(14-Bands) Application of Analytical k.p Model With Envelope Function

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Application of analytical k.p model with envelope functionapproximation to intersubband transitions in n-type IIIVsemiconductor quantum wells

C. W. Cheah and L. S. TanCenter for Optoelectronics, Department of Electrical and Computer Engineering, National Universityof Singapore, Singapore 119260

G. KarunasiriDepartment of Physics, Naval Postgraduate School, Monterey, California 93943

Received 14 August 2001; accepted for publication 12 December 2001

A 14-band k.p model combined with an envelope function approximation has been developed forthe analysis of IIIV semiconductor quantum wells by including the six 7 , 8conduction bandsnonperturbatively. With appropriate approximations, the envelope functions associated with the 7 ,8bands can be expressed in terms of the two 6 conduction band envelope functions, which arethe most important components in the electron wave function of an n-type direct-gap IIIVcompound semiconductor quantum well of zincblende structure. The Schrodinger-type equations forthe 6 conduction band envelope functions are derived, together with the energy-dependenteffective mass that includes the effect of band nonparabolicity, as well as the eigenenergy-dependenteffective potential for the envelope wave functions. The Schrodinger-type equations and theboundary conditions for the conservation of probability flux in the 14-band k.p model are found tobe different from those of the conventional effective mass model. The 14-band model is then appliedto the study of intersubband transitions due to transverse magneticTMand transverse electricTEmode infrared radiation in n-type quantum wells, and the calculated absorption spectra arecompared with those computed using an equivalent 8-band k.p model. It is found that the TMabsorption spectra calculated using the two models are very similar, but the TE absorption spectracalculated using the 14-band model is up to 6 times higher than that calculated using the 8-bandmodel. A design of the quantum well structure for enhancing TE absorption is also discussed. 2002 American Institute of Physics. DOI: 10.1063/1.1448890

I. INTRODUCTION

Quantum well infrared photodetectors QWIPs havebeen very well researched for more than two decades. Due tothe maturity of MBE growth technology, ease of processing,and high electron mobility, n-type direct-gap IIIV com-pound semiconductor material systems such as AlGaAs/GaAs and GaAs/InGaAs quantum wells QWs have beenthe popular choice for QWIPs.1 These devices employ theelectronic behavior around the valley. Since a zincblendesemiconductor possesses the symmetry of the tetrahedralgroup Td, the lowest antibonding conduction bands ex-hibit spherical symmetry and transform according to the 6double group with the inclusion of spin states, and thus in-frared radiation polarized in the transverse electric TE

mode is not expected to excite intersubband transitions in the QW. However, it is also expected that the selection rulegoverning intersubband transitions will be relaxed due to theeffect of band mixing,2 4 i.e., the basis set that describes theelectronic states in the conduction bands should also includethe p-like valence band, or even the p-like conduction bandthat transforms according to 7 and 8 double grouprepresentations,5,6 as a result of the truncation of the periodicbulk structure. Furthermore, the spatial variation of the ma-terial parameters band energies and interband momentummatrix elements manifested in the spatial dependent effec-

tive mass,7 as well as the formation of subbands,8 also causethe selection rule to relax. The possibility of TE mode inter-

subband transition is significant to QWIPs, as it means thatnormal incidence can then be achieved in a focal plane arraywithout the need for complex fabrication processes such asthe grating coupling scheme.8

Over the years, experimental observations of normal in-cidence absorption have been reported.911 Peng et al.12,13

reported observing a very large TE mode absorption that iscomparable in magnitude to transverse magnetic TMmodeabsorption in GaAs/AlGaAs QWs, and invoked a 14-bandk.p model5 to account for the large TE mode absorption.However, the calculation was later questioned by Lew YanVoon et al.14 and a debate ensued.15,16 Flatte et al.6 con-

ducted a theoretical study on the GaAs/AlGaAs superlatticeby including the upper p-like conduction band states in the8-band k.p model perturbatively and concluded that the up-per limit for the ratio of TE to TM absorption is about 20%.Liuet al.17 performed further experiments and found that theratios of TE to TM absorption in GaAs/AlGaAs and InGaAs/AlGaAs QWs are no more than 0.2% and 3%, respectively.However, the issue of how large the TE absorption can ac-tually be remains largely unresolved.

