Calculation of Transmission Tunneling Current Across Arbitrary

7/24/2019 Calculation of Transmission Tunneling Current Across Arbitrary

1/7

Calculation of transmission tunneling current across arbitrary potential barriers

Yuji Andoand Tomohiro Itoh

Citation: Journal of Applied Physics 61, 1497 (1987); doi: 10.1063/1.338082

View online: http://dx.doi.org/10.1063/1.338082

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2/7

Calculation

of

transmission tunneling current

across

arbitrary potential

barriers

Yuji Ando and Tomohiro itoh

Microelectronics Research Laboratories

NEC

Corporation 4-1-1.

114iyazaki

Miyamae-ku Kawasaki

213,

Japan

(Received 11 September 1986; accepted for publication 5 November 1986)

This pape r presents a simple method for accurately calculating quantum mechanical

transmission probability and current across arbitrary potential barriers by using the multistep

potential approximation. Th is method is applicable to various potential balTiers and wells,

including continuous variations of potential energy and electron effective mass. Various

potential barrier structures and a hot-electron transistor are analyzed to show the feasibility

of

this method.

I

INTRODUCTION

Recently, from the viewpoint of high-speed and new

functional device application,I-3 there has been an increas

ing interest in resonant tunneling in quantum-wen

and

su

periattice structures. The WKB (Wentzel-Kramers-BriI

louin) approximation,

the

conventional method used to

calculate the transmission coefficient across potential bar

riers, fails to explain the resonance phenomena. Further-

more, the WKB method is inaccurate in regions where the

potential profile varies abruptly,4 Abruptly varying poten

tial functions are, however, frequently encountered at the

interface between two different materials in heterojunction

structures.

Another method for calculating the transmiss ion coeffi

cient

is

to solve Schrodinger s equations

through

potential

barriers. Chandra and Eastman

5

calculated the transmission

coefficient for a triangular barr ier by solving SchrOdinger s

equations via the numerical method.

On the other

hand,

Gundlach

6

calculated

the

tunneling

current

for a trapezoidal

barrier

by

connecting the Airy functions, exact solutions for

Schrodinger s equations,

at

two interfaceso

The

same proce

dure has been applied

to

triangular barriers

by

Christodou

lides

et al.

7

Lui and Fukuma

8

showed this calculation to be

applicable to use with arbitrary piecewise linear potential

barriers. These calculations are, however, unsuitable for de

signing quantum-well and superlattice structures, because of

the complicated treatment involved. Furthermore, varia

tions

of

electron effective masses have never been taken into

account in these analyses,

This paper presents a simpler method for accurately cal

culating the t ransmission coefficient

and current

across arbi

trary potential barriers.

In

the present method, we approxi

mate variations of potential energy and electron effective

mass by multistep functions (multistep potential approxi

mation).

The

transmission coefficient is calculated by con

necting

momentum

eigenfunctions.

9

The details of the cal

culation procedures are described in Sec. II. As mentioned

above, various potential barriers, including continuous var

iations of potential and effective mass,

can

be analyzed easily

by using the present method. For example, rectangular and

parabolic potential barriers, fabricated with

GaAs/

AIGaAs

heterostructures, are analyzed in Sec.

III.

Section

IV

de-

scribes

the

analysis for hot-electron transistors

HETs)

an application

to quantum

size effect devices.

II. CALCULATION PROCEDURE

A Transmission probability across arbitrary potentia

barriers

In the present calculation, instead of dealing with co

tinuous variations

of

potential energy, we split the potenti

barrier

up into

segments, in

which

potential energy

can

b

regarded as a constant. In the limit as the divisions becom

finer and finer, a continuous variation will be recovered.

Let

us assume

the

potential barrier

to

be a sequence of

small segments.

An

example, in

the

case where N

=

10,

shown in Fig. 1 where

the

potential

barrier

U x),

the

effe

tive mass m* x) ,

and the

permittivity

E X)

are approxima

ed by

the

multistep functions

U x)

= =

U[(x

j

._ +x j ) /2 ]

,

m* x)

=mj=m*[ x j

.

