Kinetic energy operator for strained heterostructures

~ )

Solid State Communications, Vol. 93, No. 1, pp. 97-101, 1995 Elsevier Science Ltd Printed in Great Britain. 0038-1098[95 $9.50 + .00

Pergamon

0038-1098(94)00411-0

KINETIC ENERGY OPERATOR FOR STRAINED HETEROSTRUCTURES A. Brezini International Centre For Theoretical Physics, Trieste, Italy and F. Behilil Laboratoire de Physique Electronique des Solides, Dept. de Physique, USTO B.P. 1505, Oran El'Mnaouer, Algerie

(Received 8 November 1993; in revised form 18 May 1994 by E. Molinari) An exact model calculation for band structure based on the one dimensional Kronig-Penney model including the local potential, induced by the monolayers is developed and compared to the conventional one, although it is formulated to strained superlattices, it may be generalized to any abrupt strained heterostructures.

1. INTRODUCTION

in this context Burt [15] suggested a new-envelopefunction method, derived the corresponding exact envelope-functions equations and managed an effective-mass equation. For a slow grading, EMA may be applied with reasonable satisfaction. In the treatment of abrupt grading, the situation is more confused since the interface between materials introduces new problems: the conduction band edge Eo(z) and effective-mass re(z) become position dependent. Actually, no rigorous derivation exists leading to a unique form of the kinetic operator. Several forms supported by arguments have been proposed [4-6] and most of them may be written as

THE EFFECTIVE-MASS Approximation (EMA) is a useful tool to calculate physical quantities when the desired accuracy did not justify a more sophisticated theory (for a review see [1, 2]). Although originally formulated to treat impurities in an otherwise perfect crystal [3], i.e. weak perturbations such as shallow donors, EMA has been extended recently to applications in semiconductor heterostructures [4-6], or the so-called graded crystals. The EMA which works so well for microstructures has been examined in two ways: the phenomenological approach of the Hamiltonian (or equivalently the boundary condition problem) and the envelope function method [7]. Much of this work has been devoted to the boundary condition problem: namely how to connect the solutions on either side of an atomically abrupt interface [8-13] or equivalently trying to determine the Hamiltonian phenomenologically. The lack of a derivation of the effective mass equations has hampered progress in this field: namely a rigorous derivation that starts from the microscopic Schrfdinger equation as well as providing at each stage an estimation of the error in any approximation made. The origin lies in the heuristic nature of the conventional envelope-function method [14]. Recently,

Hkin = ¼[m'~pm~m~ + m~pm~pm'~],

(1)

as suggested by Von Roos [5], with a +/3 + 3' = - 1 and p is the momentum operator. The basic requirement of this kinetic operator is the hermiticity. The question of the possible value of the parameters a, /3 and 3' is still controversial. However, it has been demonstrated recently [16] that a and -y are equal unless divergences of the energy eigenvalues for the abrupt heterojunctions occur and the case a ~ 7 corresponds physically to the presence of an impenetrable barrier [17]. Therefore, the effective-mass Hamiltonian transforms into 97

98

ENERGY OPERATOR FOR STRAINED HETEROSTRUCTURES

a single term Hamiltonian H = ½m'~pm~pm ~ + E c + V(z),

the strict sense since even/3 = 0 fits well experimental data on GaAs/AlxGal _xAs and Si-Ge heterostructures [18]. Therefore, the problem of the correct effective-mass Hamiltonian has still to be achieved. Up to now, most of theoretical models have neglected the mismatch between the lattice parameters of the two materials. Thus, it turns out to be an attracting goal to deal with strained heterostructures in order to generalize the kinetic operator via:

(2)

where V ( z ) denotes the potential. Morrow [17] analyzed with details different models within the framework of EMA. Indeed two classes of parameters have been predicted = 0

and

/3 = -1,

(3)

[16, 18-24] and c~ = - ½

and

/./kin = I

/3 = 0,

(4)

maaapmBa-2Apm,~aa,

2. EXACT MODEL We consider an abrupt semiconductor heterostructure constituted of two alternating materials A and B. In the growth direction axis, the widths of the material A ( B ) regions are nAa~ (nnaa) where aA(aB) are the lattice parameters in the material A (or B) and n,~(ns) are integers describing the number of monolayers in each material. The problem may be handled analytically by simulating each of the regions by a one-dimensional Kronig-Penney lattice. For the first heterostructure unit cell [0, nAa A + nnan], the potential may be written

I VA - VA ~gtA6(Z -- (n -- 21-)aA) nB

f o r 0 < ,~ <

nAaA,

"= J

ve - ,,e E n=l

wells argued for /3 = - 1 to be the best candidate. Recently, within the framework of the effective-mass theory Fu et al. [26] have derived the subband structures of several GaAs/Alx-Gal_xAs multiple quantum-well samples. When they fit their calculated intersubband transition energies to optical data by adjusting the parameter/3 and the conduction-band offset coefficient Q they found that the fit is insensitive to/3; furthermore these results appear to be dependent on the value of Q. In fact the value/3 = -1 cannot be conclusive in

(5)

where a is the lattice parameter. The purpose of this paper is to propose a possible theoretical approach in determining the numerical values of a, fl and A. Towards this end we compare an exact solvable model with EMA results parametrized by a, fl and A.

