On the theory of envelope functions in lattice-matched heterostructures

08 (1YYl) Physica Xorth-tic,liand

5S-66

On the theory heterostructures

Received 5 May 1990 Revised manuscript received

of envelope

Y August

functions

in lattice-matched

IYYO

Starting from the envelope function formalism of Luttinger and Kohn (LK) for a three-dtmensional crystal. exact one-dimensional envelope function equations are derived valid for lattice-matched heterostructures. It is shown that the transition to one-dimensional envelope functions generally leads to more than one one-dimensional envelope function per hand index n, in contradistinction to what is generally assumed. In the important cast of [O 0 11 and [I I I] grown heterostructures it is shown that for parallel wave vectors contained in a region around the I‘-point the number of one-dimensional envelope functions reduces to one, in accordance with the usual applied theory in the literature. Exact equations are derived governing the one-dimensional envelope functions. It is indicated to which form the one-dimensional envelope functions reduce for a perfect bulk crystal. This is of interest in a flat band approach in which such functions arc used to construct wave functions which have appropriate continuity properties cvcrywhcre.

1. Introduction The use of envelope functions in the quantum mechanical description of slightly perturbed crystals has been advocated by Peierls [l], Wannier [2], Slater [3], Luttinger [4], Luttingcr and Kohn [S] and many others. The general idea is to express the solutions $(I-) of the Schriidinger equation for the perturbed system in terms of Bloch solutions of the unperturbed system with amplitudes (envelopes) which vary slowly in space. An often discussed system in this connection is a crystal with a shallow impurity center [6]. Under some general assumptions concerning the slow variation in space of both the perturbing potential and the envelope functions themselves, the above functions appear to satisfy ver!’ simple SchrGdinger-like equations in which the properties of the host crystal arc present in terms of all effective mass only. The advantage of an envelope function formalism is therefore obvious in such cases. The situation is less favorable in systems where the perturbing potential shows relatively large variations in space over short distances. An extreme example of such a case is present in an abrupt heterostructure composed of two dissimilar semiconductors. Here, the potential function exhibits strong variations over a few atomic layers [7]. such that the necessary ingredients for a successful application of the envelope function formalism seem to be absent. Surprisingly. in some noticeable cases. such as quantum wells composed of GaAs and AlGaAs, there are reports [X-13] on apparent succcsscs when working with envelope functions. The main purpose of this article is to reformulate the basic equations of the envelope theory as applied to lattice matched heterostructures. To this end we will first derive in section 2 the ~XUCI Luttinger-Kohn envelope function equations for a general perturbing potential U(r). This is done for a three-dimensional lattice from the outset, contrary to for instance Burt’s treatment [14]. We claim that it is important to take the three-dimensional case as the starting point in view of an interesting subtlety 0921-4S26/Y1/$03.50

0

1991 - Elsevier

Science

Publishers

B.V. (North-Holland)

J. P. Cuypers,

W. van Haeringen

I Envelope functions in lattice-matched heterostructures

59

in the process of obtaining one-dimensional envelope functions which depend on the coordinate z only. This subtlety is met in section 3 where we specialize to perturbing potentials U(x, y, z) in which the dependence on z is general, while there is periodicity of U in the X, y-directions. This situation is met in the class of lattice-matched heterostructures. In most treatments of such systems the separability of the envelope functions in a z-dependent and a X, y-dependent part is loosely introduced and assumed from the outset. We will explicitly show, that strictly speaking more than one z-dependent envelope function may have to be introduced in enveloping a Bloch wave solution I,!J~~,,(~).Here II is a band index, while k, is some wave vector in the first Brillouin zone (1BZ) of the underlying host crystal. We will indicate under which conditions the general practice of using only one z-dependent envelope function can be justified. In section 3 we give the exact equations which the z-dependent envelope functions have to satisfy. Since it is common practice to treat the layers in a heterostructure as being completely bulklike (flat band approximation) [8-131, we also illustrate the application of the envelope function equations to perfect bulk crystals. In section 4 we summarize our main conclusions.