In this article, a 14-band k.p model with envelope func-tion approximation is developed with the 6 upper 7 , 8

JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 8 15 APRIL 2002

51050021-8979/2002/91(8)/5105/11/$19.00 2002 American Institute of Physics


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conduction bands included nonperturbatively. The envelopefunctions associated with the p-like valence and conductionbands are expressed analytically in terms of the envelopefunctions associated with the spin-up and spin-down 6con-duction bands, and Schrodinger-type equations for the twoconduction band envelope functions are derived. The energy-dependent effective mass that includes the effect of bandnonparabolicity as well as the energy-dependent effective po-

tential for the envelope functions can be derived easily fromthe equations. The boundary conditions that couple thespin-up and spin-down 6envelope wave functions are alsoderived. Calculated absorption spectra for TM and TE modeinfrared excitations for several QW structures are presentedand compared with those obtained using the 8-band k.pmodel. Based on these results, a possible design of the QWstructure for improving TE absorption is proposed and dis-cussed.

II. 14-BAND k.p MODEL WITH ENVELOPE FUNCTIONAPPROXIMATION

Since the bulk periodicity in the in-plane direction isassumed to be preserved, the electron wave function of amultiple quantum well MQWcan be expressed as1822

nr,k,q eikr

lfl

nz ,k,qu lr, 1

where l runs through the set of Bloch functions chosen toform the set of basis functions for the Hilbert space, q is thesuperlattice wave vector, k and r are the in-plane wavevector and position vector, respectively. u l(r) is the pointzone center basis Bloch function of the lth band, andfl

n(z,k,q) is the z i.e., growth direction-dependent enve-lope function associated with the lth band and nth subband.

The term e ik

ru l(r) preserves the periodicity of the crystalin the in-plane direction for any off-zone center state. The 14zone center basis functions chosen are shown in Table I.Following the argument of group theory, it is well knownthat under the symmetry operation of the Td group for allzincblende type crystals with noninversion symmetry, thenonzero interband momentum matrix elements are23

P iSpxXv iSpyY

v iSpzZv,

QiXvpyZc iXvpzY

ciYvpzXc

iYvpxZc iZvpxY

c iZvpyXc ,

R iSpxXciSpyYc iSpzZc .

The nonrelativistic time-independent single electronSchrodinger equation including the spinorbit coupling con-sidered in this study can be written as

12m0p2Vr

4m02c2

Vrp nr,k,qEnknr,k,q , 2

where p i, is the modified Plancks constant, m 0isthe free electron mass, c is the speed of light, is the spinoperator, En(k) is the eigenenergy of the nth subband, and

V(r) is the electrostatic potential field of the crystal at equi-librium. By substituting Eq. 1 into Eq. 2, followed byprojecting the resultant equation onto each of the 14 basisBloch functions, and integrating over a unit cell, the follow-ing secular equation is obtained:

l l0Enk,q

pz2

2m0

2k2

2m0 fl

nz,k,qml

1

m0pz km0 flnz ,k,q ml 0, 3

where l0is the band edge energy at point for thelth band,ml is the Kronecker delta,

ml1

um*rp Vr

4m0c2 u lrd3r

and is the volume of the unit cell. In the above derivation,it has been assumed that the envelope functions vary muchmore slowly in real space compared to the Bloch functions,22

i.e.,

V

flnz,k,qu lrd

3r

V

flnz ,k,qd3r

1

u lrd3r, 4

and that the spinorbit coupling between the different band edges is negligible, so that

ml1

u m*rpu lrd

3rpml.

Equation 3 is used to express the envelope functionsassociated with the 7 , 8valence and conduction bands interms of the 6conduction band envelope functions fcandfc . In order to simplify the algebraic operation, the freekinetic energy terms of all the envelope functions associatedwith the 7 , 8 bands, as well as the terms involving the

coupling between the 7 , 8 valence and conduction bandsthat are directly proportional to the in-plane wave vector ki.e., terms involving Qk), are neglected from the opera-tion. The free kinetic energy contributed by the 7 , 8bandstates, (2/2m0)(

2/z2) f7,8 , is expected to be small. Asfor the coupling terms involving Qk, the electrons are ex-pected to occupy the states close to the valley for thetypical doping and temperature concerned, and hence onlysmallkare considered. Moreover, such coupling terms arezero at the zone center (k0), which is the case in manytheoretical studies. Therefore, the 12 envelope functions ar-guments of which are omitted for brevitycan be expressedas

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TABLE I. 14-band basis Bloch functions at point zone center of IIIV compound semiconductor.

