Mesoscopic Description of Heterojunctions

HAL Id: jpa-00247251

https://hal.archives-ouvertes.fr/jpa-00247251

Submitted on 1 Jan 1996

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Mesoscopic Description of Heterojunctions

R. Balian, D. Bessis, G. Mezincescu

To cite this version:

R. Balian, D. Bessis, G. Mezincescu. Mesoscopic Description of Heterojunctions. Journal de Physique

I, EDP Sciences, 1996, 6 (10), pp.1377-1389. �10.1051/jp1:1996142�. �jpa-00247251�

Mesoscopic Description of Heterojunctions

R. Balian

(*),

^{D. Bessis and G-A- Mezincescu}

^(**)

CEA

/DSM/Service

de Physique Théorique, CE Saclay, 91191 Gif-sur-Yvette Cedex, France

(Received

9 February1996, accepted 18 June 1996)

PACS.73.40.Lq ^Other semiconductor-to-semiconductor contacts, p-n junctions, and heterojunctions

PACS.71,10.-w Theories and models of _many electron systems

PACS.73.20.Dx Electron states in low-dimensional structures

(superlattices,

quantum

well structures and

multilayers)

Abstract. Starting from the microscopic current associated with the single-electron _wave

functions and using Wannier functions suitably

generalized

for nonperiodic heterostructures _we

construct mesoscopic

(either

discrete _or

continuous)

conserved currents associated with states

having components only in the considered band. This provides _a rigorous ^basis to the _ap- proximate envelope function approaches. As _an application we analyze general one-dimensional heterojunctions, mapping explicitly ^the microscopic asymptotic states enta mesoscopic

(one-

band

envelope)

_ones. This proves that, apart from the effective _masses and the band offsets, the connection rules _are characterized in general by three parameters, ^{the values of which} are

unconstrained, confirming the results of phenomenological analyses.

1. Introduction

The recent

developments

in trie

technology

of nanostructures

require

trie calculation of trie

electronic

properties

of devices which involve either

slowly varying compositions (graded

lat-

tices)

_or

sharp

^{interfaces between dilferent materials,}

especially

semiconductors.

Using directly

the

single-electron microscopic Schrôdinger equation,

with _a

potential

which is

periodic

_in each homogeneous piece ^of

crystal,

would be

uselessly complicated.

Current

approaches

get ^{nid of} the oscillations that _occur in the _wave function at the atomic

scale, by replacing

^{the micro-}

scopic

_wave function with _a smoothed _mesoscopic effective _wave function. The

simplest

such

approach

is the effective-mass

approximation.

^As

onginally

introduced

iii

it is based _on the

assumption

that within each

piece

^of material the energy of the electron lies close to _a band

edge,

in which _case the

microscopic

wave function is the

product

of the Bloch _wave associated

with this band

edge

and _a

slowly

_varying ^continuous

envelope

function

(for

reviews and refer-

ences, ^see [2,3] ^For theoretical _purposes we shall make _use of _{a more}

general procedure,

valid

throughout

^{the relevant}

^band,

^and

^relying

on a discrete mesoscopic wave function. This func- tion is here defined _as the set of coefficients in the expansion of the

microscopic

wave function

(*j

Author for correspondence

(e-mail: zaf@spht.saclay.cea.fr)

(**) Present and perrJanent ^{address: institutul de Fizica} ài

Tehnologia

Matenalelor, C.P.

MG-1,

76900

Bucureàti, Màgurele, România

@ Les

Éditions

_{de Physique 1996}

on the Wannier basis of the

band,

at least for each piece ^of

homogeneous

_material

[4,5].

The continuous

envelope

function then _{appears as a natural}

interpolation

of this _discrete function.

The problem then arises to build the effective equations which _govern trie

mesoscopic,

dis-

crete or

continuous,

_wave function. This bas been achieved for

crystals

with _a

slowly varying

composition

[6,7].

It _was shown _in that _case that trie discrete _wave function _is trie solution of

an effective

Schrôdinger

equation. It _was also

shown,

in trie special _case of _a

graded

semicon- ductor with _a

non-degenerate band,

when trie electronic _energy lies _near trie local band

edge E(r),

that trie continuous

envelope

function is governed

by

_an effective Hamiltonian of trie form

H~R ~2

=

-~V

~

~V+E(rj, ^~l.lj

where

1/m"(r)

is trie local inverse effective _{mass tensor}

(reducing

_{to a} scalar for _a

locally isotropic band).

If _a

slowly varying

externat potential is

applied,

it should

simply

be added to

(1.1).

As noted

above,

trie

hypothesis

of _a slow _variation for trie composition of trie material is violated in _{a case} of practical interest. Across _a

heterojunction,

trie composition bas _a

jump

over a

length

scale comparable with trie lattice size,

especially

when interfaces _are built

by growing

a

crystal layer by layer.

In trie

slowly varying

_case trie existence of

generalized

Wannier

functions bas been _proven

ii,

_8].

