Vertical transport through laterally periodicallymodulated heterostructures

D. Embriacoa,*, G.C. La Roccab,c

aI.N.F.M. and Dipartimento di Fisica, UniversitaÁ di Pisa, Piazza Torricelli, 2, I-56126 Pisa, ItalybDipartimento di Fisica, UniversitaÁ di Salerno, via S. Allende, I-84081 Baronissi (SA), Italy

cI.N.F.M., Scuola Normale Superiore, Piazza dei Cavalieri, 7, I-56126 Pisa, Italy

Received 17 April 2000; received in revised form 16 October 2000; accepted 16 November 2000 by E. Molinari

Abstract

The ballistic transport in a laterally periodically modulated heterojunction connecting two reservoirs is studied: this can

model the presence of an ordered layer of self-organized quantum dots. The problem is solved in the envelope function

formalism using a three-dimensional (3D) generalization of the transfer matrix approach that exploits the two-dimensional

(2D) lattice periodicity. Illustrative numerical results are presented showing qualitatively different behaviors of the transmission

coef®cients due to the multichannel nature of the propagation. In particular, deep minima of the transmission occur due to

destructive interference between different conducting channels, a situation that cannot be realized in the absence of the lateral

modulation. q 2001 Elsevier Science Ltd. All rights reserved.

Keywords: A. Nanostructures; A. Semiconductors; C. Electronic transport; C. Tunneling; C. Electronic band structure

PACS: 81.10.Aj; 81.15.Hi; 68.35.Ct

1. Introduction

Modern nanometer growth techniques have been used to

manufacture devices in which phase-coherent electrons can

travel ballistically at low temperatures. Novel examples are

arrays consisting of one or more layers of ordered quantum

dots (QD) [1, 2]. At present, good samples of two-dimen-

sional (2D) lattices of QD with typical lateral spacing of the

order of some tens of nanometers can be grown [3±5]. As

we can expect that this length scale, at least at low tempera-

tures, is smaller than the electron phase breaking mean free

path (that we can assume to be of the order of 1 mm) the

coherence effects of electron motion become important and

may affect the conduction in such devices. Recently, vertical

transport measurements on disordered arrays of QD have

been presented [6]. We consider here theoretically the

conduction problem through a heterojunction characterized

by a potential barrier modulated periodically in the planes

perpendicular to the growth direction (vertical axis z). This

potential can model the presence of an ordered layer of self-

organized QD [1±5].

The vertical transport is investigated calculating the trans-

mission coef®cient T(E) of the ballistic junction connecting

two reservoirs as a function of the energy E of the incident

electron; the low ®eld current ¯ow J is then obtained via the

formula

J /Z

T�E� 2F

2EdE �1�

that relates the quantum-mechanical T to the macroscopic J

by means of the thermal distribution function F charac-

terizing the reservoirs.

The focus of our work is the solution of the quantum-

mechanical problem for a (nearly arbitrary) laterally

periodically modulated potential barrier based on a suitably

symmetrized generalization of the usual transfer matrix

approach.

2. The model calculation

We take the electron motion inside the region of the

Solid State Communications 117 (2001) 407±411

0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.

PII: S0038-1098(00)00492-0

PERGAMONwww.elsevier.com/locate/ssc

* Corresponding author. Tel.: 139-50-911-284; fax: 139-50-

482-77.

E-mail address: embriaco@cibs.sns.it (D. Embriaco).

junction to be ballistic, neglecting all phase breaking

scattering mechanisms. The system is thus described in the

usual envelope function approach [7] by the Hamiltonian

H�3D� � p2z

2mp�z� 1p2

x 1 p2y

2mp�z� 1 V�x; y; z� �2�

where mp�z� is the effective mass and V�x; y; z� is the

potential determined by the band edge offsets. All the

information on the junction is contained in mp�z� and

V�x; y; z�. We take V periodic in the x, y-direction parallel

to the plane of the junction to describe the lateral modu-

lation. In general, mp may also depend on x and y giving an

additional modulation of identical periodicity, however, for

technical reasons, we here neglect this effect that does not

change our result qualitatively.

Thanks to the in-plane periodicity, the solutions can be

labeled with a given kk belonging to the 2D ®rst Brillouin

zone of V: during the transmission the in-plane wave vector

can vary only by a vector G belonging to the 2D reciprocal

lattice of V. In other words the incident electron, if its energy

is high enough, can be transmitted (or re¯ected) in different

directions (it is ªdiffractedº by the lattice potential). In this

sense the transmission, but also the propagation inside

the barrier, is a multichannel one, where the different

channelsÐboth conducting and outgoingÐare singled out

by the Gs.