Associated band Bloch functions

Conduction band ucrr12 ,1

2irS

ucrr12 ,1

2irS

Heavy hole band uhhrr32 ,3

2v

1

2rXviYv

uhhrr32 ,3

2v

1

2rXviYv

Light hole band ulhrr32 ,1

2v

23rZv

1

6rXviYv

ulhrr32 , 12v

23rZv 16rXvi Yv

Spinorbit valence band usohrr12 ,1

2v

1

3rZv

1

3rXviYv

usohrr12 ,1

2v

1

3rZv

1

3rXviYv

Heavy electron band uherr32 ,3

2c

1

2rXciYc

uherr32 ,3

2c

1

2rXci Yc

Light electron band ulerr32 ,1

2c

23rZc

1

6rXciYc

ulerr32 ,1

2c

23rZc

1

6rXciYc

Spinorbit conduction band usoerr12 , 12c 13rZc 1

3rXciYc

usoerr12 ,1

2c

1

3rZc

1

3rXci Yc

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fhhkP

m0

1

hhfc

Q

3m02

1

hh le so eHc2

2 leso e22Hchh3kPQ

m0

2

z2fc

2 leso e22Hci2kR

z

fc

2 le so eHc2R

2

z 2fc, 5

flh1

3m0

1

lh so hHv2 so h2Hvhe

i2kQR

m0

zfc so h2HvkP fc

2so hHviPz

fc , 6

fso h1

3m0

1

lh so hHv2

2 lhHvhe

i2kQR

m0

zfc2 lhHvkP fc

lh2HviP

zfc , 7

fhhkP

m0

1

hhfc

Q

3m02

1

hh le so eHc2

2 leso e22Hchh3kPQ

m0

2

z2fc

2 leso e22Hci2kR

zfc

2 le so eHc2R 2

z 2fc, 8

flh1

3m0

1

lh so hHv2 so h

2Hv

he

i2kQR

m0

zfc so h2HvkP fc

2so hHviP

zfc , 9

fso h1

3m0

1

lh so hHv2

2 lhHvhe

i2kQR

m0

zfc2 lhHvkP fc

lh2HviP

zfc , 10

fhekR

m0

1

hefc

Q

3m02

1

he lhso hH2

2 lhso h22Hvhe3kQR

m0

2

z2fc

2 lh so h22Hvi2kP

zfc

2 lh so hHv2P 2

z 2fc, 11

fle1

3m0

1

leso eHc2so e

2Hchh

i2kPQ

m0

zfc so e2HckR fc

2 so eHciR

zfc , 12

fso e1

3m0

1

le so eHc2 2 leHc

i2kPQ

m0

1

hh

zfc2 leHc

kR fc le2HciR

zfc , 13

fhekR

m0

1

hefc

Q

3m02

1

he lhso hH2

2 lhso h22Hvhe3kQR

m0

2

z2fc

2 lh so h22Hvi2kP

zfc

2 lh so hHv2P

2

z 2fc, 14



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fle1

3m0

1

leso eHc2so e

2Hchh

i2kPQ

m0

zfc so e2HckR fc

2 so eHciR

z

fc

, 15

fso e1

3m0

1

le so eHc2 2 leHci

2kPQ

m0

1

hh

zfc2 leHckR fc

le2HciR

zfc , 16

where, in the above equations, l l0 (2k

2/2m0)EHl , and H

l is the deformation potential correction to thelth bandedge due to strain.24 H

vis the term due to strain that couples the light hole and spinorbit valence bands, and Hcis the term

that couples the light electron and the spinorbit conduction bands via strain.25

Projecting Eq. 4onto u c and integrating over a unit cell results in the following equation:

ckfc2

2m0

2

z 2fc23

iP

m0

zflh

1

3

iP

m0

zfso h

kP

m0fhh

1

3

kP

m0flh23

kP

m0fso h

kP

m0 fhe

2

3

iR

m0

zfle

1

3

iR

m0

zfso e

1

3

kR

m0 fle

2

3

kR

m0 fso e0. 17

By substituting Eqs. 516into Eq. 17, a Schrodinger-type equation involving fc is obtained as follows:

cfc2

2m0

2

z2fc

iP

3m02

z

1

lh so hHv2 lh2 so h22HviP zfc

2 lh so hHvhe

2kQR

m0

zfc2 lhso hHvkP fc kP

3m02

1

lh so hHv2

2 lhso hHviP zfc2 lh so h22Hv

he

3kQR

m0

zfc2 lh so h22HvkP fc

kR

3m03

1

he lhso hHv22 lh so hHv2PQ 2

z2fc

2 lh so h22Hvhe

3kQ2R

m0

2

z 2fc

2 lhso h22Hvi2kPQ

m0

zfc 2kkR2

m02

fciR

3m02

z

1

le so eHc2

le2 so e22HciR zfc2 le so eHc

hh

2kPQ

m0

zfc2 le so eHckR fc

kR

3m02

1

le so eHc2 2 leso eHciR zfc

2 le so e22Hchh

3kPQ

m0

zfc2 leso e

22HckR fc kP3m0

3

1

hh le so eHc22 le so eHc2QR 2

z2fc

2 le so e22Hchh

3kPQ2

m0

2

z 2fc2 le so e22Hc

i2kQR

m0

zfc 2kkP2

m02

fc0. 18

In each layer of homogeneous material, Eq. 18can be reduced to

2

2mz

2

z 2fcvi

zfcVeffEfc0, 19

where



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1

mz

1

m0

2

3m02 P

2 lh2so h22Hv

lh so hHv2

R2 le2 so e22Hc

leso eHc2 2

2kkQ2R2

3m04

2 lh so h22Hvhe

2 lhso hHv

2

2 le so e22Hc

hh2 leso eHc

2 ,

v

2kxkykxkyPQR

3m03

2 lh so h22Hv

he lh so hHv2

2 le so e22Hc

hh le so eHc2 , 20

VeffEcHc

2kk

m * ,

1

m

1

m0

P2

3m02 3hh

2 lh so h22Hv

lhso hHv2 R

2

3m02 3he

2 leso e22Hc

le so eHc2 .

It is interesting to note that in the 14-band k.p model, Eq.19, which resembles the Schrodinger equation in the effec-tive mass model, contains a term of first order in derivativewhich diminishes at the zone center as well as along the110and equivalent directions. This term does not exist inthe 8-band k.p model as the crystal is assumed to be ofinversion symmetry. In addition, the effective mass, mz, inthe growth direction also possesses additional contributionsproportional to the in-plane wave vector which is not presentin the 8-band k.p model and the conventional single bandeffective mass model. It is also worth noting that the in-planeeffective mass, m , converges with mz at the zone center

only if there is no strain and spinorbit coupling.The Schrodinger-type equation for the spin down 6conduction band electron envelope function, fc , can be de-rived following the same procedure and the resultant equa-tion within each homogeneous material layer is similar toEq. 19, i.e.,

2

2mz

2

z 2fcvi

zfcVeffEfc0. 21

Although the functions fcand fcare decoupled fromeach other within each homogeneous layer, they are actuallycoupled at the abrupt interface, which is apparent in Eq. 18.In order to arrive at the boundary conditions, it is assumedthat the envelope functions fc and fc are continuousacross the boundary between two different layers of material.This assumption implies that the basis functions of the 6bands are sufficiently similar in each material. The boundarycondition that governs the probability flux can be arrived atby integrating Eq. 18 across the boundary over an infini-tesimally small region, which results in the continuity of

i

mz0z

fc 2

km

z

fc k

mfc , 22

where

1

mz0

1

m0

2

3m02P 2 lh2 so h22Hv lhso hHv2

R2 le2so e22Hc

leso eHc2 ,

1

m

2PQR

3m03 2 lhso hHvhe lhso hHv2

2 leso eHc

hh leso eHc2 , 23

1

m

2

3m02 P22 lh so hHv lh so hHv2

R22 le so eHc

le so eHc2 .

Similarly, the other boundary condition that originatesfrom fc is the continuity of

i

mz0

zfc

2k

m

zfc

k

mfc . 24

As seen in Eqs. 22and24, the spin-up and spin-down6 states are coupled at the interface, and the coupling isproportional to the spinorbit split off energy as well asstrain. Therefore, fc and fc are to be solved as a pair ofcoupled solutions for the total wave function as defined inEq. 1. The two functions are only decoupled from eachother at the zone center, and this approach is different fromthat of Ref. 3, in which fc and fc are considered decou-pled even at off-zone center states. The inclusion of the up-per six p-like conduction bands enhances the coupling effect



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at the interface. The eigenstates can now be solved by thestandard transfer matrix method with periodic boundary con-ditions. If the terms proportional to v in Eqs. 19and 21,as well as the terms inversely proportional to m in Eqs.22and 24, are neglected as they are generally quite smallcompared to the other terms, then the solutions for fc and

fcare in simple plane wave representation. In addition, theresultant transfer matrix is of rotational symmetry about thegrowth axis such that all the states at the same distance fromthe zone center are degenerate.