2. The exact Luttinger-Kohn

envelope

function

equations

In this section we will derive the Luttinger-Kohn (LK) equations the non-periodic part of the potential U(r) is slowly varying. We start with the one-electron time independent Schrodinger non-periodic perturbing potential U(r):

w, + flew

without

assuming,

equation

for

as LK did, that a crystal

= ‘%w

with

a

(1)

The Hamiltonian H, contains apart from the kinetic energy term all the periodic parts of the potential; it is assumed to lead to the exact band structure of the pure crystal. This asks generally for potential terms in H,, which are non-local and energy dependent as well (quasi-particle picture). We assume in what follows the non-periodic part of the potential to be local and energy independent. The wave in the complete set of orthornormal functions function $(r) is now expanded, following LK [5],

x,&9 = ei(k-fd.r+nk,(r)

,

system whereh,,W = exp(ik,. W,k,,(r > is the Bloch function solution belonging to the unperturbed for band n and some wave vector k = k, which is not necessarily but usually chosen at the location of a band extremum. In what follows we will choose k, = 0. The wave function is quite generally written as

(3) where the summations run over all band indices n’ and wave vectors k’ in the 1BZ. In all subsequent summations over k the restriction to the 1BZ is assumed unless explicitly mentioned. Substituting (3) in (1) and taking the inner product with the ~,,~(r) one finds

T ;{(~klf&ln’k’) + bkl~b’k’)bLW) Following

LK [5] without

carrying

through,

= EA,(k).

however,

any of their approximations

(4) we may rewrite

(4) in

60

J. f’. Cuypers.

W. van Haeringen

I Envelope

functions

in lattice-matched

heterostructures

the form

En(0)+zk2-E A,(k)++,,, i + c c VB,,. 11’ m

(K,)

c

. kA..(k)

fi(k - k’ + K,)A,.(k’)

= 0,

(5)

k’

where fi( q) is the Fourier given by

coefficient

of U(r),

V the crystal

volume

and the p,,,, and the B,,,.(K,,)

B,,,z.(Km) = (nOle’Kprl‘In’O) .

(7)

K, denote the set of three-dimensional reciprocal direct space representation by introducing envelope F,(r)

= c

A.(k)

arc

lattice vectors. functions

We proceed

by writing

eq.

e’k’r.

(5) in

(8)

k

Multiplication

of eq. (5) with exp(ik.

(E,(O) -g

v2 -

r) and summing

over k gives

z+(r)+2 c p,,,,.VF,,.(r) n’

+ c n’

c

I/B,,.

(Km) 1 d”r’ emiK”“‘U(C)

A(r - r’)F,,.(r’)

= 0

,

(9)

m V

where A(,. _ r’) = $ c

e’k.(r-r’)

(10)

k

The only “approximation” we have made so far is to assume the perturbing energy independent. Within this approximation eqs. (9) are exact. The relation between the envelope functions F,,(r) and the wave function the definition (8) and the expansion (3) and reads

potential

U to be local and

cc/(r) can be obtained

using

(11) The A(r - r’) function occurring function with, however, a finite

I

d3r’ A(r - r’)F,(r’)

= F,(r)

in (10) when integrated over all space gives 1. It is therefore width. One easily verifies the relations

,

a S-like

(12)

V

and

I

V

d3r’ A(r - r’) em’Kfn’r’= SKK,,, .

(13)

J. P. Cuypers,

W. van Haeringen

I Envelope functions in lattice-matched heterostructures

61

It is, however, the occurrence of the function U(r’) in the integral in (10) which strictly forbids to use the function A(r - r’) as the S-function. It makes the set of equations (10) essentially non-local, which may complicate the envelope function approach, especially when considering strongly varying functions u(r).

3. The exact Luttinger-Kohn

envelope

function

equations

for potentials

which are non-periodic

in one

dimension

3.1. Separation of the envelope functions We will now specialize to cases in which the potential function U(r) shows general behavior in one (z-) direction only, while having the periodicity of the host lattice in the perpendicular X, y-directions. In lattice-matched heterostructures to be discussed later on the perturbing potential U(r) is of this kind. Translational symmetry in the X, y-direction will be used to separate the envelope functions F,,(r) in parts depending on p = (x, y) on the one hand and z on the other. Due to the x, y-periodicity of U(r), solutions #l(r) of (1) can be attributed to a two-dimensional wave vector label q,, . Applying Bloch’s theorem we may generally write e(r) = $q,,(r)=

eiqil”uq,,(r),

(14)

with

where R,, is a Bravais lattice vector of the three-dimensional lattice lying in the X, y-plane. The vector q,, may be any two-dimensional wave vector but there is no loss of generality if, from now on, we choose q,, to be within the 1st Brillouin zone of the two-dimensional lattice (lBZ(2D)). Namely, by is in the lBZ(2D) and where G, is some two-dimensional writing 411= qred + G,, where qred reciprocal lattice vector, satisfying exp(iG, . R,,) = 1, this only leads to a redefinition of the function u,,,(r) for which (15) holds as well. The complete set of G, vectors can be obtained by projecting all three-dimensional lattice vectors K,,, on the X, y-plane since exp(iK, . R,,,) = exp(iG, *R,,) = 1, where K, = G, + K,,,,i ,