Assuming

that _{one can} define

generalized

localized Wannier functions in trie _presence of trie

interface,

which _seems

plausible

but to _our

knowledge

bas been achieved

only

in one-dimensional _cases [9,

loi,

_{one can} still _use trie continuous

envelope description

wherever trie

corresponding

^discrete

envelope

coefficients _are

slowly varying.

For any reasonable

generalization

of trie Wannier functions _m trie _presence of _a

sharp

interface

between

homogeneous media,

these _will tend to the Wannier functions of the

corresponding homogeneous

material

sufficiently

far from the interface.

Here,

trie _evanescent

(non spectral)

solutions which _are needed for

matching

trie

microscopic

_wave function and its normal derivative

across trie

junction

_{in two or more space dimensions will bave}

decayed.

Trie microscopic

potential

is

(locally) periodic

and trie

heterojunction

_can be felt _{at most}

through

trie

slowly

varying potential of _an eventual space

charge

associated with trie interface.

Thus, if trie _two media separated

by

the

sharp

interface _are

homogeneous,

_an electronic

wave function _within each _{one is}

simply

a superposition of Bloch _waves. More

generally,

if trie two media bave

slowly varying compositions,

trie

envelope

function _in each'of them will

satisfy

a

Schrôdinger equation

with _an effective Hamiltonian _{of trie} form

(1.1). However,

trie form of trie effective equation in trie slab

containing

trie interface is still _an open problem.

Even trie _case when trie slab's thickness is

microscopic (comparable

_to trie

spatial

extent of trie Wannier

functions),

_so that _we bave

only

to match

envelope

functions _{over trie}

interface,

is not yet

completely

settled and bas been

intensely

discussed

(see,

for instance

[2,3]) By

using without

justification

trie effective Hamiltonian

il.l

_across the

heterojunction,

_one

might

infer (1) that the

envelope

function F must be continuous

(since

otherwise trie product m"~~ _{VF of}

a step ^function

by

_a à-function would be

meaningless);

and

iii)

that m"~~ _{VF must} take trie _same values _on both

stries,

which

yields

_a well-defined

jump

_m trie normal derivative of F

(as

seen

by integrating

trie effective

Schrôdinger

equation _{over an} mfinitesimal segment

lying

across trie

interface).

Trie

heterojunction

would then be characterized

only by

trie materai

constants

m",

E of trie _two

homogeneous

media.

However,

it is easy ^to see that this procedure is

certainly

incorrect.

In

slowly graded materials,

trie

validity

conditions for trie form

il.l)

of trie effective Hamil- tonian _are not violated if trie kinetic term is

replaced by

jm"°vm"-~°-ivm"°;

/~2

11.2)

trie value of _{a is} immaterial

ii, iii

_as

long

_as trie variations _are

sulliciently

slow.

Nevertheless,

a blunt _use of

il.2)

_{across a}

sharp junction

would

provide

_as

boundary

conditions: _{(1) trie con-}

tinuity

of m"°F, and

iii)

trie

continuity

of m"~~°~~Vm"°F.

Obviously,

since these conditions

depend

_{on a,} the results which holà for

slowly graded

materials cannot be extended to

sharp

interfaces. A separate

study

^of the

boundary

conditions at the interface is needed. There have been attempts to _use the irrelevant in the

slowly graded

case parameter _a ^for

describing

the

boundary

conditions. A dilferent _answer has

recently

been

given

on

phenomenological grounds il ii

in the _case of unstrained

interfaces,

for which the

crystal

_groups ^{of both}

homoge-

neous media have in _common a

subgroup

of two-dimensional discrete translations

parallel

to the

interface. It bas been shown that the

general matching

conditions for F and VF then

depend

on a Hermitean 2 _x 2 matrix associated with the energy of the

sharp junction.

Trie interface is thus characterized

by

four real parameters

(three

in the _case of time-reversai

invariance).

The 2 _x 2 transfer matrix

describing

this connection rule has the property ^of

being

the

product

of _a real unimodular matrix and _a pure

phase

factor. The latter property ^{had earlier been noticed}

to

imply

current conservation at the

junction

_[12]. Let us also mention _a dilferent

approach involving

a non-conventional type ^of

envelope

functions, which _are continuous

together

^with

their

gradient

at an interface [3], ⁱⁿ contradiction to the results below.

We return below to this

problem, focusing again

on unstrained

junctions

but

relying

_now

on the

microscopic theory.

In Section 3, we show

by analyzing

^the

general

one-dimensional

equations

that the

envelope

function and its normal derivative _are discontinuous across a hetero-

junction.

^{Their values} on both stries _are related

by

_means of _a transfer matrix which represents

an intrinsic property ^{of the}

junction (Eqs. (3.14, 3.21)

for discrete

envelope

functions,

(3.22)

and

(3.24)

for their continuous

interpolation).

In

particular,

for _a

heterojunction

between two semiconductors _near a band

edge,

the transfer matrix _is net determined

by

the effective _masses

on both sides and the

jump

in the band

edge.

