The envelope representation

Superlattices and Microstructures, Vol. 23, No. 3/4, 1998

The envelope representation Peter J. Price IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, U.S.A.

The question of the nature and calculation of an electron envelope function in semiconductors is revisited. c 1998 Academic Press Limited

Key words: envelope function.

This contribution concerns a question that first arose in the 1950s development of semiconductor physics, about accounting for the spatial variation of electron wavefunctions on a scale larger than the lattice constant. The electron state of a neutral ‘shallow’ donor† was the natural instance. It was understood to be like that of a hydrogen atom with effective mass m ∗ (as in E B = E 0 + h¯ 2 k 2 /2m ∗ for the Bloch energy function E B (k) near the applicable band edge, instead of the free-electron kinetic energy) and with confining potential energy W (r) = −e2 /εr where ε is the dielectric constant of the semiconductor. The questions that this posed were: 1. Why was this so, and what was the wavefunction, actually? 2. How should the deviations of the donor binding energies from the ‘hydrogenic’ Rydberg m ∗ e4 /2h¯ 2 ε2 be accounted for? Since the free electron wavefunctions were the Bloch functions, φ(k, r) = u k (r) exp ik · r where u( ) has the lattice periodicity, it was perceived that the donor wavefunction was something like F(r)u 0 (r) where u 0 is the periodic function at the band edge (k = 0 or k = k0 ), with F(r) to be the ‘wavefunction’ of the hydrogenic model, with a Bohr length constant 2h¯ 2 ε/m ∗ e2 . The latter being large compared with the lattice constant, accordingly F(r)—the ‘envelope function’—was slowly varying on the scale of the lattice constant. What happened in the immediate neighborhood of the donor impurity atom (accounting for the ‘chemical shift’ or binding energy, dependent on the specific donor) was not so clear. More precise development of the actual wavefunction ψ(r) in terms of an envelope function equal to ψ/u 0 leads to a series of correction terms centered on the band edge. There is, however, a more general formulation: We may in any case expand the wavefunction ψ in a Bloch representation X b(k)φ(k, r) (1) ψ(r) = k

† Donor rather than acceptor is meant to single out a non-degenerate band edge of the band-energy function, E B (k), for simplicity at this point.

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(where the φ(r) will be taken as normalized in unit volume, while ψ is normalized in the macroscopic system of volume V and containing N lattice cells† . If ψ(r) is spread over many lattice constants, a correspondingly small range of k values in eqn (1) will be important (and their u( ) functions should accordingly be close although not identical). With ψ varying on a smaller scale as in nanostructures, on the other hand, one may prefer an expansion in a Wannier representation: X (R)w(r − R) (2) ψ(r) = R

where the sum is over the translation vectors of the crystal lattice, and the (spin-independent, real) Wannier function for this set of Bloch functions is X φ(k, r) (3) w(r) = (N V )−1/2 k

with the inverse relation φ(k, r) = (V /N )1/2

X

w(r − R) exp(ik · R)

(4)

R

In formal terms eqns (1) and (2) are of course equivalent. For eqn (2) the Schr¨odinger equation becomes the Koster–Slater equation in terms of the (R), which is modelled by ‘tight binding’ formulations. Equivalent to ψ(r) in (1), and suitable for the present purpose, is the ‘wavefunction’ X ¯ b(k) exp ik · r (5) ψ(r) = k

—providing an Envelope Representation. The lattice-scale variation due to the u(k, r) function is thus removed. ¯ of the actual Schr¨odinger operator H = HB + W (r) for ψ? Let this surrogate What then takes the place, for ψ, ¯ be H such that (6) (ψ¯ 1 | H¯ |ψ¯ 2 ) = (ψ1 |H |ψ2 ). We find ¯ ¯ + H¯ ψ(r) = E B (−i∇)ψ(r)

Z Z

¯ 2) d 3 r1 d 3 r2 K (r; r1 , r2 )W (r1 )ψ(r

The kernel, K ( ), to be discussed below, satisfies Z Z d 3 r1 d 3 r2 K (r; r1 , r2 ) = 1

(7)

(8)

¯ and it is expected to smooth out W and ψ¯ in the neighborhood of r, in (7). The kinetic term, E B (−i∇)ψ, is familiar. It applies, here, anywhere in the zone (for example, for the quantum levels in heterostructures). ¯ With a uniform magnetic field, the When E B (k) = E 0 + (h¯ 2 /2m ∗ )k 2 , it becomes [E 0 − (h¯ 2 /2m ∗ )∇ 2 ]ψ. usual elaboration applies. Near a degenerate band edge, where we have an array of these wavefunctions, it generalizes with the usual ‘off-diagonal’ elements (as does the potential term). ¯ given by eqns (1) and (5), is The relation between ψ and ψ, Z ¯ 0 )d 3 r0 (9) ψ(r) = 0(r, r0 )ψ(r and its inverse ¯ ψ(r) =

Z

ψ(r0 )0(r0 , r)d 3 r0

(10)

† Vagueness here about the domain of the summation in eqn (1) is for simplicity of presentation. The summation could be limited to one symmetry ‘wedge’ of a Brillouin zone, or be over one whole zone, or in more than one band, with ψ ¯ where appropriate, becoming arrays of functions. and ψ,

Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 where

507

1 X φ(k, r1 ) exp −ik · r2 V k

0(r1 , r2 ) =

In terms of the Wannier function given by eqn (3) and its ‘empty lattice’ equivalent X exp ik · r w0 (r) = (N V )−1/2

(11)

(12)

k

eqn (11) becomes 0(r1 , r2 ) =

X

w(r1 − R)w0 (r2 − R).

