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