Whereas the

knowledge

of the local effective

mass m"

jr)

and the local band

edge E(r)

is sullicient to characterize _a

slowly graded

medium, the presence of

sharp

^{interfaces introduces additional} parameters. ^{It should be noted that trie}

envelope

functions

satisfy

_a

Schrôdinger

equation with the effective Hamiltonian

(1.1) iocaiiy

on either side of the

junction,

but not

giobaiiy. Indeed,

_we shall _see

by

the end of Section 3 that the

matching

conditions described

by

the transfer matrix cannot be

represented by adding

to a Hamiltonian of the form

(1.1)

_a

potential

which would be _a distribution localized at the interface. Nor _can

they

be

represented by

_a

single

parameter o

entering

_a unique Hamiltonian

as m

(1.2).

We shall find _no other restriction _on the transfer matrix than the constraints

imposed by

current conservation

(or self-adjointness

^{of the} Hamiltonian from which the

probability

current conservation stems in quantum

mechanics) and,

in the absence of

magnetic ^fields,

time-reversai invariance. The transfer matrix that _we obtain

by mapping

the exact

microscopic

solution onto the

envelope description

has the form and number of parameters

predicted

in

il ii by adding

_a

phenomenological

contact energy ^to the kinetic _energy.

Trie

replacement,

^{useful in}

practice,

of the

microscopic

_wave ^function

requires

^of course that the

physical macroscopic quantities

which _are calculated

using

the latter coincide with trie

ones obtained from the former. This

correspondence

^{within reasonable estimable} errors

is inherent to the construction of the

envelope approximation

as

regards

the energy

eigenvalues

and scattering ^data. When

calculating

^transition

probabilities

under the influence of electro-

magnetic

^fields

by

means of the

envelope description

we assume

implicitly

that the

probability (or charge)

density ^and current obtained from the microscopic and

mesoscopic approaches

also coincide. As _a

preliminary

step, we therefore _express, in Section 2, within _a

general framework,

the conservation of

probability

at the

mesoscopic level,

for

homogeneous crystals

as well _as for

Don.homogeneous

^{materials. This will allow} us to define _{a current}

density

which is associated

with the

mesoscopic,

discrete _or continuous, _wave function, and which will

play

_a central rôle in the discussion of Section 3.

2.

Mesoscopic

Current

Let _us first wnte, ⁱⁿ a uniform

crystal,

the correspondence between the

microscopic

and the mesoscopic

description

for electrons

belonging

to a

single

band. We shall treat the electrons _as

independent partiales

and shall

disregard impurity

and

space-charge

elfects. If the considered band is

non-degenerate,

the Wannier functions

wj(r)

_are deduced from _one another

through

the lattice

translations,

and each _one is locahzed _around

a lattice site

Ri,

where denotes

a set of three

integers;

_we

drop

the band index

throughout.

For _a

degenerate band,

the energy

spectrum _ek» and the Wannier functions

wj~(r)

involve _in addition the branch index _{/t or u,} and these functions _are related to _one another

through

operations of the lattice _group.

They

are

expressed

in _terms of the Bloch _waves

~fik»(r)

as

where _~J denotes trie volume of trie Brillouin _{zone over} which k is

integrated.

Time-reversai invariance

implies

_ek»

" e-k», 1b(~ ^"

il-kV-

A suitable choice of trie

unitary

matrix

U(k)

makes trie Wannier functions

subject

to trie symmetries of trie crystal _group, real and localized

in as much _as

possible

around

Rj~,

within _a range of trie order of _a few lattice cells. Trie

center Rj~ _may

depend

_{on u,} for _instance for Wannier functions located around lattice bonds

and symmetric _or

antisymmetric

with respect to Rj~

[13,14].

A

microscopic

_wave function in trie band bas trie form

lbir, t)

₌

~j 4livit)Wivir)

,

(2.2j

and its components 4liv can be

regarded

_{as a} discrete mesoscopic _wave

function,

defined _{on trie}

sites Riv rather than _on trie fuit space r

[î].

Trie

Schrôdinger

equation for trie set 4l reads

~hj4~iv

⁼

~ (ÎU)H)Î'U')

Ah,v,,

(2.3)

1,v,

where trie matrix elements

(ÎUÎHÎfU~)

"