[6, 25]. Thus, literature provides various attempts to determine the correct symmetrization procedure, including an elegant method based on exact solvable models as support. However, the limitation of such approach are that they assume the existence of a unique symmetrization procedure, such as the effective potential energy can be represented by the band edge Ec. Furthermore, the problem of the relation of the envelope function to the wavefunction is left open. In such situations for which there is no universal agreement, one has to turn to experiments to determine the appropriate values of and/3. In this respect, valuable information may be obtained through the optical absorption, the luminescence and the photoreflectance measurements. In this spirit, Galbraith et al. [22] from a comparison of effective-mass results with photoluminescence excitation spectra from GaAs/A10.95Ga0.65As quantum

V(z) =

Vol. 93, No. 1

-

(n -

(6)

for n4a A < z < n.4aA + nBaB ,

where z is the growth axis, 6 the delta function and v A the strength of the delta function respectively. Here Vi is a constant potential in region i and vi is given by 2 t t h ai/rnoai where a/ measures the strength of the 6function in each region and m0 the free electron mass; h is Planck's constant. Using a standard transfer matrix for the wave function and its derivative, we get the total matrix for a unit cell r =

(nB)(kB)Ts

(hA)(kA).

T)4

(7)

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ENERGY OPERATOR FOR STRAINED HETEROSTRUCTURES

Here T)"3(ki) denotes the transfer matrix in the region i and is given by

Oti .

I

T(n')i (ki) =

cos ki - ~ ! sm ki

sin ki k;

-k i sin k i - ~(1 + cos ki)

99

where E/(z) denotes the bottom ofthe local conduction band at position z and m i the effective-mass respectively.

a_~'. _ .] k2 (1, cos k,)/, (8)

cos ki - kii sin ki

with

and the interface transfer matrix between material A and material B is

For the abrupt heterojunction the electron wavefuncfion ~b(z) must satisfy the continuity of m'~aX~(z) and m~a-2;~pm'~a~(z) at the interface. As done previously by transfer matrix method and setting c~'i = 0, since in the same material i the potential is constant, T[(qi) transforms as

rq =

T/=

h2 2moa2i k 2 = E - Vi,

[; o]

(9)

.

(10)

ai/a:

[ COS(n,qi) |

sin (niqi) ]

I1_--qi sin (rtiqi)Oj

cos (niqi)

The allowed energy bands in the heterostructure correspond to complex eigenvalues of the transfer matrix T which implies the conditon

where

ItrTI _< 2.

2m~a 2 q~ = E - E~"

(11)

Diagonalization of the transfer matrix T{ n') and direct product of equation (7) yields

qi

(18)

(19)

The matching conditions at the interface yields a transfer matrix of the form

]tr T I = 12 cos (n,,64) cos (n,~6B) - OAOBfAB sin (n~/54) sin (nBSn)1,

(12)

T/)(a, ~ ) (=m J\mi] ')a[10

(~iii¢ai/a]]

,

(20)

where anwn sin (64) a,twA sin (6B) f , tB = aAW4 sin (84) "~ aBwB sin (64) ,

(13)

T ' = T i n ( a , 13)T~(qs)T~4(o6/3)TJ(q4),

I ai sin ki , 0"< cos (63 = I cos k~ - T, - 6 -< rr

As seen before the total transfer matrix over a unit cell of the heterostructure is

(14)

(21)

for which the trace is given by tr T' = 2 cos (n4qA) cos (nBqB)

sin ki

wi -- ki

a'i (1 - cos ki),

-~i

--fAB sin (n.aq4) sin (nsqs) ,

(15)

(22)

where

and 0i=sgn

coski-ai

.

(16)

Therefore, the band structure of the structure is described by bands of energy for which the trace (12) does not exceed two.

fAB

\anJ

qs \ r o B /

\a~]

q4 \ r a n ]

" (23)

This last step enables one to look at the equivalence between EMA results for the allowed energies obeying to the condition Itr T'] < 2 and the exact model calculations given by equation (11).