(16)

f being the unit vector in the z-direction. We now generally express the wave function $,,,(r) as in (3): (17) In view of (14) we observe that only those k’ vectors in the lBZ(3D) can contribute, which can be written in the form k’ = q,, + Gi + kii, Gj being a two-dimensional reciprocal lattice vector. This enables

us to rewrite (17) as

$q,,(r)= exp(iq,, * p)

T

[z(q’$G”

A,,( q,, + G, + kli)

eikJ* eiGl’P]u,P,(r) ,

(18)

where the index q,, + G, in the summation over ki indicates the restriction to values such that (q,, + G, + kii) E lBZ(3D). In view of q,, E lBZ(2D) there is at most a limited number of G, vectors contributing to (18). In the appendix it is shown for the important case of the zincblende lattice, that for [ 1 0 0], [ 1 1 01 and [ 1 1 l] planes the set of G, vectors contains at mosf rwo vectors, one of which is the vector G, = 0. In the [ 1 0 O] and the [ 1 1 l] case there is a restricted q,,-area in the lBZ(2D) such that G, = 0 is the only G,-vector. Such a restricted q,,-area does not exist, however, in the [l 1 0] case. The second G,-vector. if present, depends of course on the particular value of q,, For [n, 1. m] planes with higher values of n. I or m the number of G, vectors contributing to (18) will generally be larger. By defining +<;,I

l4,,

A,,( q,, + G, + kii) elhi’ ,

L.,,,iGI (2) = :

(1’))

and comparing (18) and (11) we thus have to conclude expression for 4q,,(r) read

F,,(r) =

that the envelope

functions

F,,.(r) needed

c f,,, q,,+c;,(z)e”q~~+“““’

in the

(20)

G,

This

implies

that

in the most

general

asks for more than

functions

one envelope

tiq,,(r) = C C L.,,,+,,(z) 11, G,

One-dimensional

3.2.

Since

U(r’) =

C

is periodic

U,,.(z’)

function

of $q,,(~) in terms

per band

index

of z-dependent

envelope

n, or (21)

c”q~+C;op~,,~O(r)

Luttinger-Kohn

the potential

case a description

equations

in the X. y-direction

we may write

e”‘-”

(22)

G’

Substitution

of (22) and (20) in (9 ) and ustng

(6,,(O)+ ;

+; +

(q,, +

G,)‘~

.( c (P,,d

qll +

(16) can straightforwardly

be shown

to lead to

2 $ - +i,.q, +,$4

G,) + ;

P,,,,,,

&)f;,, q,, t,,(z)

z i z V~,,,~.(K_)Idz’l/,“,~~;;,,.(z’)e~’”~.’;’~q,,+.,(z - ~‘)f;,~.,,,+~;;(z’)=(I, (23) I.

where

L is the length

Llq,_,;r(z

-

z’)

=

of the crystal

/_

’

c k:

and

e’k:(; 2”

(24)

J. P. Cuypers,

W. van Haeringen

I Envelope functions in lattice-marched heterostrucrures

63

is a one-dimensional g-like function. The resulting eqs. (23) are coupled equations for the envelope functions f,,,,,+G,(z) with varying II (all bands) and G,. Note that coupling between f-functions with different Gi is induced only through the last term in (23). In conclusion: there are generally more than one envelope functions needed per band index when specializing to z-dependent envelopes, while eqs. (23) form the exact set of equations for them. 3.3. Application to a perfect bulk crystal When considering heterostructures it is common practice in the literature [8-131 to work in the flat band approximation and to use for each material layer of a heterostructure the envelope function equations for the corresponding bulk crystal consisting of material (j). We therefore focus on the application of eqs. (23) to perfect bulk crystals. Let us suppose that the quantities p:,,, , and E!,(O) are all known for the respective materials (j). It will be clear, according to eqs. (23), that the corresponding envelope functions f L,r,,+G,(z) have to fulfill the relatively simple set of equations

(E:(O) +u:,+& (q,,+ Cl’- 2 $i- E)f’,,l,+.,(4 -(q, + Gi) + + PL & > f !C4,,+c,(4 + k c (P;,d n’

= 0 .