A second independent solution of the eigenfunction atk is the time reversal of the eigenfunction at k, i.e.,

r,k,q Rzr,k,qRzeikr

lflz ,k,qu lr

e ikrfc* z ,k,qu crfhh* z ,k,qu hhrflh* z,k,q u lhrfso h* z ,k,qu so hr

fc* z,k,qu crfhh* z ,k,qu hhrflh* z,k,q u lhrfso h* z ,k,qu so hr

fhe* z,k,quherfle* z,k,q u lerfso e* z ,k,q u so erfhe* z,k,q u her

fle* z ,k,q u lerfso e* z ,k,qu so er, 25

where is the time reversal operator, and Rzis the rota-tional operator about the z axis by . From the rotationalsymmetry of Eqs.22and24as shown in Appendix A, Eq.25is obtained by taking

fcz,k,qfcz,k,q, 26

fcz,k,qfcz,k,q , 27

and substituting Eqs. 26 and 27 into Eqs. 516. Theoperations of Rz on the basis functions are wellknown. As shown in Appendix B, the wave function in Eq.25is orthogonal to the wave function defined in Eq. 1andthe two wave functions are therefore independent of eachother.

III. RESULTS AND DISCUSSION

To demonstrate the application of the 14-band k.p modelthat has been developed here, the absorption spectra of sev-eral MQWs are calculated. In the calculations, the followingassumptions about the electromagnetic field are made.26

1 The electromagnetic field is small.2 It has a relatively long wavelength compared to the di-

mensionin the growth directionof the MQW, and thusis constant over the structure.

3 It obeys Coulomb gage, i.e., A0AppA,where A is the vector potential of the magnetic field.

By Fermis golden rule, the transition rate from the ithstate to the jth state is given by

Wi j

2i j

2(EjEi)

where

i je

m0i*r,q ,kApjr,q ,kdr

3

is the transition probability due to the electromagnetic per-turbation of (e/m0)Apbetween theith and jth states, and eis the electronic charge. To include the effect of relaxationand line broadening due to scattering, the Dirac delta func-tion is replaced by a Lorentzian function in Wi j, i.e.,

EjEi/2

EjEi2/22

where is the linewidth. If the magnetic vector potential isgiven by A eA0cos(kort), where e is the field direc-tion, k

0 is the field wave vector, and is the angular fre-

quency, then the absorption coefficient can be expressed as

()2c

nr2A0

2

1

Vk E(k)2

EGE(k)

2 /2

EkEG2/22

[f(EG)f(E(k)) ]

c

2nr2A0

2L

k

E(k)

EG(k)E(k)2

EkEGk2/22

[f(EG(k))f(E(( k)) ](E(k))dE(k)dk2 28



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for a transition between the ground state at energy EG(k) tothe excited state at (E(k), as the transition preserves thein-plane crystal momentum k because a photon carrieslittle momentum. In Eq. 28 is the permeability and n risthe refractive index of the material, respectively, the functionf is the Fermi distribution function and the density of states(E(k)) is given by8

Ek 1N

q

EqkEqk

1

N

qqq

dEk/dq

1

N

q

dEk/dq

1

N

L

d dy

dEk 1

d y

dEk , 29

where yqd, LNd, dis the length of each period, and Nis the number of periods in the entire QWIP structure.

The MQWs analyzed in this article are assumed to ben-type, uniformly doped in the well region at a concentrationof 21018 cm3 and undoped in the rest of the device. Theoffset in the conduction bands of the barrier and well regionsis taken to be 65% of the total band gap difference. Theabsorption spectra are computed by taking five subbands intoaccount, and a linewidth of 20 meV in the Lorentzianfunction is assumed.

The absorption spectra of a square MQW with a periodicstructure of 40 In0.3Ga0.7As/300 GaAs calculated usingthe present 14-band model as well as the equivalent 8-bandmodel are shown in Fig. 1. This structure has only one boundstate in the well and the transition is from bound to con-tinuum. As can be observed in Fig. 1a, the TM absorptionstrengths calculated with both the 14-band and the 8-band

models are very similar. Both the absorption spectra exhibitasymmetry with a slope decreasing more slowly into the highenergy region due to the nature of bound to continuum tran-sition. The TM absorption spectrum calculated with the 14-band model is slightly narrower, and the peak is slightlyhigher, than that of the 8-band model, because the C2 sub-band produced by the 14-band model is closer to the barrierconduction edge and the states are closer to being quasi-bound. The TE absorption spectra in Fig. 1b calculatedwith the two models are very different from each other. Thepeak absorption strength obtained with the 14-band model isapproximately 4 times as large as that obtained with the8-band model. The additional contribution to TE absorption