3. EFFECTIVE-MASS A P P R O X I M A T I O N For heterostructures, the conduction-band envelope function ~(z) satisfies the effective-mass Schrfdinger equation [ i1m i ~a iAp m ifla i-2Ap m ai a A i + Eic(z)]~b(z) = E3b(z),

(17)

4. DISCUSSION In the exact model calculations the eigenvalues of the matrix Ti are 4-exp (+ iSi) while for EMA case there are exp (4- iqi). Therefore, one can conclude the consistency between the two approaches if the

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E N E R G Y OPERATOR FOR STRAINED HETEROSTRUCTURES

equality 6i = [qi[ is fulfilled. Such condition may be reached only for energies near a band edge, in this limit the energy may be written

Vol. 93, No. 1

while for even-n bands 3 = - 1

and

A = - 1,

(34)

in the asymptotic limit of vanishing band offset. Obviously, one must be careful in comparing zmoa i conclusions obtained from one-dimensional systems where n is the index for the band. We recall that for with realistic three dimensional systems. However, in the Kronig-Penney lattice the band minima are most cases even such over simplified models have some bearing in describing at least qualitative insights. k[, = n ~, ( n = 1,2,...). Expansion of the E - k In this spirit, the present model "predict" the domain dispersion relation near the band minimum k = k C of validity in treating strained heterostructures by gives the effective-mass means effective-mass calculations. .', Let us consider a typical strained heterostructure m i = m 0 n27r 2 . (25) constituted by well material of lattice parameter aa Thus, near a conduction band edge the energy (24) embedded between two barrier materials with lattice goes to the limit parameter as < aA. The limit of interest is the widths less than the critical layer thickness in such way that a hE i,n _ _ 62. (26) dislocation-free strained layer will be energetically E = E~ + 2moa2i stable [27]. Under these circumstances, all the strain On the other hand for EMA the energy is will be included in the well layer which is forced into the tighter layers in the two in-plane directions and h2 Ei, n 62 (27) then will relax along the growth direction. E= c "t- ~--SS'-~2 i . •~ m o a i Thus, the net strain in plane is Thus, as the energy goes to the conduction band edge (as-a,~ a.ss_ 1, (35) the condition for consistency is reached (6,. = Iqil). Taking into account these considerations, w; becomes alL= k aA / = aA -2mi/m o for n odd (28) and in the growth direction

E = Vi +

k2i = E i'n +

(k 2 - n27r2), (24)

,

~vi =

(mo62i)/(27r2n2mi) for n even

a± -

and equation (11) reads 12 cos

(nA6A) cos (ns6s)

--fA(~) sin(nA6A) sin (ns6s) [ < 2,

(29)

for n even and where

fA(t, S-----

(a-~A)2;~+16a(m-~a) ~

q

a,,t~2~+16s(mA) as,] 6,4 -~s ' (30)

and 12 COS(nA6A) COS(ns6s) _¢(2) J A B sin(nA6A) sin (ns6s) I < 2,

(31)

for n odd and where fA(2)

(aA)Z;~+I6A

(ms)

1,

(36)

aA

(as'~2:~+16s (mA) (32)

These couple of equations have to be compared with the effective-mass condition (22). As main result for odd-n bands the agreement can only be fulfilled for 3 = - 1 and A = 0, (33)

where aJ stands for the new lattice parameter in the growth direction. Typically for tetrahedral semiconductors a± ~ -all since aA > as in our situation this implies in turns a] > aA > as. The difference aA x -- a8 is approximately twice the difference in pure materials. Within the framework of the effective-mass theory, the strained layer-lattice parameter to be involved in the boundary conditions at the interfaces is probably the lattice parameter aJ and not aA, i.e. the lattice parameter of the pure material. This feature enhances the sensitivity of the choice of boundary conditions. The question of interest concerns the appropriate value of 6 for realistic systems. The Bitch states at conduction band edges in standard semiconductor crystals exhibit an anti-bonding nature [28]. The wave function presents a mode between adjacent atoms and appears in that respect similar to the n = 2 case for Kronig-Penney lattices. Indeed, the value of " 6 " is governed by the behavior of the band-edge wave function at the interface. This suggests that the even-n boundary condition, i.e. 3 = - 1 , 6 = - 1 , appear relevant for conduction band edge states in strained

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ENERGY OPERATOR FOR STRAINED HETEROSTRUCTURES

superlattices where both conduction-band minima in question are of the same type, namely InxGa~ _xAs/ GaAs systems. In general, a simple one-band effective mass is not suitable in describing valence-band states; a multiband approach is usually required. However, for significant particular cases the set of equations reduces to the simple effective-mass version for the hole states [21]. In particular, valence-band wave functions at the zone center present a bonding character [28] without a mode between neighbouring atoms and then behaves like the n = 1 case for the Kronig-Penney lattices suggesting that the odd-n boundary conditions, fl = - I and 6 = 0 should be used in treating the hole states by the effective-mass theory.

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