(25)

Note that we added an unknown constant Ui in eqs. (25) for each material ( j). These constants have to account for valence band offset parameters between the respective material layers 17,151. In (25) there are no non-local coupling terms. The set of functions {f:.,,,(z)} is furthermore decoupled from the sets {f i,,,,+c, (z)} with G, # 0, though both kinds of functions may, at given energy E, be present in the representation of a general wave function (db,,(r) for an infinite perfect lattice material (j). Solutions f L.,,,+,, (z) of (25) at fixed q,,, Gj and E exist which are of the exponential type. In vector notation they can be written as f:,;‘+,(z) = (A:;:,

A:::, . . . , AA:‘,,

.). eik:“’ ,

(real or complex) ki,“. The various coefficients apart from an overall constant, by solving the determinantal equation of (25). The k,-values to be considered are to be such that q,, + G, + Re(k,)i E lBZ(3D). They belong either to a true Bloch wave (k, real) or a truly evanescent wave (k, complex). Though the above attribution of k,-values to specific (q,, + G,)-vectors is indeed unique, it will be a serious question, whether at a finite number N of energy bands taken into account a reliable determination of k,-values and AL:-coefficients can indeed be carried through. This will ask for a calculation of complex band structures for a variety of q,,-values, [16]. The general solution of (25), at given q,, , Gi and E, is some linear combination of functions fi,s+G,(~) with various real or complex ki’“-values. Boundary conditions at interfaces will put restrictions on the above derived general solutions in the sublayers due to the fact that the function tiq,,(r) of (21) and its derivative are both continuous. where (s) indicates the various solutions with different

AL: at given j, G, and ki_.”can, in principle, be calculated,

3.4. Discussion of the obtained equations We have succeeded in eqs. (23) to obtain the exact coupled set of equations for the z-dependent envelope functions for the important class of lattice-matched heterostructures. Incidentally, in deriving

eqs. (23) we have based ourselves on the general envelope function expression (7), without any further restriction to k-values Ikl 6 2~ria. The allowance of such a further restriction can be judged from cast to case. This restriction has, however, not been carried through in this work. A striking feature of eqs. (23) is the occurrence of non-local terms containing the potential although the equations have been derived starting from a local disturbing potential in the Schrodinger equation (1). This non-locality, due to the A(z - z’)-functions in eqs. (23), has a reach of typically several atomic layers since the amplitude of the 3(z - z’)-functions falls off rather slowly. The non-locality due to U(r) in the envelope function equations in contrast to the local U(r) in the Schrodinger equation seems to give rise to a paradox. However, as we will show below, there is no real paradox. First we have to realize that, in deriving (23), we have started the derivation by using basis functions x,,(r) which are related to one perfect periodic host lattice for the entire system, e.g. either GaAs or AlAs in a GaAsiAlAs based heterostructure. In order then to correctly describe clr,,,(r) in both the GaAs and the AlAs regions in terms of slowly varying envelope functions, one is of course confronted with envelope functions reflecting properties of both materials. More specific, these envelope functions will inhibit among other things GaAs properties in the AlAs regions and vice versa. This mutual influence is formally governed by means of the non-local coupling terms present in eqs. (23). This explains the seeming contradiction between the fact that $q,,(r) is locally connected to U(r) while the envelope functions are not. Due to this non-locality phenomenon it might at first seem not attractive at all to use envelope functions in the description of abrupt heterostructures, as we have to solve a very complicated set of coupled integro-differential equations. An evasion of these difficulties can be accomplished by assuming the layers to be entirely bulklike (the so-called flat band approach). It is then sufficient to determine proper envelope functions for the involved perfect bulk crystals. This leads us to the simplified set of equations (25). Boundary conditions at interfaces will then have to fix the solution valid for the entire heterostructure. 4. Conclusions The introduction of one-dimensional envelope functions in the Luttinger-Kohn scheme for latticcmatched heterostructures asks generally for more than one one-dimensional envelope function per band index n. For the important [0 0 l] and [l 1 I] directions in zincblende based heterostructures this number of functions reduces to one only if the parallel component q,, of the electron wave vc:is)r is taken within a restricted area of the lBZ(2D). In hitherto known GaAsiAlAs based devices the involved q,,-vectors happen to fulfill this condition. Exact equations have been derived for these one-dimensional envelope functions. It has been pointed out that, even for a perfect bulk crystal, the second z-dependent envelope function may be present in the representation of the Schrodinger wave function. The obtained envelope function equations, though exact, are still in need of further simplifications. One of these will be the reduction to a smaller number of bands, most naturally achieved by using Lowdin renormalisation [17]. Furthermore, it remains to be shown that the above mentioned connection rules at interfaces indeed reduce to the generally used simpler forms. These two points will be dealt with in a forthcoming paper. Acknowledgement This work is part of the research which is financially Materie (FOM)“, pelijk Onderzoek (NWO)“.