in the 14-band model comes from the coupling between the6conduction band and the 7 , 8conduction bands via theinterband momentum matrix R, where the contribution fromthe spinorbit coupling energy between the 7 , 8conduc-tion bands is introduced. Furthermore, there is also the con-tribution from the coupling between the 7 , 8 conductionbands and the 7 , 8 valence bands via the interband mo-mentum matrix Q .

The absorption spectra of a bound to bound type squareMQW with a periodic structure of 60 In0.3Ga0.7As/300 GaAs are shown in Fig. 2. The TM absorption spectra in Fig.2aobtained with the 14-band and 8-band models are againvery similar. The slightly different resonant energies in the

two spectra are due to the additional coupling to the 7 , 8conduction bands in the 14-band model, which results in aslightly different effective mass and effective potential forthe envelope wave functions. The TM absorption spectra ofthis MQW are much stronger than those of the bound-to-continuum type shown in Fig. 1a. The peak of the TMabsorption in the 8-band model is larger than that of the14-band model because the strong bound to bound TM ab-sorption is dominated by the contribution of the s-like 6conduction band envelope wave functions. In the frameworkof the 8-band model, the contribution of the 6 conductionband envelope wave functions is relatively larger in the totalnormalized wave function as compared to the normalized

wave function in the framework of the 14-band model, inwhich the wave function is the linear combination of 14components instead of 8. In Fig. 2b, the TE absorptionspectrum calculated with the 14-band model is also muchlarger than that of the 8-band model. Several prominentpeaks are clearly observable in both the TE absorption spec-tra and it is worth noting that the maximum peak is due tothe transition from the C1 to C4 subbands, instead of thetransition from the C1 to C2 subbands. This is because thestates in the bound C2 subbands exhibit strong odd parityand therefore the transition from the C1 to C2 due to TEpolarized field is supposedly forbidden by the parity selec-tion rule. However, the spatial variation of the material pa-

FIG. 1. Absorption spectra of a 40 In0.3Ga0.7As/300 GaAs square QWcalculated using the 14-band and 8-band k.p models in response to aTMmode radiation, bTE mode radiation.



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rameters, the finite subband width and the coupling to the

7 , 8bands relax the selection rule and actually allow thetransition to occur, albeit with a rather small magnitude. TheTE absorption strength due to the C1 to C2 transition in the14-band model is approximately twice that in the 8-bandmodel. On the other hand, the transition from C1 to the sub-bands in the continuum in the 14-band model is about 4times as large as that of the 8-band model. This is likely dueto the contribution from the more significant coupling to the7 , 8conduction bands of the excited states in the 14-bandmodel because the excited states in the continuum are ener-getically closer to the upper conduction bands. From thesecalculations, it can be concluded that for a symmetric squareMQW, the TE absorption is dominated by the contribution of

the transitions to the excited states in the continuum. There-fore, the TE absorption spectrum is always broader and itspeak position deviates from that of the TM absorption spec-trum.

The absorption spectra calculated with both the 14-bandand 8-band models of another MQW is shown in Fig. 3. Theperiodic structure of this MQW is an asymmetric steppedQW with a structure of 300 Al0.15Ga0.85As/40 GaAs/40 In0.15Ga0.85As, and the intersubband transitions are of thebound-to-bound type. As shown in Fig. 3a, the TM absorp-tion spectra calculated with both the 14-band and 8-bandmodels are again very similar, with the peak absorption ob-tained from the 8-band model being slightly larger for the

same reason as that of the bound-to-bound square MQW.

The TM absorption strength of the stepped MQW is largerthan that of the bound-to-continuum square MQW due to thebound-to-bound nature, but is smaller than that of the bound-to-bound square MQW due to the breaking of inversion sym-metry. The bound states in the C2 subband do not exhibit anydefinite parity. Consequently, as shown in Fig. 3b, the TEabsorption strength is almost twice as large in the steppedMQW than in the square MQW. This is in agreement withthe theoretical study conducted by Yang et al.7 The peak TEabsorption calculated with the 14-band model is approxi-mately 6 times larger compared to that of the 8-band model.It is also worth noting that the TE absorption spectra of thestepped MQW is dominated by the transition from the C1 to

C2 subbands, similar to that of the TM absorption spectra.

