1. ◎K・P理論を用いた光吸収計算 ◎中間バンドを介入したキャリア再結合 ◎ 中間バンドに関するバンド理論とバンド計算
曽我部東馬＠東大先端研　07/21, 2011

2. OUTLINE ( ,)) ● Intermediate band calculation ● BASICS:
K•P method and light absorption ^
●　Light absorption and carrier dynamics
( ,))
●　Intermediate band calculation

3. KP perturbation method
The muscling up scheme for understand optical dynamics in semiconductor
Fermi’s golden rule
Absorption coefficient
The Schrödinger equation
●E-k dispersion
●En and density of states
●Ψ: Wavefunction
Radiation transition lifetime
Non Radiation transition lifetime
Electron cooling:
Bi-exciton recombination:
KP perturbation method
Lutinger-Kohn Hamiltonian
Plane wave expansion

4. Schrödinge equation Bloch function Note: is not included^
K・P　perturbation method :
Schrödinge equation
Bloch function ^ ^
Note: is not included

5. Kane model Luttinger- kohn model^
K・P : Kane model and Luttinger –Kohn model
Kane model
Group A
Group B
L-K parameter:
Luttinger- kohn model
S like
P like
Kane parameter P :
K・P
K・P
K・P

6. Initial basis set: (electron eigen states) are given as:^
K・P :　　Kane model
Initial basis set: (electron eigen states) are given as: , ^
(R(r) part will be solved by using K P method)
S like
P like
P like
Hamiltonian used for calculation (SO effect included)

7. Explicit form of Hamiltonian
Kane model :
Explicit form of Hamiltonian
Block diagonalisation
Kane parameter (predetermined:
where:
Spin orbit interaction (0.3~0.6eV):
Det(H)=0
Eigenvalue
Eigenvalue

8. CB band HH band LH band SO band K・P : Kane Model: Energy level (k=0)^
K・P : Kane Model:
Energy level (k=0)
Wavefunction (k=0)
CB band ● Eg ●
HH band
LH band Δ
SO band ●

9. Following the Löwdin’s renormalization theorem and taking
K・P : Luttinger –Kohn model
Hamiltonian:
Initial basis set:
Following the Löwdin’s renormalization theorem and taking
the Kane’s solution at k=0 for valence bands, we write:
Löwdin’s 　perturbation method:
Wavefuction of B is included but
not included in the calculation.
No need to calculate because L-K parameter is used

10. Explicit form of Hamiltonian:
K・P : Luttinger –Kohn model
Explicit form of Hamiltonian:
L-K parameter (predetermined):

11. K・P : strain effect based on Luttinger –Kohn model (my work)
In1-xGaxAs on InP
In_1-xGa_xAs x= E-Kx dispersion
In_1-xGa_xAs x= E-Kz dispersion z z z
Kx (Å-1 )
Kz(Å-1 )

12. K・P : Exercise: strain effect based on Luttinger –Kohn model
In_1-xGa_xAs x=0.4 (reference)
In_1-xGa_xAs x=0.3 (my work)
PRB,
Kx (Å-1 )
Kz(Å-1 )

13. K・P : Constant energy contour
Unstrained In1-xGaxAs on InP (x=0.468)
Kx (π/a)
Kz (π/a) HH LH

14. Total transition rate per unit volume
Transition rate using Fermi’s Golden Rule
Absortption
<b|
<a|
Emission
Absorption
Emission
Total transition rate per unit volume
Net upward transition rate per unit volume

15. ●H’(r) ●Wavefunction ●Energy level ① ④ ⑤ ② ③
Calculate net transition rate ① ④ ⑤ ②
●H’(r)
●Wavefunction
●Energy level ③

16. Light Origin of H’(r): electron and photon interaction②
Light
(Is not used for calculating transition rate)
Atom
Scale of　electric field distribution is too big than atom size
Dipole approximation

17. TE mode: TM mode: CB HH LH SO
Momentum matrix elements of bulk and quantum well/dot semiconductor ③ CB ● Eg HH ●
TE mode: LH Δ
TM mode: SO ●

18. Kane parameter (from experiment)
Momentum matrix elements of bulk and quantum well/dot semiconductor ③
Kane parameter (from experiment)
TE MODE CB ● Eg Px Py HH ● LH
TM MODE Px Py Pz

19. Delta function: ④ =6
=10
=20
=30

20. = No. Photons absorbed per unit volume per second
Absorption coefficient:
No. Photons injected per unit area per second
No. Photons absorbed per unit volume per second
Absorption
coefficient =
Momentum transfer matrix

21. 𝟐 𝑽
term and Joint (reduced ) density of states (JDOS): (1) ①
3-D density of states
Conduction band
Density of states for electrons
Valence band
Density of states for hole

22. CB VB term and Joint (reduced ) density of states (JDOS): (2) Ef
𝟐 𝑽
term and Joint (reduced ) density of states (JDOS): (2) CB Ef ●
reduced electron
mass : mr
with Eg ● Ei VB
Reduced (joint) density of states
Under thermal equilibrium
Non-equilibrium
Carrier concentration difference

23. CB VB Absorption coefficient under thermal equilibrium: BulK material● Eg ● VB

24. 　Light absorption and carrier dynamics
( ,tby tomic Stanko)

25. Truncated pyramid Quantum dot InAs (QD)/GaAs
Calculation model:
Truncated pyramid Quantum dot InAs (QD)/GaAs
QD superlattice model ● ● ●
d=2~10nm
h=2.5nm
b=10nm
K P plane wave method
●L-K hamiltonian
●Fourier transformation ● ● ●