~j

~~~

/

_dk

e~~'~~i~~~"w')U~~ (k)ek»U],~(k) (2.4)

~ Bz

of trie Hamiltonian bave trie _same range in

)Riv

Ri,

v, as w

The

probability

_)4livÎ~ satisfies the conservation law

fi

^)4liv)~

~j ImÎ4l(~ (iu)H)i'u')

4lj,~,

= 0

(2 5j

i,v>

~ '

where each _term

in the _{sum can} be

mterpreted

_{as a current} of

probability,

which flows from Ri~

to _a site Ri,v,

lying

at a distance that does not exceed À. Since the mesoscopic wave function

is defined _at the points Ri~, it is natural to introduce _a discrete mesoscopic

density P(r)

₊

~j

_14~ivl~à

_{(r Riv)}

12.6)

located at these

sites,

and _a

mesoscopic

current

density

j(r)

_e

~j ilt~~4l(~ (lu

H

i'u)

_Ah>v,

_(ILi,v, ILiv) /~ ^d(

^à

^{jr (ILiv Il Î)Ri,v,) (2.7)}

1v1'v' °

concentrated

along

the links which connect

neighbouring

sites. The conservation law

()

+ div J ₌ 0

(2.8)

is then _a consequence of

(2.5)

and of the

identity

à(r

IL) ^à

(r IL')

₌

/ ^~ _d(

^~ _ô

_{(r (IL}

₍₁

_()IL')

dl

=

jR' R)

Vr

/~ ^dl

^à

^{jr jR il j)R') (2.9j}

o

The

mesoscopic

relation

(2.8)

is not

directly

^related to the conservation law

ôpmi~/ôt

+ div Jmi~ ₌ 0 between the

microscopic density

_{pmj~ =} ~fi"~fi and the

microscopic

current

density

Jmj~ ₌ Im hm~~~fi"i7~fi. However, the microscopic

description

and the less detailed

mesoscopic description

become

equivalent

at the

macroscopic

level, when ail

quantities

_are ^smoothed

by

means of _a convolution with _a function which varies

slowly

_on the scale À. In

fact,

_pm~~ and _p

yield

the _same result after such _a

smoothing;

this

equivalence

holds for the current densities Jm,~ and J, at least for

non-degenerate

bands

(a simple proof

_uses the

identity lt~m~~

ⁱ⁷ =

ix, Hi

and the translational

invanance).

The above definitions contain _some arbitranness. On the _one

hand, depending

on the

crystal

group, the Wannier functions _are not

always

defined

umquely

_[14]. On the other

hand,

the discrete nature of trie _wave function 4liv ^bas

suggested

trie definition of the localized

density (2.6).

We

might alternatively

have

spread

the

probability

_)4livÎ~ over some

region surrounding ILiv,

for instance

spread

^it

uniformly

over the

corresponding

Voronoi cell. We

might

also have

associated with the discrete set 4liv the smooth

interpolating

function

Fuir)

₌

~

x

jr Jtiv)

4liv,

12.10)

which coincides with 4li~ at the lattice sites within _a normalization

factor,

and which has _no Fourier component ^{outside the Brillouin} zone. This

requires

to take _as a

weight

x(r)

e

((2r)~~J]

^~~~~

/

_{dk e~~'~,}

_(2.Il)

BZ

which is real and satisfies

j

_dr

x

jr Riv)

_x

(r Ri'v)

" ôi _i' "

ix lRiv Ri'v) 12.12)

u~

This continuons

mesoscopic representation

_maps a pure Bloch _wave

~fik»(r)

^{onto trie}

plane

wave

F)"(r)

₌

(2r)~~/~U~~(k)e~k'~

^{With the} mesoscopic density

p(r)

₌

£~ )F~(r))~

we can

associate _a conserved current

J(r)

=

~j ih~~ /

_an

_{dr[F] (ri}

x

(ri

_ILiv

(lu

H

i'v')

_x

(r[

_{ILi,~, F~,}

(r[

ivi,v,

x

(ri

_ri

/ _dl

à

(r tri Ii f)rl)

o

~

=

~j _h~~(2r)~~ /

ds

/

_dk

_{/~ d(}

_e~~~'~

~~, _~ BZ 0

xF](r il ()s)F~,(r

+

(s)i7k (U~~(k)ek»U],~(k)]

,

(2.13)

constructed

by using (2.3), (2.9)

and

(2.10)-(2.12). Although

several

dilferent,

discrete _or

continuons,

_mesoscopic representations _can thus be

defined, they

ail lead _{to a} conservation law

(2.8),

with _a mesoscopic current

density,

such _as

(2.7)

_or

(2.12),

which is

macroscopically

equivalent

to

Jmi~.

Whatever trie choice trie mesoscopic

description

_requires

only

trie informa- tion about trie _set 4livi while trie

microscopic

_current

density

_Jm~~ involves interband matrix

elements _{since i7~fi is not}

a superposition of Bloch _waves of trie considered

band,

_a

_mesoscopic

current is defined _{within this} band

only.

Trie existence of _a

mesoscopic

conserved current becomes useful when _{we turn to} non-uniform crystals. Consider first _a