To solve the three-dimensional (3D) problem we dis-

cretize the potential along the z-direction and indicate its

dependence on x and y in each block with zj , z , zj11

by Vj�x; y�, the mean value of V taken along the z-direction

from zj to zj11. The Hamiltonian describing the electron in

each block j is now separable:

H�3D�j � H�2D�

j 1p2

z

2mpj

�3�

with

H�2D�j ;

p2x 1 p2

y

2mpj

1 Vj�x; y� �4�

mpj being the effective mass of the semiconductor in the jth

block. We can now solve H�2D�j c�j;p�kk �x; y� � Ep

j;kkc�j;p�kk to

®nd all eigenstates labeled by p in the block j with a given

kk in the 2D ®rst Brillouin zone using a standard plane wave

expansion. This is easily done taking advantage of the

symmetry properties of Vj: the eigenfunctions form ªmini-

bandsº of proper symmetry each labeled with p as shown in

Fig. 1. The conservation of kk for each block j assures that

we need only the discrete set of eigenfunctions with a given

kk as determined by the incident wave-function of energy E.

From now on, we assume a given kk for the electron wave-

function through the junction and omit this label from all the

following expressions.

The symmetry considerations enable us to write the more

general solution of (3) in the form:

Cj�x; y; z� �XNp�1

c �j;p��x; y��A�p�j eik�j�p z 1 B

�p�j e2ik

�j�p z� �5�

where

k�j�p ;

�������������������2mp

j

"2�E 2 E�p�j �

sif E 2 E�p�j . 0 and

k�j�p ; i

����������������������2

2mpj

"2�E 2 E�p�j �

sotherwise. The truncation to N terms takes advantage of the

rapid convergence of the series at least in the energy range

of interest. In fact, the eigenfunctions whose energy E

D. Embriaco, G.C. La Rocca / Solid State Communications 117 (2001) 407±411408

Zj zZ0 ZM

modulatedpotential

freeelectrons

freeelectrons

j=0 J=M

π/ a

π/ a

X

Γ

M

kx

yk

p=2,3

p=1

p=4

Fig. 1. Schematic representation of the junction. The ®rst and last blocks are described by free electron levels (right panel); the inner part of the

barrier is divided into blocks in which V is kept constant along z and it presents minibands due to the laterally periodically modulated potential

(the shaded region). In the left panel are shown the energy levels for a block in the shaded region in the ®rst Brillouin zone of the 2D square

lattice shown in the inset. The energy unit is E0 ; �"2=2mp��2p=a�2; the potential is that of a circular well of radius 20% of the 2D lattice

constant and depth 2E0.

exceeds that of the incident carrier by several tens of meV

are not needed; the typical separation between different E's

is of the order of E0 ; �"2=2mp��2p=a�2, with a the 2D

lattice constant of V (for typical values a � 300 �A,

mp � 0:0665Ði.e. for GaAsÐthis factor is 10±20 meV).

Typically we use some tens of minibands in the expansion

of Eq. (5) (N < 40±60) which is enough to assure the

numerical convergence. We impose the continuity of Cand �1=m p�z���2=2z�C between one block and the previous

one: this gives linear relations between the Aj and Bj coef®-

cients of two adjacent blocks. So if Xj is the vector formed

by these coef®cients in the jth block

X j ; �A�1�j ; B�1�j ; ¼ ; A�p�j ; B

�p�j ; ¼ ; A�N�j ; B�N�j � �6�

we can write

Xj � T�j�´Xj21 �7�where T(j) is the 2N £ 2N matrix determined by the conti-

nuity relations. The coef®cients are explicitly given by

T�j�2p21;2q21 �

1

2

pj

���qj21

� 1 1

k�j21�q

k�j�p

mpj

mpj21

!

� exp�i�k�j21�q 2 k�j�p �zj� �8�

T�j�2p21;2q �1

2

pj

���qj21

� 1 2

k�j21�q

k�j�p

mpj

mpj21

!

� exp�2i�k�j21�q 1 k�j�p �zj� �9�

T�j�2p;2q21 �

1

2

Dpj

���qj21

E 1 2

k�j21�q

k�j�p

mpj

mpj21

!

� exp�i�k�j21�q 1 k�j�p �zj� �10�

T�j�2p;2q �1

2

Dpj

���qj21

E 1 1

k�j21�q

k�j�p

mpj

mpj21

!

� exp�2i�k�j21�q 2 k�j�p �zj� �11�

where

pj

�����qj21

�;Z�c �j;p��x; y��pc �j21;q��x; y� dx dy �12�

If we take the matrix product T � TM´TM21¼T1 of the Tj

matrices of the blocks in which we divide the barrier, we

have the transfer matrix of the whole barrier that contains all

the quantum-mechanical information about the barrier at the

given energy and kk of the incident carrier. The problem of

transmission is solved like in the one-dimensional case,

imposing the appropriate form for the coef®cient vectors

in the ®rst and last blocks of the junction

X0 � �1; r1; 0; r2; ¼ ; 0; rN� �13�i.e. in the ®rst block we have the incident wave-function and

the re¯ected ones propagating in the negative z-direction in

different channels 1; 2; ¼ ; N; and

XM � �t1; 0; t2; 0; ¼ ; tN ; 0� �14�i.e. in the last block there are only transmitted waves which

D. Embriaco, G.C. La Rocca / Solid State Communications 117 (2001) 407±411 409

Fig. 2. Single laterally periodically modulated block: the left panel shows the transmission probability for normal incidence for a single layer of

