an introduction to electrons and photons in semiconductor...
Post on 30-Sep-2020
10 Views
Preview:
TRANSCRIPT
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
Electronic Materials Group, Massachusetts Institute of Technology
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?)
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
V(r)
n=1
n=2
n=3
Energy levels
Allowed band
Gap
Distance between atoms
Si real band diagram
Electronic Materials Group, Massachusetts Institute of Technology
Courtesy of IBM. Used with permission.
Electronic Materials Group, Massachusetts Institute of Technology
Courtesy of IBM. Used with permission.
Photon Band structures
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
IffEE > and TkEE Bf >>−
( )[ ]TkEEEf Bf /exp)( −−≈
Boltzmann approx.If TkEE Bf >>−fEE < and
( )[ ]TkEEEf Bf /exp)(1 −−≈−
Carrier concentrations (thermal equilibrium)
Electronic Materials Group, Massachusetts Institute of Technology
[ ])(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)
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
• 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)
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
“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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
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
,)(
Electronic Materials Group, Massachusetts Institute of Technology
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…
Electronic Materials Group, Massachusetts Institute of Technology
)()(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
Electronic Materials Group, Massachusetts Institute of Technology
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
Electronic Materials Group, Massachusetts Institute of Technology
Momentum in discretizedEnergy is discretized
n=1n=2n=3Particle in a box
( )*
22
2/
mdnEn
π=d
Suggested readings
Electronic Materials Group, Massachusetts Institute of Technology
• Photonics, Saleh, Teich (chapters 9, 12, 15)
• Fundamentals of Semiconductors, P.Y.Yu, M.Cardona, Springer ed. (chapters 7, 9)
top related