graded

material with _a slow variation _on trie scale À. This _case has been

elucidated I?i

by

_means of _a representation

(2.2)

in _terms of

generalized

Wannier functions.

These _{are still} localized _{in a}

range of _a few lattice constants, ^but are no

longer

deduced from _one another

by translations,

_except

approximately

for

neighbouring

sites. The discrete conservation law

(2.6)-(2.8)

remains

valid,

_since the equation

(2.3)

_on which it is based does _not

rely

_on trie existence of _{any order}

in trie material. Trie

only

requirements _are trie localization of trie functions _{wiv, and trie} fact that these functions

wiv are

solely

connected to _one another

by

trie Hamiltonian

H,

_1-e-,

they

_{span an} invariant

subspace

of H. Trie alternative representation

(2.10)-(2.13)

bas _{a more} limited _range of

vahdity,

due _to trie

long

tait of trie

smoothing

function

(2.Il)

and to trie

underlying

periodic structure. It

froids, however,

in

asymptotically

uniform

regions.

We _{now turn to}

sharp

interfaces. Trie existence of

generalized

Wannier functions

satisfying (2.2), (2.3)

and

having

ail

a short range is no

longer granted although

it is

suggested by

_some

works

[9,10]. Nevertheless,

trie above

analysis

still froids far

enough

from trie

junction,

with two dilferent _sets of localized Wannier

functions,

relevant _to either side. Both trie

microscopic

and the mesoscopic currents _are conserved _across trie interface. Trie values of 4lj~ at

large enough

distances from trie interface should therefore be related in such _a way that trie smoothed asymptotic values of

(2.7)

_or

(2.13)

_are trie _same

on both stries for

stationary

states. We shall check this property in trie _{next section.} While trie

mesoscopic dynamics

_is

governed by

equations of trie form

(2.3), (2.4)

at _a sullicient distance from trie

interface,

_we shall _write trie _{connection rules}

across it, and shall _see that

they obey

_no other constraint than trie time-

reversal invariance and trie conservation of trie

mesoscopic

current

(2.7)

_or

(2.13).

3. One-Dimensional

Heterojunction

From _{now on, we} restrict ourselves to _a one-dimensional system. This is

physically

relevant for actual

heterojunctions

when the whole lattice structure is invariant under discrete translations

along

two directions

parallel

to trie interface. In trie asymptotic

regions

_x

- +cc, we have

two band structures, characterized

by

lattice constants

a+,

band energies

et,

_Bloch

waves

~fi)(x)

and Brilloum _zones )k) < na+ _{Since the}

heterojunction

_is

sharp,

_{we may assume} that

electrons move in _a

potential

^{which is}

periodic

in _a

region

_x > b+ and in _a

region

x <

b~,

and that the interface

region

b~ < x < b+ has _a width of _a few lattice cells. The Wannier functions

w/(x)

_are localized within _a distance of order around the sites

Xj+

e b+ ₊

ia+,

_>

Àla+,

and the functions

w/ (xl

around

Xl

e b~ + ia~, <

-Àla~

A

microscopic

wave function

~fi(x,

t)

has _on both stries the form

£~~~ 4l/(t)w/ ix)

,

x > b+ ₊

,

i~jX,t)

"

13'i)

£j~~ 4l/(t)w[(x)

,

x < b~

,

where 4l satisfies the _mesoscopic

Schrôdinger equation

ià ~

~f

=

~ _{(i (H+ 1')} ~), 13.2)

~~

i,

(1

H+ fj

=

É /

_{dk ~~ka~}

_Il-1')e)

,

(3.3)

2r Bz

for1 _>

Àla+

and for1 <

-Àla~.

In order to describe the intermediate

region,

_we have to return to the

microscopic

time-

independent Schrôdinger equation

at energy E. For the time

being,

we assume that E lies within both bands which _are relevant for _x > b+ and _x < b~ This defines two

quasi-momenta k+,

such that

e)~

=

ej-

= E. For z > b+ _and

x < b~, _an

eigenfunction

~fi(x) ^{of H is} a linear combination of Bloch _waves,

~llx)

₌

/&

(A+~l)+ lx)

+

B+~l±~~ (x)j

,

x

é

b+

,

13.4)

where A+ and B+ _are related to A~ and B~ The

corresponding mesoscopic

wave function

(3.1)

has for1 _>

Àla+

and <

-Àla~