(13)

R

From the first of the identities

Z 0(r1 , r2 )d 3 r2

=

Z 0(r1 , r2 )d r1 3

=

φ(0, r1 ) R

(14)

(1/V ) φ(0, r)d r 3

¯ ¯ we see that if ψ(r) is substantially constant on the relevant scale in eqn (9) then ψ(r) ≈ ψφ(0, r), as with the traditional Luttinger–Kohn envelope function. However, our freedom in choice of a zero for k makes this approximation applicable to any small region k ≈ k0 of the Brillouin zone where the |b(k)| are appreciable, ¯ in a form ψ(r) ≈ ψφ(k 0 , r). By substitution of eqns (9) and (10) into eqn (6), we find K (r; r1 , r2 ) = 0(r1 , r)0(r1 , r2 ) By eqn (14) we then have

(15)

Z J (r, r1 ) ≡

K (r; r1 , r2 )d 3 r2 = φ(0, r1 )0(r1 , r).

(16)

R Then eqn (11) and the orthogonality property of the φ(k, r) gives J (r, r1 )d 3 r1 = 1, and hence the normalization (8) of K ( ). If ψ¯ is slowly varying, the second term on the right of eqn (7) becomes Z ¯ (17) ≈ ψ(r) J (r, r1 )W (r1 )d 3 r1 in which W (r1 ) is averaged around r accordingly. ¯ The localization of ψ(r) and of W (r) when integrated over K ( ) and 0( ) evidently depend on properties of w( ) and w0 ( ) in eqn (13).† From the identity Z Z (18) w(r)d 3 r = (N V )−1/2 φ(0, r)d 3 r ≈ (V /N )1/2 and w(0) = (N /V )1/2 we have 1 w(0)

Z w(r)d 3 r ≈ V /N ≡ v

(19)

† The actual Wannier function (3), and hence the localization in eqn (13) etc., will depend on the disposable choice of phases represented by a factor exp iθ (k) in the k sums. It was elucidated by Kohn that these phases may be chosen such that w(r) falls off finally at a maximum rate like exp −κr where iκ is the imaginary part of the k distance to the (nearer) singularity of E B (k) connecting to a neighboring band. This does not apply, however, to w0 (r). Obviously, the actual ¯ r) can wavefunction ψ(r) is invariant (apart from multiplication by a constant) under the choice of the θ(k), while ψ( ¯ r). This question invites further vary to the extent that the potential term in eqn (7) deviates from the localized W (r)ψ( investigation.

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where v is here the unit cell volume, so that the effective range of w( ) in integrals should be in the order of the lattice constant. Similarly Z (20) w0 (r)d 3 r = (V /N )1/2 and w0 (0) = (N /V )1/2 , and so 1 w0 (0)

Z w0 (r)d 3 r = V /N ≡ v.

(21)

If W (r) is not slowly varying (even though ψ¯ is), it is replaced by an effective potential, say W¯ (r), equal ¯ The electron state for to the coefficient of ψ¯ in eqn (17), so that the potential term of eqn (7) becomes W¯ ψ. a neutral shallow donor can be expected to deviate from the ‘hydrogenic’ result because (a) for the lattice cells(s) at and near the impurity atom the potential term of eqn (7) from W (r) due to the Coulomb attraction ¯ and (b) there is an additional ‘central cell’ contribution to W due to the is not well approximated by W ψ, differing chemical identity of the donor atom, and in fact some resulting coupling to the states of another band or bands† . Obviously, in this regime, the inner part of the donor electron state may be calculated by means of the Koster–Slater equation for the (R) of eqn (2). This may be joined to the WKB solution of that equation ¯ for the outer part‡ , or to the equivalent ψ(r) by means of the relation ¯ ψ(R) = (N /V )1/2 (R)

(22)

for the ψ¯ values at cell-center locations. With 0 in eqns (9) and (10) considered as an operator, the single-electron density matrix ρ, specified in a homogeneous crystal lattice, transforms to ρ¯ = 0 † ρ0

(23)

in the envelope representation. The usual description in Boltzmann transport theory is, in fact, in terms of this ρ¯ (that is, in terms of the corresponding Wigner function: ρ(k ¯ 0 , k00 ) → f (r, k)). An ‘evanescent wave’ state as in tunneling theory, in particular combining the Bloch states of one band only, can equally well be described by a Koster–Slater version of the WKB formula. The envelope function would then be given by the companion to eqn (2), X ¯ (R)w0 (r − R) (24) ψ(r) = R

