An introduction to electrons and photons in semiconductor ...€¦ · ELECTRONS AND PHOTONS IN...

ELECTRONS AND PHOTONS IN SEMICONDUCTORS

Electronic Materials Group, Massachusetts Institute of Technology

An introduction to electrons and photons in semiconductor structures

Luca Dal Negro

Contents


Electron and light waves

Band structure of semiconductors

Density of states

Electrons in the bands – Fermi-Dirac function

Carrier in semiconductors

Photons in Semiconductors

Joint Density of States

Transition Rates

Absorption in direct and indirect semiconductors

Low dimensional semiconductors structures

Electron are waves (or not?)


riknknk eru ⋅−= )(ϕBloch waves:

Electron Waves

*

2

2mkE =

Quadratic dispersion

kcEck=⇒=

πν

2

Optical Waves

V(r)=V(r+R) e(r)= e(r+R)

Linear dispersionPropagation group velocity depends on the wave momentum Propagation velocity is a

constant of the medium

Light versus electrons


022 =+∇ ϕϕ kScalar Maxwell or SchrödingerEquation can be cast in the same form

22 )(2 VEmk −=

Electron Waves

2

)(2)( rmVr =σ

E-V can be negative!

Frequency independent )(20

2 rkk σ−= Space dependent kDeviation from a uniform background

2

22

ck ω

=

⎥⎦

⎤⎢⎣

⎡−= 2

0

2

)(1)(c

rr ωεσ

Optical Waves

always positive!

w2 frequency dependence!

Band structure of solids


V(r)

n=1

n=2

n=3

Energy levels

Allowed band

Gap

Distance between atoms

Si real band diagram


Courtesy of IBM. Used with permission.


Courtesy of IBM. Used with permission.

Photon Band structures


From: J.D.Joannopoulos, R.D.Meade, J.N.Winn, Photonic Crystals, Univ.Princeton Press, 1995

Image removed due to copyright considerations. Image removed due to copyright considerations.

Density of States I


Standing waves in a box

πndk =⋅

Boundary condition:

dx

y

zDirect space

⎟⎟⎠

⎞⎜⎜⎝

⎛=

dn

dn

dnk zyx πππ ,,

The k vector is discretized:

),( zyx nnn positive integers

Density of States II


dπ

3

3

34

812

)(⎟⎠⎞

⎜⎝⎛

⋅⎟⎠⎞

⎜⎝⎛⋅

=

d

kkN

π

π

kkNdkddkk ∆=∆ )()/1()( 3ρ

K-space

Unit volume

spin Unit volume in k space

# of k states in volume d3

2

2

)(π

ρ kk =

r(k)Dk = # per unit volume of electron states with wavenumber within k and k+ Dk

Density of k states

Density of States III


cc m

kEE2

22

+=

Electron Dispersion relation:

VV m

kEE2

22

−=

Holes Dispersion relation:

Density of energy states:

dkkdEEvc )()(/ ρρ =

( ) ( )

( ) ( ) 2/132

2/3

2/132

2/3

22)(

22)(

EEmE

EEmE

vv

v

cc

c

−=

−=

πρ

πρ

Eg

E

Ev

Ec

r

Occupation probability: the Fermi-Dirac function


Electron (Holes) occupation probabilities under thermal equilibrium T: equilibrium temperatureEf : Fermi energy (Fermi level)

0.0 0.2 0.4 0.6 0.8 1.00.00

0.25

0.50

0.75

1.00

f(Ef)=0.5

E=Ef

f(E)

Energy (eV)

T=0K T=300K T=500K

f(E) = probability of occupancy by an electron

1-f(E) = probability of occupancy by a hole (valence band)

Ef

Ec

Ev

?

Approximations


IffEE > and TkEE Bf >>−

( )[ ]TkEEEf Bf /exp)( −−≈

Boltzmann approx.If TkEE Bf >>−fEE < and

( )[ ]TkEEEf Bf /exp)(1 −−≈−

Carrier concentrations (thermal equilibrium)


[ ])(1)()( EfEEp v −= ρ)()()( EfEEn cρ=

Electron densityDensity of states

Occupancy probability

∫∞

=cE

dEEnn )(

Concentrations (populations/volume):

∫∞−

=vE

dEEpp )(

electrons holes

In an intrinsic (pure) Semiconductor, at any T :

pn =

The Boltzmann case (non-degenerate case)


When the Fermi function can be approximated by the Boltzmann onewe can calculate explicitly the concentrations (non-degenerate case):

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

TkEE

NnB

fcc exp

2/32 )/2(2 hTkmN Bcc π=Effective density of States in the conduction band :

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

TkE

NNnpB

gVc exp

Constant, independent of Ef !

Law of Mass Action

Intrinsic Semiconductor


inpn == 2innp =Intrinsic concentration

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−≈

TkE

NNnB

gVci 2

exp2/1 The intrinsic concentration of electronsAnd holes increases with temperature at an exponential rate!