the form

4l)

=

ÀÀ (A+e~~~

^{(~~+~~~) +}

B+e~~~~(~~+~~~)j (3.5)

of _a

superposition

of two discrete

plane

_waves ^with opposite momenta within the Brillouin

zone.

Accordingly

the

asymptotic interpolating

functions

(2.10)

reduce to

F+ ix)

₌

^A+e~~~~

+

B+e~~~i~ (3.6)

In each

asymptotic region,

^both mesoscopic current densities

(2.î)

^and

(2.13)

take the _same value _as trie

microscopic

current, and trie

matching

^{of the} two forms of

(3.5)

^should

^imply

conservation,

namely

J "

(iA~i~ iB~ _i~j ^~~

"

(iA~

^~

iB~i~)

^v ,

13.7)

where

~+

e

de)~ /hdk+

denotes the group

velocity

on one side _or the other.

Let _us assume that this

matching

bas been achieved

by solving

^the microscopic

Schrôdinger

equation

for ~fi(x) ^{in trie} region b~ < x < b+. _For

our purpose, ^we need

only

to know trie values of ~fi(x) ^{and its derivative at trie}

point

^{b+ in} terms of their values at b~, a relation characterized

by

trie

microscopic

2 x 2 transfer matrix t of trie interface at trie energy E. For convenience _we

define it in terms of trie two-component vector

~l(x) (x

_1b)

ilblx)i

_OE = ,

13.8)

(h/2m)~l'Ix) 12h)~~ (xl ix, Hillb)

which allows _{us to express} trie

microscopic

_current

density

_as

Jmiclx)

=

i~llx)i~a211blx)1 13.9)

(a2

is trie usual Pauli

matrix).

Trie form of trie

Schrôdinger

_{equation entails that the transfer}

matrix t, ^defined

by

ll~

l~~)1

" ll~

l~~ll

'

(~'~°i

conserves the current, or

equivalently

that the Wronskian _{of two} solutions is _{a constant.} More-

over, time-reversai invariance

implies that,

if [~fi(x)] ^is a solution, [qfi(x)]" ^{is also} a solution.

Thus,

₍₁₎ the matrix _t satisfies _the

identity t~°2t

= a2

,

(3.Il)

expressing current conservation, and

iii)

it is real. These two properties

equivalently

_mean that _t is _a real matrix with unit determinant. Otherwise t is

arbitrary,

_{as one can}

easily

check

by solving

mortel

Schrôdinger

_{equations. Likewise the values}

(3.4)

of [~fi(x)] ^{at the} periodic sites

Xj+

₌ ^b+ + ia+

ii

>

0)

and

Xp

₌ b~ + ia~

ii

<

0)

_are related _{to one another}

by

the two

transfer matrices

titi

(xiii _ii

=

t+

titi

ixi+)1

,

13.12)

_~~

~fi" ~fii

j~j j~~ /fij

~fi

(~jj2

~ ~ ~ 2~fi k k k

t = cos ka ₊

j

sin ka

,

j

₌ j

,

(3.13)

~~ _~(Î~/~R~)

_{ÎIbÎ(~)Î~ ~~}

_lb~~Î(~)

which _are real and have _a unit determinant

(we dropped

in

(3.13)

trie index + in

k+, a+,

b+ _and

~fi)).

Translational invariance _{in each}

region

is

expressed by (j+)~

= -l, so that trie matrix

j+ _depends

on two real parameters. If trie potential _is symmetric with respect to trie _sites b+ ₊

ia+,

_its

_diagonal

_elements

_vanish,

_{and it}

_depends

on one real parameter.

We wish to match trie mesoscopic _wave functions

(3.5)

_across trie junction. To this aim _we relate them to trie microscopic functions for which this

matching

is achieved

through (3.10)

and

(3.12).

On

analogy

with trie last form of

(3.8),

_we introduce _for

sulliciently large

_)i) trie

two-component

vector

~ ~~~

14lf

[4~i ⁺

(~~)

,

(3.14) (2h)~~ ~i, (Xl _{X~Î) (1)H+ )1') 4l(}

which _again

provides

trie

mesoscopic

_current

density, _equal

_{to trie} microscopic _one Jmi~, as

[Ah]~ a2 (Ah]. ^Since trie

eigenfunctions

of trie mesoscopic Hamiltonian

(3.3)

bave _trie form

(3.5)

which

imphes

that

£

_(X~+

_X()

(1

(H+ (1') 4l(

= ih

~+ÀÀ

(A+e~~~ ~Î B+e~~~~ ~Î

,

(3.15)

~,

they

_are

generated

in trie asymptotic _regions

by mesoscopic

transfer matrices T+

according

to

[~+

~n+

j~+j

1+1 l ,

(3_16j

0 2/~+

T+

= cos

k~a~

₊

J~sin k~a+

,

J+

=

(3.17)

-~+

/2

0

Elimination of A+ and

B+

_between

(3.4), (3.5), (3.8), (3.14)

^and

(3.15)

allows _us to define conversion matrices K+ which relate trie microscopic ^and

mesoscopic descriptions

^for

large

_ii

l~tl

"

~~

ll~

l~tll

'

(~'~~) (h/m~)Im ~fi[e~~kb~) -(2/~)Im ~fike~~kb~)

~~ ~

-(h/2m)Re ^~fi[e~~kb~

^{Re ~fike~~kb~}