26. Charge density contour and band dispersion
Quantum dot array
Single Quantum do e0 h0
GaAs
InAs e0 e2 e2 e1 e1 e0 e0

27. Due to shape anisotropy, QD is more TE polarized
Absorption : IB=>CB
Band structure and Potential profile of InAs (QD)/GaAs
Absorption spectrum (cm-1)
Shape anisotropy and polarization
e0e1
e0e2
---TE mode
---TM mode ③ ② ①
Barrier Continuum
Bragg type confinment miniband
e1 ,e2 band : p like e2 e1 TE ③ Py ① ② Px e0 TM Pz
Due to shape anisotropy, QD is more TE polarized ● ● ● ●
IB=> CB transition

28. Absorption spectrum (cm-1)
Absorption : VB=>IB
Band structure and Potential profile of InAs (QD)/GaAs
Absorption spectrum (cm-1)
Polarization dependence
5x104 e1 e0 e2
Barrier Continuum ① ② ③ h0
Due to highly biaxial compressive strain and the hydrostatic component makes h0 state heavy hole character. HH Px Py TE ● ● ● ● TM
VB=> IB transition

29. Sommerfeld factor get extremely large at band edge
VB=>CB transition and many body effect
VB=> CB transition
E (eV)
Band edge
Sommerfeld factor (absorption enhance effect) e1 e0 e2
Barrier Continuum ① h0
Sommerfeld factor get extremely large at band edge
●　Many body effect (Coublum interaction) is not important because exciton is almost ionized , but it becomes pronounced when at low temperature.

30. =52 Radiative transition times (radiative recombination)
●　Life time depends on energy difference
Single QD ns ns
QD SL(Kz=0) ns
●　Life time depends on moment um matrix
e1,e2 =>e0 (intraband transition)
=52
e0=>h0 (inter band transition)
Single QD ns
QD SL(Kz=0) ns

31. Single InAs QD InAs QD SL Auger recombination: electron cooling
Coulomb integral
Single InAs QD
InAs QD SL

32. Auger recombination: biexiciton recombination
Single QD
Single QD
QD SL
QD SL
Single QD
QD SL

33. Electron phonon interaction: absorption and emission
e1-e0
Absorption
Emission
36meV

34. Summary: ① Radiative recombination. CB=>IB
②　Radiative recombination. IB=>VB
⑤　Auger electron cooling between CB=>IB
③　Phonon emission assisted CB=>IB
④　Phonon absorption assisted IB=>CB
⑥　Auger biexciton relaxation between IB=>VB

35. Appendix:
Kronig-Penny Model

36. GaAs/AlxGa1-xAs quantum well superlattice band calculation:
Well width
Barrier width
Al 0.3 Ga 0.7 As
Ga As
Eg１
Eg2
ΔEc
Ga As
Ga As
ΔEv
井戸幅(ドット直径）
バリア層幅
Eg１
Eg2
ΔEC
ΔEv
2nm-30nm
2nm-10nm eV
1.80 eV
0.25 eV
0.125 eV
Calculation method: Transfer matrix method　

37. Bound state & Continuum miniBand
井戸幅：10nm バリア層幅　4nm
(eV)
(eV) C8 C1 C7 C6
バリアポテンシャル(eV) B2 C5 C4 B1 C3 C2 C1 B0 Г Х Х Х Г Х

38. Bound state miniBandが一つだけ存在する条件：
井戸幅：5nm バリア層幅　4nm
(eV) C7
(eV) C2 C6 C1 C5
バリアポテンシャル(eV) C4 Г Х B0 C3 C2 C1 Х Г Х

39. 伝導帯Bound miniband の障壁幅/井戸幅依存性 ：
バリア層幅　
2nm
バリア層幅　
4nm
バリア層幅　
8nm
バリア層幅　
10nm
(eV) B2 B2 B2 Г Х B2 Г Х B4 B6 B4 B4 B6 B4 B6 B6
井戸幅：
10nm, 20nm , 30nm
井戸幅：
10nm, 20nm , 30nm
井戸幅：
10nm, 20nm , 30nm
井戸幅：
10nm, 20nm , 30nm B5 B5 B5 B5 B3 B3 B3 B3 B4 B4 B4 B4 B1 B1 B1 B2 B1 B2 B2 B2 B3 B3 B3 B3 B2 B2 B2 B2 B0 B1 B1 B1 B1 B0 B0 B0 B1 B1 B1 B1 B0 B0 B0 B0 B0 B0 B0 B0 Х Г Х Х Х Х Г Х

40. Light hole Bound miniband の障壁幅/井戸幅依存性 ：
バリア層幅　
2nm
バリア層幅　
4nm
バリア層幅　
8nm
バリア層幅　
10nm
(eV)
LH0
LH0
LH0
LH0
LH0
LH0
LH0
LH0
LH0
LH1
LH1
LH1
LH1
LH0
LH0
LH0
LH2
LH2
LH1
LH1
LH2
LH2
LH1
LH1
井戸幅：
10nm, 20nm , 30nm
井戸幅：
10nm, 20nm , 30nm
井戸幅：
10nm, 20nm , 30nm
井戸幅：
10nm, 20nm , 30nm
LH2
LH2
LH3
LH3
LH3
LH2
LH2
LH3
LH4
LH1
LH1
LH4
LH4
LH4
LH1
LH1
LH3
LH3
LH3

41. 伝導帯 miniband & 価電子帯 miniband：
バリア層幅：4nm
井戸幅: 4nm
バリア層幅：4nm
井戸幅: 20nm
(eV)
(eV) B3 B2 B0 B1 B0
LH0
HH0
HH0
LH1
HH2
LH0
LH2
HH3
HH1
HH4
LH3