
e
2y zm v m v
k T k Ty z
k Te dv e dv
m
2
e ax dxa
heat transfer associated with electron flow
Heat Flux
Flux of energy
→ similarity to current density
2
e12
xx x x y z
m vq r v fdv dv dv
F B xE k T J
e
: average energy of the “hot electron”
F BE k T
,E x x xJ q fv d
2
e12
xx x y z
m vr v fdv dv dv
Derivation of Heat Flux
2
e12
xx x x y z
m vq r v fdv dv dv
F
B
,0
33e1 22x
E E
k Tx ex y zv
m v mr e dv dv dv
h
2 22
e eeF
B B B B
,0
42 2 23e
31y zx
x
m v m vm vE
k T k T k T k Tx x y zv
mr e v e dv e dv e dv
h
Let2
e B
B e2x x
x
m v dv k Tt
k T dt m v
2
e ax dxa
F
B
,x
Ev t t
k T
heat transfer associated with electron flow
F F
B B B
4 2e B F B B3 2
e B e
2( ) 21 e e e
E E
k T k T k Tx
m k T E k T k Tq r
h m k T m
B
22e BF B
3
41 k Tm ekE k T
r T ee h
B
22e B
3
41 k T
x
m ekJ r T e
h
F B xx
E k T Jq
e
Field Emission and Electron Tunneling
Tunnel
Electron wave
Electronenergy
Potential U
Potential barrier (hill),
( )U x
E
2 3/ 2
exp/
VJ C
L V L
L0 x
( )x
When the field strength is very high, electrons at lower energy levels than the height of the barrier can tunnel through the potential hill.
Thermionic emission may be enhanced or even reversed by an applied electric field
Current density
Fowler-Nordheim equation
Field emission
(x) : width of potential at E
Current Density
Electron motion: governed by Schrödinger’s wave
equation
Wavefunction form
2 e2
2mk E U
Time dependent Schrödinger equation2 2
2e
( , ) ( , )( ) ( , )
2
x t x tU x x t i
m x t
e2( , )
im E Ui t ikx i tx t Ae Ae e when E > U
e2( , )
im U Ei t ikx i tx t Ae Ae e when E < U
Tunneling current density
kinetic energy in the x direction, E
energy at the top of potential barrier, Emax
reference energy, Emin
number of available electrons, n(E)
max
mint ( ) ( )
E
EJ e E n E dE
Transmission probability or transmission coefficient
0e
e0
e
222 ( )
2 ( )Right
2 ( )Left
( )
idx m E U i
dx m E U
idx m E U
J eE e
Je
e0
2( ) exp 2 ( )E dx m U E
Energy barrier with two electrodes
Fowler-Nordheim tunneling
chemical potential
current density by various approximations
e0
2( ) exp 2 ( )E dx m U E
Left Left( )/
e Ex
e V L
Left Left( )V
U x e e xL
electric field, /V L
Potential U
Tunnel
( )U x
LefteE
L0 x
Left
Right
( )x
Left Left ( )V
E e e xL
Current density
positive constant,
C
positive constant,
2*
e e3/ 2 1/ 2Left Left3
4 2 2 24exp ( ) ( )
(2 ) 3
L m L memJ e e
e V e V
2 3/ 2 3 / 22 *e Left
e Left
4 2exp
8 3 /
m ee m V
m L e V L
2 3/ 2
exp/
VJ C
L V L
Electrical Transport in Semiconductor Devices Number Density, Mobility, and the Hall Effect
Number density of electrons and holes determines the electrical, optical, thermal properties of semiconductor materials.