IV. CONCLUSION

In this article we have presented an analytical form ofthe 14-band k.p model for the study of intersubband transi-tions in QWs by including the upper 7 , 8 conductionbands nonperturbatively. It is found that the effective massSchrodinger-type equations, as well as the boundary condi-tions for the probability flux for the envelope functions, arenot exactly the same as those of the conventional effectivemass model that can be easily derived from the 8-band k.pmodel. The expressions for the effective masses in the

FIG. 2. Absorption spectra of a 60 In0.3Ga0.7As/300 GaAs square QWcalculated using the 14-band and 8-band k.p models in response to aTMmode radiation, bTE mode radiation.

FIG. 3. Absorption spectra of a 40 In0.15Ga0.85As/40 GaAs/300 Al0.15Ga0.85As stepped QW calculated using the 14-band and 8-band k.pmodels in response to aTM mode radiation, bTE mode radiation.



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growth and in-plane directions are also derived. It is foundthat the inclusion of the 7 , 8 conduction bands, althoughnot important in the theoretical study of TM intersubbandtransitions, is crucial in the study of TE absorption. However,the computed TE absorption strength remains very smallcompared to the TM absorption, and is typically less than1%. The breaking of symmetry in the design of the QWIP isalso found to significantly improve the magnitude and the

shape of the TE absorption spectra.

ACKNOWLEDGMENTS

This work was supported by the Singapore National Sci-ence and Technology Board under Grant No. NSTB/17/2/3GR 6471 Project 4, and the National University of Singaporein the form of a research scholarship for C. W. Cheah.

APPENDIX A

By omitting the terms involving m in Eqs. 22 and24, the boundary conditions become the continuity of

imz0

zfc

k

mfc A1

and

i

mz0

zfc

k

mfc , A2

together with the continuity of fc and fc at the abruptheterojunction. The continuity of Eq. A1across the bound-ary ensures the continuity of

eiimz0

zfc

k

mfc , A3

whereei is a constant phase. EquationA3can be rewrit-ten as

i

mz0

zeifc

kei

mfc A4

and Eq. A2can also be rewritten as

i

mz0

zfc

kei

meifc. A5

Therefore, (eifc ,fc) is the pair of solutions for theenvelope wave functions at the state ke

i, which is thein-plane state at an angle to the in-plane state at k .

APPENDIX B

By substituting Eqs.26and27into Eqs.516, wehave

fhhz,k,qfhhz ,k,q, B1

flhz ,k,q flhz ,k,q, B2

fso hz,k,qfso hz ,k,q , B3

fhhz,k,qfhhz ,k,q, B4

flhz ,k,q flhz ,k,q , B5

fso hz ,k,qfso hz,k,q, B6

fhez ,k,qfhez,k,q, B7

flez,k,qflez,k,q, B8

fso ez ,k,q fso ez,k,q, B9

fhez ,k,qfhez,k,q , B10

flez,k,qflez,k,q, B11

fso ez ,k,q fso ez,k,q. B12

Equations 26, 27 and Eqs. B1 to B12 are the 14envelope wave functions for the state k, and are the re-sultant envelope wave functions under the operation ofRz() on the envelope wave functions for the state k.

The time reversal operatoronly takes the rotated enve-lope wave functions into their complex conjugation.2729

However, due to the spin states in the basis Bloch functions,is an antiunitary operator that assumes the form UK,

whereUis a unitary operator, and only operates on the spinstates. K is an antiunitary operator and performs complexconjugation on its operand. Uis chosen as the Pauli matrixand mixes spin state, i.e.,

Uy 0 ii 0 .

Since spin state transforms like spinor according to therepresentation D1/2 in a full rotation group, the rotationRz() for spin spinor is

30

D1/20,0,

i 0

0 i

and for the p-like functions that transform according to therepresentation of D1 of the full rotation group, Rz() pos-sesses the form

D10,0,1 0 0

0 1 0

0 0 1.

Therefore, the total Rz() for the basis Bloch functionis simply

RzD1/20,0,D10,0,.