~~~~~

the Bloch function is taken at trie

point b+,

and _we

dropped

in

(3.19)

trie index + in

k+,~+

and

~fi).

^{Like ail trie transfer matrices, the conversion matrices}

K+

_are real and have _a unit determinant. Moreover,

they

_are related to t+ and

T+ through

K+t+

=

T+K+ (3.20)

Altogether,

for and _)1')

sulliciently large,

1' _< 0, _{we can} thus relate trie

asymptotic

^forms

of

4l/

and

4lp

on the two sides of the

junction by

means of

(3.10), (3.12), (3.18)

^and

(3.20),

which

yield

the connection rule

l~tl

"

~~

l~~l~ l~~l~' l~~l~~ l~il ^l~'~~)

=

(cos ^k+la+

⁺

^J+sin k+ia+) ^K+ta2K~a2 (cos

^{k~i'a~ J~sin}

k~i'a~) [4ljj

The identity ^{of trie}

microscopic

and

mesoscopic

currents on either side is

implied by

trie fact that ail the 2 _x 2 matrices involved _are real and have _a unit determinant. This was the _reason

why

_we ^introduced trie mass m in trie definition

(3.8)

and trie factor

(a+)~~/~

in trie definition

(3.14).

The

correspondence (3.21)

also froids for trie continuous envelope ^function

(3.6),

^{defined in}

such _{a way} that

F+lx)

[F+(x)]

e

(3.22)

(~+ /2k+) ^dF+ ix) /dx

coincides with

[4l)]

^{at trie sites} x =

Xj+

We _can now introduce trie

mesoscopic

transfer matrix T of the

junction.

^{We first choose} a

point

b between b~ and

b+,

which charactenzes

a

precise

location for the

junction.

We then

extrapolate F+(x)

clown to b,

F~(x)

_up to b,

by imagining

that the two

crystals

_are

homogeneously

^{continued. The}

^mesoscopic

^{wave function}

F(x)

is then

governed by

the

equations resulting

from

(3.21),

~~~

₊

_(k+)~ F+

= 0

,

x

é _b, e)~

= E

,

(3.23)

[F+ (b)]

= T

[F~16)]

^T =

É+ta2É~a2

,

(3.24)

where trie matrices

É+

are obtained from

K+ through

trie

replacement

^{of b+}

^by

^{b in trie}

exponentials

of

(3.19).

We thus

find,

in

conjunction

with trie obvions

mesoscopic

_wave equa-

tions

(3.23),

_a

boundary

condition

(3.24)

which govems ^the discontinuities of

F(x)

and of its derivative across trie

junction

b between trie two

crystals.

Trie _mesoscopic transfer matrix T, ^{like trie}

microscopic

transfer matrix t and trie transfer matrix

(3.21)

for discrete

envelope

functions, ^{is real with unit} determinant but bas _in

generaJ

no other property. ^{It is thus} characterized

by

three real

parameters,

^which moreover

depend

on trie energy E.

They

also

depend

on our somewhat

arbitrary

choice of trie precise location of

b within the

junction,

albeit in _a trivial and smooth _way. If _we release time-reversai

invariance,

the transfer matrix T is the

product

of _a

global phase

factor and _a real matrix with unit

determinant;

this is the

general

form for

matching

conditions that

satisfy

current conservation [12]. ^{Unlike the} wave

equations (3.23),

the mesoscopic transfer matrix T does not

depend only

on the two materials in contact. It is _a characteristic of the

heterojunction itself, depending

_on

its

microscopic

structure. It is

directly

related _to the S-matrix

describing

scattenng ^{of Bloch}

waves

by

the

junction,

and is

experimentally

accessible

through

transmission and reflection

measurements. Theoretical evaluations would

rely

_on the determination of the

microscopic

transfer matrix t and of

(3.19), (3.24).

The equations

(3.23), (3.24)

refer _{to an} electron with well-defined energy.

They

_can be extended to

time-dependent

_wave

packets having

a

sufficiently

small

spread

in energy around

a value Eo. For two

metals,

when the Fermi energy Eo lies within the bands,

(3.23) yields

the

time-dependent Schrôdinger

equation

ih~~

=

-£~~~

+

V+F+ (3.25)

m+ ~

m[

_e

hk) /~)

,

V+

e

Eo m[ (~))~ /2 (3.26)

For _a

semiconductor,

the average electronic _energy

Eo

lies _{near a} baud

edge El,

_we have

et

_ct

et

₊

h~k~/2m[,

and the equations

(3.25), (3.26)

^still hold.

They

would also be

easily

extended for two

slowly graded

materais with _a

sharp

contact. As

regards

the

boundary

condition

(3.24)

at x = b, ^{it is}

legitimate

to

disregard

its

dependence

_on

E, replacing ~+ /2k+

by h/2m+

in

(3.22)

and E

by

Eo in

T,

_{as soon as} the

spread

in energy is small

compared

to the band

width,

and

(3.25)

is thus

simply supplemented

with

(3.24).

In this

approximation,

the mesoscopic ^current

density [Fi ta2 (Fi

reduces to the

expression

for free

partiales

with _mass

mi.

In the

special

_case when T has the form

Il

0 T =

,

(3.27)

9

a

single Schrôdinger

equation with _a

à-potential,

ih)

=

) )

+ VF ₊

hgô(x b)F

,

(3.28)

where m" and

ôF/ôx

_{are step} functions _with

a

jump

at x = b, makes the

synthesis

of

(3.24)

and

(3.25). Likewise,

if the transfer matrix T is

diagonal

and positive, the two materials and their

junction

_can be

represented by

_a

single Schrôdinger equation

with _a

singular

kinetic

energy of the form

il.2),

trie parameter a

being

defined _{as o}

In(m[ /m[

= In Tii

" -In T22.

However,

_in trie

general

_case, there _{is no}

single Schrôdinger

equation that accounts for trie

mesoscopic dynamics

_across the

junction.

The _occurrence of _a

discontinuity

for the

mesoscopic

wave

function,

while trie

underlying microscopic

_wave function is continuous, is

analogous

to a behaviour exhibited

long

_ago for the gap function in

sharp normal-superconductor

junctions Î151.

In the _case of _a

heterojunction

between two semiconductors, when the energies are such that the

approximations et

_ci

et

₊

h~k~/2m[

hold _on both stries. the

envelope

function F satisfies

(3.24), (3.25)

with V+

=

et,

the real matrix T

having

_a unit determmant. The above microscopic

study

then confirms the result of _our

phenomenological approach il ii,

which

however used _a

slightly

dilferent definition of T.

This

analysis

is aise relevant for surface bound states

lying

astride trie

heterojunction,

_pro-

vided

they

extend _{over a} suilicient number of lattice cells. The values of ~+ _in

F+(x)

c~

exp

[~~+(x

_{b)] are} then

given

for T12

#

0

by

~^~

~~.

^~

~).

₌

Et _{Ei 13.29)}

~ ~ ~

lÎÎÎ~

^{~~ ~}