T=300K

ni (cm-3)

Si 1.5 x 1010

GaAs 1.8 x 106

Interactions of Photons with e-h


• Band to Band (Interband) Transitionsvertical or phonon assisted

• Impurity to band transitionstransitions between donor or acceptor levels and a band in doped semiconductorsthe transitions may also be assisted by traps in defect states

• Free carrier transitionsphotons impart their energy to electrons in the bandsi.e. electrons in conduction band are promoted higher energy levels in the bandand a phonon assisted thermalization process relaxes the system down to the band minimum

• Phonon Transitionslong wavelength photons release the energy by lattice vibrations excitation

• Excitonic Transitionsthe photon absorption results in the formation of a bound (theough Coulomb interaction) e-h pair (exciton) . Photons are emitted by excitons annihilation

Interactions of Photons with e-h (II)


Ec

Ev

νhEE =− 12

Momentum conservation

Energy conservation

K

E E

K

Need assistance of other particles (phonons, ….)

DirectBandgap

IndirectBandgap

GaAs Si

k∫k’

phel

ph

kak

kkk

>>=

=≈

/2

/212

π

λπ

Conditions for Absorption and Emission


Conservation of energy and momentum requires that a photon of frequency n interact with electrons and holes of specific energies and momentum determined

by the E-k dispersion relation

Eg

E2

E1

hn

( )

( ) νν

ν

hEEhmmEE

EhmmEE

gv

rv

gc

rc

−=−+=

−+=

21

2

Energy levels with which the photon interacts:

( )cvr

gr

vcg

mmm

Ehmk

hmk

mkEEE

/1/1/1

222

22

2222

12

+=

−=

=++=−

ν

ν

Optical Joint Density of States


“Constrained” density of states / Density of states for photon interaction r(n)under the conditions of energy and momentum conservation / couples theConduction and valence band density of states

( ) )(/)()()( 2222 EddEddEE cc ρννρννρρ =⇒=

( ) ( ) ggr EhEhm

≥−= ννπ

νρ ,2)( 2/12

2/3

hn

r(n)

Eg

Absorption and Emission Rates


We consider direct band-gap semiconductor (vertical transitions)

What is the photon emission/absorption probability density ?

Occupancy probabilityDepends on 3 factors: Transition probability

Joint Density of StatesEmission Condition:

E2 filled with one electron , E1 empty (filled with one hole)

[ ])(1)()( 12 EfEff vce −=νAbsorption Condition:

E1 filled with one electron , E2 empty

[ ] )()(1)( 12 EfEff vca −=ν( fc and fv are the Fermi functions for theSemiconductor in quasi-equilibrium)

Direct band-gap semiconductors


ω

Eg Eg

EFcEF

EFv

THERMAL EQUILIBRIUM NON EQUILIBRIUM

Fermi level Quasi-Fermi levels

kTEEF FeEEfEf /)(1

1),()( −+==

kTEEvc vFcvceEf /)(/ //1

1)( −+=

Actual emission/absorption


Satisfying the emission/absorption occupancy conditions does not assure thatthe emission/absorption process actually takes place! photons/(sHzcm3)

)()(1 ννρτ e

rsp fr =TRANSITION RATES (Spontaneous emission)

At thermal equilibrium (only 1 Fermi function) and in the non-degenerate case:Tkh

eBef /ν−≈

( )( ) Tk

E

r

r

gTkEh

gsp

B

g

B

g

emD

EheEhDr

−

−−

=

≥−≈

τπ

νννν

2

2/3

0

2/10

2

,)(


Direct-gap absorption (or -gain)

[ ] [ ] phvcphcvabs EfEfAEfEfAkr ρρ )(1)()(1)()( 01101001 −−−=

0

1

[ ]ρ)()()( 10 EfEfAkr cvabs −=

UP DOWN

Particle Statistics + Fermi’s golden rule

Density of statesTransition matrix element Occupation StatisticsFermi if Fermions (e-h)Bose-Einstein if Bosons(phonons)

More processes…


)()(8

)(

)()(8

)(

2

2

ννρπτλφν

ννρπτλφν

ν

ν

ar

abs

er

st

fr

fr

=

=

Photon flux spectral density

Stimulated emission rate:

Absorption rate:

Direct band-gap absorption (Boltzmann approx.) :

( ) 2/1)( gEhconst −×= ννα

Indirect-gap absorption


3 body problem Phonons are involved to conserve momentum

Low probability / second order process

pha

phe

EEEh

EEEh

−−=

+−=

12

12

ν

ν

11

/ −= TkEph Bphe

N # of phonons

( )( )2

2

)(

)(

phgphE

phgphA

EEhNconst

EEhNconst

−−×=

+−×=

ννα

νναA

E

Eg Eg+ EphEg- Eph

T2>T1

T1

Quantum confined structures


Momentum in discretizedEnergy is discretized

n=1n=2n=3Particle in a box

( )*

22

2/

mdnEn

π=d

Suggested readings


• Photonics, Saleh, Teich (chapters 9, 12, 15)

• Fundamentals of Semiconductors, P.Y.Yu, M.Cardona, Springer ed. (chapters 7, 9)

An introduction to electrons and photons in semiconductor ...€¦ · ELECTRONS AND PHOTONS IN...

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