program of the “Stichting voor supported by the “Nederlandse

Fundamenteel Onderzoek der organisatie voor Wetenschap-

.I. P. Cuypers,

W. van Haeringen

I Envelope functions in lattice-matched heterostructures

65

Appendix In fig. 1 the first Brillouin zone of a fee-lattice is given. When viewed from the [0 0 11, the [l lo] or the [l 1 l] direction, projections of this Brillouin zone may be obtained on the respective planes perpendicular to these directions. The outer contours of these projections are indicated by lBZ(3D) in figs. 2,3 and 4, respectively. Also given in these pictures are the respective two-dimensional Brillouin zones,

Fig. 1. The first Brillouin

zone of a fee-lattice

61

(lBZ(3D)).

B2

Fig. 2. Projected three-dimensional Brillouin zone on a plane perpendicular to the [0 0 II-direction. The outer full drawn lines are the borderlines of the projection of the lBZ(3D) on the [0 0 II-plane. The first Brillouin zone of the 2D lattice (lBZ(2D)) is denoted by broken lines. The vectors B, and B, are basis vectors of the 2D reciprocal lattice. The meaning of the hatched area is explained in the main text of the appendix. lBZI3D)

Fig. 3. As fig. 2. The view is now taken

from the [l 1 1] direction.

lBZ(3Dl --

lBZl2D) 62

-

@ 4

Fig. 4. As fig. 2. The view is now taken

from

the [l lo]

direction.

There

is no hatched

area in this case.

66

J. I’. Cuypws.

W. vun Haeringerl

I Envelope

firnctions

in lattice-matchc4

heterostructures

indicated by lBZ(2D). Basis vectors B, and B,, generating the set of two-dimensional reciprocal lattice vectors are given as well. Finally, in the [0 0 l] and [l 1 11 case a hatched area within the lBZ(2D) is indicated, defined by the property that for all qll within this area, it is impossible to find a two-dimensional reciprocal lattice vector G f 0 such that q,, + G E lBZ(3D). Vectors q,, outside these but within lBZ(2D) are such that precisely one such G-vector exists. This can easily be regions, checked from figs. 2 and 3. Note that there is no such hatched area for the [l 1 0] case in fig. 4. The possibility that both q,, and q,, + G lie in the lBZ(3D) opens the possibility of having two z-dependent (2) per band index 12 (see eq. (19)). envelope functions f,,,,, (2) and f, ,,*,,+c;

References [I ] 121 [3] [4] [S] [6] 171 [Xl [Yl

K. Pcierls. Z. Physik X0 (lY33) 763. G.H. Wannicr, Phys. Rev. 52 (1937) IYI. J.C. Slater. Phys. Rev. 76 (194’)) 15Y2. J.M. Luttinger. Phys. Rev. 84 (1951) 814. J.M. Luttinger and W. Kohn. Phys. Rev. Y7 (IYS5) X6Y. W. Kohn. Solid State Phys. S (lY773 257. C.G. Van de Wallc and R.M. Martin. PhyL. Rev. B 3.5 (19X7) X154. G. Bastard. Phys. Rev. B 24 (LYXL) 56Y3; 25 (lYX2) 7584. M. Altarelli, Phys. Rev. B 2X (I%+) 842. M. Altarelli, Physica B 117 & I I8 (19X3) 747. [IO] M.F.H. Schuurmans and G.W. ‘I Hooft. Phys. Rev. B 31 (19%) X041. [ 111R. Eppenga. M.F.H. Schuurmans and S. Colak. Phys. Rev. B 36 (19X7) 1554 1121 G.A.M. Hurkx and W. van Haeringen, J. Phys. C 18 (1985) 5617. 1131 S.R. White and L.J. Sham. Phys. Rev. Lett. 47 (19X1) 87Y. 1141 M. Burt. Semicond. Sci. Technol. 3 (1988) 73’3. [IS] A. Baldereschi, S. Baroni and R. Resta. Phys. Rev. Lett. 61 (lY,XX) 734. [Ifi] Y. Chap and J.N. Schulman, Phys. Rev. B 25 (1’382) 3Y7.5. 1171 P. Liiwdin. J. Chem. Phys. 14, (IYSI) l3Y6