¯ The equivalent b(k) function, as in eqn (5), is the ‘momentum wavefunction’ corresponding to this ψ(r). In the Zener tunneling regime, there are two such b(k) functions (or more than two, with band-edge degeneracy), connected by an interband Franz–Keldysh matrix element X (k). They are similarly the ‘momentum’ version of the Envelope Representation. With nanostructures, we no longer may assume a ‘homogeneous crystal lattice’ substrate, with a superposed macroscopic potential and/or weak or locally isolated scattering forces: symmetries are broken in first order. The usual instances are gradients of composition (inhomogeneous alloys), and abrupt changes of composition as in the familiar heterostructure wells and barriers. For graded semiconductors, there is experience that the effect of alloy disorder is not so drastic, so it may as well be set aside as an additional weak effect. If the overall inhomogeneity of composition is not so strong as to generate substantial interband coupling, nor so as to vary the shape of the lattice cell significantly with position, we may propose to keep a uniform set of † In addition, in the ‘shallow donor’ case, the polarizability represented by ε in e2 /εr is not uniformly distributed over the crystal unit cell, and the ion field must vary accordingly over each cell. ‡ Otherwise, the numerical solution of this equation over both parts of its domain is a candidate for Monte Carlo computation.

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Wannier functions w(r, R) = w(r − R)—even though these are no longer rigorously related to stationary states as in eqn (4)—and of course retain the ‘empty lattice’ w0 (r − R) given by eqn (12). We may then define matrix elements Z M(R1 , R2 ) =

w(r − R1 )HB w(r − R2 )d 3 r

of the substrate electron Hamiltonian HB , and hence the Weyl transform XX M(R1 , R2 )w0 (r − Rˆ 12 ) exp iR12 · k E B (r, k) = (V /N )1/2 R1 R2

(25)

(26)

ˆ 12 ≡ (R1 + R2 )/2 and R12 ≡ R1 − R2 . The Schr¨odinger operator for an envelope wave equation may where R then be taken as E B (r, −i∇) + a potential term derived from W (r) as in eqn (7). For an abrupt change of composition at an interface, one can not assume any specific form of the resulting electron potential in the actual boundary region of transition from one uniform crystal potential to the other. The effect of the boundary layer is to convert incident Bloch amplitudes (from either side) linearly into reflected and transmitted ones, so that one may take the matrix of this transformation as the given property, and derive the consequent matching boundary conditions for the envelope function. We may limit consideration to the normal direction only (variables x, k), and consistently with the foregoing assume one Bloch band only on each side of the interface. At energies E that are in the ‘allowed’ range on both sides, the outgoing amplitudes in each direction in terms of the incident amplitudes are given by the 2 × 2 S-matrix, Snm (E). Then, if the components of S do not vary substantially over the E range of states contributing significantly to the two functions ψ¯ n (x), we retrieve the matching formula that is usually assumed for energies near the band edge: ψ¯ 2 d ψ¯ 2 /d x

= U ψ¯ 1 + Pd ψ¯ 1 /d x =

Q ψ¯ 1 + V d ψ¯ 1 /d x

(27)

with coefficients U, V, P, Q of this form of transfer matrix given, after some algebra, by the elements of S. Current conservation requires S to be unitary. The equivalent condition in (27) simplifies if we take P = Q = 0, as is often assumed. Then current conservation requires U V ∗ + U ∗ V = 2(ν1 /ν2 )(k2 /k1 )

(28)

where the νn are the normal-direction electron velocities (from ∂ E B /∂k). This generalizes the common bandedge case U = 1, V = m ∗1 /m ∗2 . When E is in the ‘forbidden’ range for one side or both sides, Bloch amplitudes in these formulas are replaced by their evanescent-wave equivalents, with linear relations that should derive from a continuation of the S-matrix, and again result in a transfer matrix as in eqn (27). Reflection at a forbidden region (electron confinement barrier in a nanostructure) may be characterized by the phase angle, η(E), of reflected Bloch amplitude relative to incident amplitude. Then the appropriate boundary condition (if η is essentially constant over the E range involved) is 1 d ψ¯ = −k tan η/2 (29) ψ¯ d x A multi-band extension of the foregoing would be the hybridization of X and 0 states at AlGaAs interfaces. We see again that the envelope description need not be confined to the energy neighborhood of a band edge associated with an ‘effective mass approximation’. The envelope function of eqn (5) is exact (apart from its representing a projection, here onto a single complete band, from the full domain of the actual wavefunction), while an ‘effective mass’ equation gives an approximate solution of eqn (7), and hence an approximate evaluation of this function. Combining eqns (22) and (24), we see that the envelope function has a finite, not infinite, number of independent values (here, one per lattice cell).

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