The operation ofRz() on the wave function in Eq. 1thus results in Eq. 25. The inner product of the wave func-tions in Eq. 1and Eq. 25is

Rze ikrl

flz,k,q u lr*e ikr

l

flz ,k,q u lrd3r. B13

By taking the approximation in Eq. 4and the orthogo-nality of the Bloch functions of the different bands, the inte-grand of Eq. B13is



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fcz ,k,qfcz ,k,q fcz,k,qfcz ,k,q fhhz,k,q fhhz,k,qflhz,k,q flhz ,k,q

fso hz,k,qfso hz ,k,qfhhz ,k,q fhhz,k,q flhz ,k,q flhz ,k,qfso hz ,k,q fso hz,k,q

fhez,k,qfhez ,k,qflez ,k,qflez,k,qfso ez ,k,q fso ez,k,qfhez ,k,qfhez ,k,q

flez,k,q flez ,k,q fso ez,k,qfso ez ,k,q0.

Hence the two wave functions are degenerate and should be included in the computation of the intersubband transitionmomentum matrix.

1 B. F. Levine, J. Appl. Phys. 74, R1 1993.2 A. Shik, in Intersubband Transitions in Quantum Wells, edited by E.

Rosensher, B. Vinter, and B. Levine Plenum, New York, 1992, p. 319.3 R. Q. Yang and J. M. Xu, Phys. Rev. B 50, 7474 1994.4 R. Q. Yang, Phys. Rev. B 52, 11 958 1994.5 L. H. Peng and C. G. Fonstad, J. Appl. Phys. 77, 747 1995.6 M. E. Flatte, P. M. Young, L. H. Peng, and H. Ehrenreich, Phys. Rev. B53, 1963 1996.

7 R. Q. Yang, Appl. Phys. Lett. 66, 959 1995.8 K. K. Choi, The Physics of Quantum Well Photodetectors World Scien-

tific, Singapore, 19979 H. S. Li, G. Karunasiri, Y. W. Chen, and K. L. Wang, J. Vac. Sci. Technol.

B 11, 922 1993.10 Z. Yuan, Z. Chen, D. Cui, J. Ma, Q. Hu, J. Zhou, and Y. Zhou, Appl. Phys.

Lett. 67, 930 1995.11 G. Karunasiri, J. S. Park, J. Chen, R. Shih, J. F. Sheihing, and M. A. Dodd,

Appl. Phys. Lett. 67, 2600 1995.12 L. H. Peng, J. H. Smet, T. P. E. Broekaert, and C. G. Fonstad, Appl. Phys.

Lett. 61, 2078 1992.13 L. H. Peng and C. G. Fonstad, Appl. Phys. Lett. 62, 3342 1993.14 L. C. Lew Yan Voon, M. Willatzen, M. Cardona, and L. R. Ram-Mohan, J.

Appl. Phys. 78, 295 1995.15 L. C. Lew Yan Voon, M. Willatzen, M. Cardona, and L. R. Ram-Mohan, J.

Appl. Phys. 80, 600 1996.16 L. H. Peng and C. G. Fonstad, J. Appl. Phys. 80, 603 1996.

17 H. C. Liu, M. Buchanan, and Z. R. Wasilewski, Appl. Phys. Lett. 72, 16821998.

18 G. Bastard, Phys. Rev. B 24, 5693 1981.19 G. Bastard, Phys. Rev. B 25, 7584 1982.20 G. Bastard and J. A. Brum, IEEE J. Quantum Electron. 22, 1625 1986.21 G. Bastard, in Physics and Applications of Quantum Wells and Superlat-

tices, edited by E. E. Mendez and K. V. Klitzing Plenum, New York,1987, p. 21.

22 G. Bastard, Wave Mechanics Applied to Semiconductor HeterostructuresHalstead, New York, 1988.

23 P. Y. Yu and M. Cardona, Fundamentals of Semiconductors: Physics andMaterial PropertiesSpringer, Berlin, 1996.

24 J. Singh, Physics of Semiconductors and Their HeterostructuresMcGraw-Hill, New York, 1995.

25 F. H. Pollak, in Strained-Layer Superlattices: Physics, edited by T. P.

Pearsall Academic, Boston, 1990, p. 17.26 S. L. Chuang, Physics of Optoelectronic Devices Wiley, New York,1995.

27 E. P. Wigner, Group Theory and Its Application to the Quantum Mechan-ics of Atomic Spectra Academic, New York, 1959.

28 V. Heine, Group Theory in Quantum Mechanics Dover, London, 1993.29 E. L. Ivchenko and G. E. Pikus, Superlattices and Other HeterostructuresSpringer, Berlin, 1997.

30 M. Tinkham, Group Theory and Quantum Mechanics McGraw-Hill, NewYork, 1963.



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(14-Bands) Application of Analytical k.p Model With Envelope Function