J

+Xj)/2]

,

x) =j

[ X

i

-

1

+ x

j

) /2J,

for x

j

. t


3/7

an electron with energy E moving normally on the barrier, is

given by

r (x)

=

Aj

expUkjx)

+

OJ exp( - ikjx)

, 2)

where

k

j

=

J [2mtCE

-10)] If i,

(3)

and

is the

reduced Planck s constant.

From the

continuity

of lh (x) and

1/

mJ')

d1h

I dx)

at

each boundary,Il the

detenniningA

j

and OJ in Eqs. 2)

can

be reduced

to

the multiplication of the following

N

+ 1

2X2) matrices:

(4)

4

=

[

+ SI

)exp[ - i(k

l

+ I -

kl )xI]

2 1

-

S[

)exp[i(k/+

1 +

kJ )x/]

l -S / )exp[

i k

l

+ I +k[)Xd]

1

+S/)exp[i k

l

+

l

- k / ) x

1

]

(5)

and

mt+l

k[

S/= .

(6)

m? k

l

+

1

By setting Ao

=

1 and B N + I

=

0 in Eq. 4) for j

=

N + 1,

we can calculate the transmission amplitude AN+- 1 and the

transmission probability D(E) as foHows:

7)

and

m* k

D(E) _ ~ I 2

- ' k N + II

,

m

N

+

1

0

8)

where

(9)

B

Transmission

current

calculation

The band diagram, used for calculating the transmission

current-voltage

I-V)

characteristics for

the

potential bar

rier, is shown in Fig. 2. The solid line denotes the potential

function for

the

barrier, whereas

the

broken line denotes

the

approximated potential function. As shown in Fig. 2, the

U x,V)

qVa

o

Ef+qVa-qV

t

>

x

qVb

t

FIG. 2. Energy band diagram for the potential barrier un der biasing condi

tions.

1498

J.

Appl. Phys., Vol. 61,

No.4,

15 February 1987

i

total applied voltage

V

is expressed as

V =

Va

+ Vb + V

d

,

where Va Vb and Vd are the voltage drop values in the

accumulation layer, the barrier, and the depletion layer, re

spectively. These values

and the

space charge ns

per

unit

area in the depletion layer, which is equal

to the

net charge in

the accumulation layer, are determined using fonowing

equations

2;

exp qValkn

- q V a l k T - l =q2

n

;/2okTN

D

, 10)

Vb =

lLb[qnJ X)]dX,

Vd =

qn;/2N t - N

D

,

11 )

12)

where N D

is the

donor concentration in

the

semiconductor

at both contacts, Lb is the barrier thickness, q is the elec

tronic charge,

k

is

the

Boltzmann s constant,

and T

is

the

temperature. The Boltzmann distribution and the depletion

layer approximation are assumed for

Egs. 10) and 12),

respectively.

The potential function U(x) in the barrier is determined

by the

superimposition of

the

zero-bias potential

and

poten

tial drop

(x)

=

[qn, l

(x) ] dx. The transmission proba

bility D(E

x

, V) through the barrier region is calculated as

described above. The accumulation and depletion regions

are assumed not to affect the transmission probability. As

suming the dependence of transmission probability only on

longitudinal electron energy Ex for oblique incidence, the

current

density is given by 13

q m ~

J(E

x

)dEx

=

--::::23 D(E ,

Vb)

21Tfi

xL: [ /o(E) -IN+1CE)JdEdE

x

(13a)

where

10

and f N + I

are

the distribution functions in

the

left

contact

and

in

the

right contact, respectively. Assuming

the

Fermi-Dirac distribution, Eq. (13a) can be rewritten as

fol

lows:

q m ~ k

JCE ) =

21T

2

fz3 D(Ex , Vb)

Xln .