Number Density
electron and hole → Fermi-Dirac distribution function
F Bh ( ) /
1( )
e 1E E k Tf E
number density of electrons and holes
( ) ( )dn D E f E dE
F BC
ee ( ) /
( ),
e 1E E k TE
D E dEn
V
F B
hh ( ) /
( )
e 1
E
E E k T
D E dEn
F Be ( ) /
1( ) ,
e 1E E k Tf E e h( ) ( ) 1,f E f E
EC: minimum of conduction band
EV: maximum of valence band
densities of states in the conduction and valance bands
3/ 2*1/ 2C e
e C C2 2
C
2( ) ( )
2
M mdkD E M E E
dE
2 2
e C *e
( )2
kE k E
m
2 2
h V *h
( )2
kE k E
m
MC: number of equivalent minima in conduction band3/ 2*
1/ 2hh V2 2
V
21( ) ( )
2
mdkD E E E
dE
effective mass for density of states
geometric average of 3 masses = longitudinal mass + 2 transverse mass
2 2e h
* 2 2 * 2 2e h
1 1 1 1,
d E d E
m dk m dk
MC: number of equivalent minima in the conduction band
Most semiconductor can be described one band minimum at k = 0 as well as several equivalent anisotropic band minima k ≠ 0
Simplified E-k diagram of silicon within the 1st Brillouin zone along the (100) direction
The energy is chosen to be to zero at the edge of the valence band.Lowest band minimum at k = 0 is not the lowest minimum above the valence bandat
( In here x = 5 nm-1)There are 6 equivalent minima, these are minimum energy. On the other hand, maxima of valence band only has one k state.
So in calculation De, we must multiply 6 (MC)
( 00), ( 00), (0 0),k x x x(0 0), (00 ), (00 )x x x
we have to consider the effective mass The effective mass of electrons is a geometric average over the 3 major axes because the effective mass of silicon depends on the crystal direction.The effective mass of holes is an average of heavy holes and light holes because there exist different sub-bands.The effective mass calculation for density of
states= The geometric average of the 3 masses:(one longitudinal mass ml, two transverse mass mt)must include the fact that several equivalent minima exist : MC in the conduction band
3
2 3e,density of state C t t lm M m m m
At moderate temperatures,C F B F V B,E E k T E E k T
approximation with M-B distribution
F B F B( ) / ( ) /e h( ) , ( )E E k T E E k Tf E e f E e
F B F B
e he h( ) / ( ) /
( ) ( ),
e 1 e 1V
C
E
E E k T E E k TE
D E dE D E dEn n
C F B F V( ) / ( ) /
e C h Ve , e BE E k T E E k Tn N n N
2/3 2/3* *e B h B
C C V2 2
2 22 , 2
m k T m k TN M N
h h
NC, NV: effective density of states in conduction band and valance band
number density for intrinsic and doped semiconductors
g C VE E E
g B g B/ /2 3e h th C Ve e
E k T E k Tn n N N N T
2th :N thermally excited electron-hole pairs per unit
volume number density of intrinsic carriers
Fermi energy for an intrinsic semiconductor when ne = nh
C V V C VBF
C
ln2 2 2
E E N E Ek TE
N
Fermi energy for an intrinsic semiconductor in the middle of the forbidden band or the bandgap
n-type
p-type
Fully ionized impurities, charge neutrality requirement
NA, ND: number densities of donors and
acceptors
impurities of donors (P, As) involved → ionization of donors increases the number of free electrons
Impurities of acceptors (B, Ga) involved → Ionization of acceptors increases the number of holes
e A h Dn N n N
Electric Field = 0so net motion =0
Electric Field ≠ 0so net motion ≠ 0
Electric Field
Electrons and holes are accelerated by electric field, but lose momentum due to scattering processes.