2D lattice quantum dots with circular base radius of 80 AÊ and height of 200 AÊ ; the 2D lattice constant is 400 AÊ and the potential is a well

30 meV deep. The transmission coef®cients in the different outgoing channels and the total one (full line) are shown; the right panel represents a

schematic view of transmission at the minimum: only two conducting channels (G1 states in the shaded region) contribute to the conduction

from left to right resulting in the destructive interference shown in the last block of the schematic barrier. We also indicate the ®rst outgoing

diffracted channel (p � 2) that would appear at an energy of E0. The energy levels are that of Fig. 1 left.

propagate in the positive z-direction if the corresponding

k�M�p is real, otherwise there are exponentially decreasing

components. The problem of transmission is reduced to

the linear system in 2N unknowns {r p, t p}

XM � T´X0 �15�that can be solved (numerically) to give transmission co-

ef®cients in the different permitted outgoing channels as a

function of the energy and of the incident direction. The

total number M of blocks as well as the total number N of

bands are limited by possible numerical instabilities; all

results presented are far from such limits. The transmission

coef®cient in the outgoing channel p is then proportional to

the square of the t p coef®cient if k�M�p is real (open channel),

zero otherwise:

Tp �k�M�p

k�0�1

mp1

mpM

tp

��� ���2 if k�M�p is real; 0 otherwise �16�

the total transmission coef®cient is the sum over all the open

outgoing channels.

3. Results and discussion

We report here some illustrative results of the trans-

mission coef®cients versus energy for a carrier with normal

incidence. First we note that the transmission is character-

ized by different outgoing channels; each outgoing channel

being determined by different 2D miniband eigenfunctions.

The transmission begins in a higher outgoing channel only

when the energy exceeds a given threshold. For normal

incidence to have an outgoing carrier diffracted by a vector

G, its energy must be greater than "2G2=2mp

M . The energy

threshold for the appearance of a new outgoing channel

depends only on the periodicity of the potential and not on

its detailed shape.

In Fig. 2 we show a simple case with only one block with

laterally modulated potential of cylindrical wells 30 meV

deep (the base radius is 80 AÊ and the height 200 AÊ ) ordered

in a 2D square lattice of 400 AÊ lattice constant. At about

9 meV a deep minimum with vanishing transmitted current

is reached; this is not possible in the one-dimensional (1D)

case and it is due to the destructive interference of waves

propagating in the junction in (at least) two different

conducting channels. This interpretation is sketched in the

right panel: the direction of incidence is along the growth

axis z, so kk corresponds to the point G in the ®rst Brillouin

zone of the square lattice; the shaded region corresponds to

the laterally modulated block whose energy levels are given

in Fig. 1. The analysis of the C wave-function in corre-

spondence of minimum shows that it has only two main

contributions belonging to the two lower eigenstates at the

G point that are fully symmetric with respect to the square

point group (G1 symmetry). There are no contributions from

the next two higher levels (G5 and G3) which are of different

symmetry. This is due to the vanishing of the factors given

in Eq. (12) because the selection rule allows only conduct-

ing channels of the same symmetry of the incident wave-

function (the normal incidence is completely symmetric).

The next G1 level has too high an energy and it does not

contribute signi®cantly.

In Fig. 3 we show results for a more realistic potential

shape: the single dot has a conical shape with height along

the z-direction of 60 AÊ equal to the base radius, the numeri-

cal computations are made dividing such cones in about ten

blocks along z. The left panel of Fig. 3 is with m p � 0:0665

and a well of 100 meV. The right one has a potential

390 meV deep (equal to the discontinuity of conduction

D. Embriaco, G.C. La Rocca / Solid State Communications 117 (2001) 407±411410

0 20 40 60 80 100Energy (meV)

0.0

0.2

0.4

0.6

0.8

1.0

Tran

smis

sion

Coe

ffici

ents

TTotal

T1

T2

T3

ε00 20 40 60 80 100

Energy (meV)

0.0

0.2

0.4

0.6

0.8

1.0

Tran

smis

sion

Coe

ffici

ents

TTotal

T1

T2

T3

ε0

Fig. 3. Transmission coef®cient for normal incidence for a lattice of conical quantum dots (the lattice constant is 300 AÊ with 60 AÊ base radius

and height [8]). The left ®gure is with a 100 meV deep potential, the effective masses are taken to be mp � 0:0665; it presents a minimum in

transmission at about 17 meV. The right ®gure has the same geometry but the potential well is 390 meV deep equal to the discontinuity of the

conduction band between GaAs and In0.5Ga0.5As; the masses are that of GaAs in the free blocks and a weighted average in the modulated ones.

band edge between In0.5Ga0.5As and GaAs [8]); the masses

are taken to be that of GaAs in the free blocks and an

average between GaAs and In0.5Ga0.5As weighted with the

in-plane dimension of potential wells in each laterally

modulated block. Notice in the left panel of Fig. 3 the occur-

rence of a Fabry±Perot like resonance at about 25 meV.

As shown by the examples in Figs. 2 and 3, the use of

laterally modulated barriers allows for a greater ¯exibility in

the tailoring of transmission coef®cients compared to the

usual 1D band engineering.

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