1 + exp Ef +

qV; -

Ex ) lkT )

1 + exp E

r

+ qVa - qV - Ex

) lkT

130)

Here, the

Fermi

level in the accumulation region is treated as

to be raised by qVa .

Y. Ando and T Itoh 1498[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

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4/7

III, EXAMPLES

Various potential barriers fabricated with

GaAsl

Alx

Gal x As structures were analyzed.

The

conduction-band

offset t:.Ec

was taken to

be 60% of

GaAs and Alx

Gal x As

r

band-gap

difference. 14 In

the

fol

lowing, the electron concentration in

GaAs

was taken to be

n

=

1 X 10

18

em

-

3,

equal to N

D

and

the

Fermi energy was

presumed

to

be

0.05

eVat

77 K. The

parameters used are

listed be1ow

I5

:

1::.E

= {O.75X eV)

c O.7Sx + O.69 x - 0.45)2 (eV)

m*lmo =

0.067

+ O.083x

lEo =

13.1 1 -

x) +

10.

Ix

,

for

xO.45 ,

14a)

(14b)

Cl4c)

where

ma and Eo

are

the

free electron mass

and

the vacuum

static dielectric constant, respectively.

Ao Transmission

probability across

quantum barrier

structures

The

transmission probability is calculated in semicon

ductor-insulator-semiconductor SIS) structures with

a

rec

tangular barrier

and

with a parabolic barrier as shown in

Figs.

3(a) and 3(b),

respectively.

Figure 4(a) shows

the

transmission probability

D

for

x =

0.05 eV across

the

rectangular barrier [in Fig. 3

(a)]

with respect

to VI> In

these calculations,

the barrier

is divid

ed into

N

segments with

N

values ranging between 10 and 80.

There

is only a slight difference in solutions for

N;;.40 and

they converge

to the

solid curve

[in

Fig. 4(a)] for

N;;.80.

For

this case,

the Airy

functions can give

the

exact solution,

6

which coincides with the solid curve.

The

oscillatory behav

ior ofD

Vb

) is presumed

to

be due

to

resonance through the

virtual states above

the

barrier.

16

Electron wavelengths

at

the

resonant states are about 80 A for

Vb =

1 V

and

50 A for

a)

Rectangular Barrier

(b)

Parabolic Barrier

AlxGal-xAs

x ~ O - O _ 5 - 0 )

f ~ 7 e v

~ E

GaAs

~ - - -

350A

--0. GaAs

FIG. 3.

Analyzed single-barrier structure

band

diagrams. (a) Rectangular

barrier. (b) Parabolic barrier.

1499

J. Appl. Phys., VoL

61

,

No.4,

15

February 1987

.,...

'"

10-2--- - 0 _ - - -

_____

_ I

a)

Rectangular

Barrier L\j

r

Ex:=:O 050V

/1

j

1

I

1

p J

J ~

~ _ J

N=20

N=40,80

0.5 Hi

1.5

Voltage (v)

-- -

~

b)

Parabolic

Barrier

Vb=O

1 0

I' i

i\

i \ /

. \ \ I.

N=9

N=15

.. \

. \ I . '

, v i.j I.

\ 1

0.0 L

- l N _ = _ 1 _ 9 _ . 3 _ 9 ~ - - _ . _____

L__J

ao

Q4

Q8

Energy eV)

FIG. 4. Transmission probability plot la)

vs

voltage for the rectangul

barrier shown in Fig.

3(aJ.

where

Nvalues

range from

10

to 80,

and

(h)

electron energy calculated for the parabolic

barrier

sh )wn in Fig.

3(b

where N values range from 9 to 39.

Vb

=

2 V whereas the width

of

each segment is about 9 A

for

N

= 40. Thus, in the present method, the exact solutio

can be obtained by choosing a segment

width

sufficient

smaller

than the

electron wavelengths

at

the resonant state

Fignre

4

b) shows transmission probability

D

acros

the

parabolic barrier

[in

Fig. 3

(b)] at thermal

equilibrium

with respect to

E

for

N

values ranging between 9 and 3

The

flat

structure

for the transmission probability profiles

a notable difference from the

structure

for a rectangular ba

rier.