Mobility
mobility: ratio of the drift speed to applied electric field
du
E
d,e d,h
e h, u u
E E
mean time between collision, / v
applied electric field to electron,
eF m a eE
drift velocity, d,ee
eEu a
m v
e he h* *
e h
, e e
m m
mobility*
e
m
conductivity2
*
ne
m
ne
Electrical conductivity of a semiconductor
e e h hen en
Average energy of semiconductors
* 2 * 2e e B e th
1 3 1
2 2 2m v k T m v
Bth
3*
k Tv
mthermal velocity at equilibrium
T
At sufficiently high T, contribution of carrier-phonon scattering
3/ 2ph T
Impurities scattering3/ 2
dd
T
N Nd: concentration of the ionized
impuritiesMatthiessen’s rule
e e e ph e d
1 1 1 1
Overall mobility*
ph d
1 1 1m
e
Hall Effect
useful in measuring the mobility of semiconductors → van der Pauw method (4 probe technique) Net current flow
Lorentz force in the y direction
dF q E u B
e, h,e, h,
e h
, y yy x y x
ev eveE ev B eE ev B
e e e, e e, h h h, h h,( ) , ( )y x y y x yn E v B n v n E v B n v
e, e h, h
e, e h, h
,
0x x x
y y y
J ev n ev n
J ev n ev n
e, e h, h 0y y yJ ev n ev n
e e e, h h h,( ) ( ) 0y x y xn E v B n E v B
e e e, h h h,
e e h h
y x xE n v n v
B n n
Hall coefficient
e, e h, hx x xJ ev n ev n
e e e, h h h,H
e e h h e e, h h,( )( )y x x
x x x
E n v n v
J B e n n n n
2 2e e h h
He e h h( )
n n
e n n
e, e h, h, x x x xv E v E
Generation and Recombination
Photoconductivity: Excitation of electrons from valence band to conduction band by the absorption of radiation increases the conductivity of the semiconductor
Generation
Conductivity at thermal equilibrium before incident radiation
0 e,0 e h,0 hen en
Relative change in electrical conductance after incident radiation
e h
0 e,0 e h,0 h
( )n
n n
0 rc gn n n n
rc: recombination lifetime or recombination time
Recombination
Relaxation process, related to electron scattering, lattice scattering, defect scattering because the excess charge is not at thermal equilibriumNon-radiative: Auger effect, multiphonon emissionRadiative: using in luminescence application (LED)Net rate of change = generation rate – recombination rate
0g
rc
n ndnn
dt
Under steady-state incident radiation
0 rc g0, dn
n n n ndt
gv v v vI A I
nhvAd hvd
: absorptance
I: spectral irradiance of incoming photon (W/m2Hz)
Sensitivity of a photoconductive detector
rc e h
0 e,0 e h,0 h
( )1
( )v
vI hv n n d
e h
0 e,0 e h,0 h
( )n
n n
The p-n Junction
Diffusion of electrons and holes → Fick’s law
e he e h h,
dn dnJ eD J eD
dx dx
Diffusion coefficients
e e h he h,
3 3
v vD D
Assume e h thv v v 2
e th e the 3 3
v vD
Diffusion coefficient, Einstein relation
e B e Be *
e
k T k TD
m e
* 2e th B3
2 2
m v k T
p n---
+++
x
N
nh(x) ne(x)
p-region n-region
ECp
EVp
EFECn
EVn4
1
2
3
1. Concentration gradient holes will diffuse right and electrons will diffuse left.
2. As they leave the host material, ions of opposite charges are left behind.
3. This results in a charge accumulation and consequently it leads to a built-in potential in the depletion region.
4. The energy in the p-doped region rise relatively
Therefore, forward bias removes the barrier for elements to diffuse, on the other hand, a reverse bias creates an even stronger barrier
→ the junction has characteristic of rectification.
1 : electron drift 2 : electron diffusion3 : hole diffusion 4 : hole drift
depletion region
Current density in semiconductor for charge transfer
ee e d,e e
dnJ n ev eD
dx
drift termdiffuse term
d,e ev E
Under equilibrium condition, J = 0
e e eB
e e e
D dn dnk TE
n dx en dx e B e B
e *e
k T k TD
m e
,0
,0
V( )
,0 ,0 biV( )
VV V( ) V( ) V
n n
p p
x x
n px - x
dE Edx d x - x
dx
Vbi: built-in potential
Derivation of Current Density
,0
,0
( ) ,0B Bbi e( )
e ,0
( )1V ln
( )n n
p p
x n x n
x n - xp
n xk T k TEdx dn
e n e n - x
Applied voltage → Non-equilibrium occurs → Elements move
bie ,0 e ,0
B
V( ) ( )expp n
en x n x
k T
bi ae e ,0
B
(V V )( ) ( )expp n
en x n x
k T
bi a a
e ,0 e ,0B B B
V V V( )exp exp ( )expn p
e e en x n x
k T k T k T
sB
exp 1eV
J Jk T
Js: Saturation current density
Optoelectronic Applications
Photovoltaic effect Incident upon a p-n junction generates electron-hole pairs → Built-in electric field in the p-n junction → Solar cell and photovoltatic detector
TPV (thermophotovoltatic) devices Incident radiation with a photon energy greater than the bandgap energy strikes the p-n junction Drift current: Electron-hole pair is generated → swept by the built-in electric field → collected by electrodes at ends of cellDiffusion current: For radiation absorbed near the depletion region → minority carriers diffuse toward the depletion region