~~~ ~ÎÎÎ[ ^~~

^~

_~~~

' ~~'~~~

with ~+ _> o, ~~ > 0. For

et

_>

_{ep, letting}

~y +

[(et ep) /2m[)~~~,

_we find

by

_means of _a

graphical

solution two such bound states for

T21/Tii

< -~y,

TiiTi2

< 0, _one bound

state for

(Tii'f

₊

T21)T12

< 0, and _none for

T21/Tii

> -~y,

TiiTi2

> 0. For T12 ₌ 0, _we

find _one bound state for

T21/Tii

_{< -~y, none} for

T21/Tii

> -~y.

However,

in the limit _as T12 - 0 with

TiiTi2

< 0, _one solution of

(3.29), (3.30) yields

_a bound state with

binding

energy

large

_as 2

(Tii/Î

₊

T22/Î)~/T)~.

_{This feature indicates}

_{that, although}

_{trie transfer}

matrix T involves three

independent

parameters, their range of variation is not

arbitrary

if trie

microscopic potential ion

which

they depend through É+

and

t)

is bounded from below. Note

finally

that

complex

solutions of

(3.29), (3.30)

_Inay describe

junction

resonances.

4.

Concluding

ILemarks

The

one-band,

mesoscopic

envelope-coefficients approach

for

describing

electromc states in

heterostructures,

with its

low-energy

version the effective _mass

approximation provides

a framework in which _{one can} compute the behaviour of

complex multijunction

structures. In

fact,

_treatments

using

more

precise approximations

exist, but

they

are

presently

restricted to few

junctions.

In the

envelope

function

apprôach

the structure is characterized

by

the local effective _masses, the values of baud

edge energies

and the connection rules for trie _wave functions at

sharp junctions.

These parameters ^of trie

phenomenological theory

should

eventually

be

derivable from trie

microscopic

calculations of

simple

non-uniform structures.

The _purpose of _our

analysis

has been _to

clarify

these issues. We

rely

_on the existence of _a localized basis set offunctions

(generalized

Wannier

functions)

which is

complete

in the energy range of _an isolated band in _a non-uniform heterostructure.

Although

_an abstract

proof

of trie

existence of

generalized

Wannier functions for _many

non-periodic

systems of interest bas been known for _some time [8],

explicit

constructions _are available

only

in _some

particular

_cases

[7,9].

We

hope

and believe

that,

while

technically difficult, exphcit

construction will be

successfully

undertaken in _more

complicated

cases.

Assuming

trie

generalized

Wannier functions to be

given,

_we bave constructed for any ^state

a discrete in-band _current

density, equation (2.7),

then _a

(non-uniquely)

smoothed _{one, equa-} tion

(2.13).

Both

satisfy

trie _exact conservation law, which holds for

arbitrary

states within trie band. This should be contrasted with trie

usually

considered expressions for trie mesoscopic currents, associated to trie

(continuons) envelope

function, which _are

only approximately

valid when trie

envelope

_is

slowly varying,

for _states localized in _{a narrow} energy interval either _near

a band

edge (semiconductors),

_{or near} trie Fermi level

(metals). Actually

_we bave shown that trie _average values of all trie current densities coincide for _states

having

components

only

_m

the _{energy ranges} relevant to each of the

envelope approximations

considered.

For

slowly graded heterostructures,

where _a

perturbative

construction of the Wannier func- tions exists

iii,

_our

analysis provides

_rigorous results which

complement

the

existing

_ones.