B. V characteristics for

double-barrier structures

The

transmission probability and

the -V

characteristi

are calculated for double-barrier structures with a rectang

lar

well

and

with a parabolic well, as

shown

in Figs. 5

(a) an

5(b),

respectively.

Y.

Ando and

T.

Itoh

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5/7

a) Rectangular

WeI

AlAs

AlAs

GaAs

I lo ev

~ - - - - - - - - Ec

GaAs ' '30A

iOOA

S OA GaAs

b) Parabolic Well

AlxGa1-xAs

x=1-0-1)

r

O.956eV

---- --

Ec

GaAs

I 100A

- GaAs

FIG.

5

Analyzed double-barrier structure band diagrams.

a)

Rectangular

well. b) Parabolic well.

Figures

6 a)

and

6 b)

show D Ex) for a rectangular

well [in Fig. 5 a)] with

N

= 32

and

for

a

parabolic well [ in

Fig.

5 b)]

with N

=

41, respectively. The peaks of D E

x

),

En n

=

0,1,2, ... ), separa te at regular intervals for a parabol

ic weIll? as shown in Fig.

6 b),

in contrast to that for a

rectangular well shown in

Fig. 6 a),

These results agree wen

with the concept that boundary state energies are expressed

as En =

n

+

112)00

for the quantum

wen

expressed as

U x) =

1/2)m*w

2

x

2

,4

Calculated J- V characteristics for both structures

at

77

K are shown in Figs.

7 a) and 7 b).

Figures

7 a) and 7 b)

correspond

to

the rectangular well

and the

parabolic wen,

respectively. In these figures, the solid lines denote the

suIts, including the effects of accumulating and depleting,

whereas the broken lines denote the results without these

effects. With the accumulation and depletion t aken into ac

count,

the current

density increases and

the

voltage shifts.

These results show

that

voltage drops at contact layers seri

ously affect J-V characteristics,

and

hence, should be taken

into

account in analyzing and designing quantum size de

vices.

IV. APPLICATION-ANALYSIS OF HETs HOT

ELECTRON TRANSISTORS)

The HET, one

of

the quantum size effect devices utiliz

ing electron tunneling through potentia] barriers, was ana

lyzed

to

show

the

feasibility of this method.

A. AnalysiS procedure for

HETs

Figure 8 is

a

band diagram

of the HET

proposed by

Heiblum. 18 The

J-

V

characteristics

ofHETs

can be analyzed

applying the present calculation to the potential function

U X,VBE,V

CB

shown in Fig. 8), where

VEE

is the voltage

applied between emitter and base, and VeE is the voltage

between the base and collector. The present calculation is the

1500

J.

Appl. Phys., Vol. 61,

No.4,

15

February 1987

10

0

0.0

0.0

(a) Rectangular Well

0.5

1.0

Energy (eV)

(b)

Parabolic Well

0.5

Energy (eV)

1.0

FIG. 6 Transmission probability vs electron energy plot

a)

calculated for

the rectangular well structure shown in Fig. 5

a)

and b) calculated for the

parabolic well struc ture shown ill Fig. 5 b )

extension

of that

used for

MOMOM

Cmetal-oxide-metal

oxide-metal) devices. 18

The transmi.ssion probability DE Ex, VEE across the

emitter barrier and Dc Ex,

V

CB ) across the collector barrier

can be calculated, as described in Sec.

II

A. Current density

JE between the emitter and

base,

was calculated as de

scribed

in

Sec. II B That is,

qm*

1

E E

x

) = ~ D E E x ) [ iE E ) iB E)]dE ,

21117 Ex

15)

Y. Ando and

T.

Itoh

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6/7

6

E

D

--

4

-;J

o-E

(/)

c

OJ

0

-

2

::J

,)

0

0.0

6

E

()

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Calculation of Transmission Tunneling Current Across Arbitrary