Superlattice and Negative Differential Conductivity in Semiconductors

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Conduction

(b)

Figure 1 Electron nergy n the valence nd onduction

bands s a function of distance in the direction of the

superlattice fo r (a) alternation of donor andacceptor im-

purities and (b) periodic varia tion of lloy com position.

Solid nddashedines epresent inusoidal ndperiodic

square-wellpotentials, espectively. The

E ,

are the mag-

nitudes

of

the energygap n the semiconductor materials

used and

VI

is theamp litude of theperiodic uperlattice

potential.

pound s an d their alloys, etc. The re may be

a

number of

ways to form a superlattice structure.* We considered

two methods: (1) a periodic variation of don or or cceptor

impurities, alternately, in

a

single semiconduc tor; and (2)

a

periodic v ariation of alloy composition, introduced during

the crystal growth. Both methodscouldbe used simul-

taneously and

it

would be advantageous

to

carry out the

epitaxial growth at a relatively low temperature' to mini-

mize the therm al diffusion of impurities o r alloy constit-

uents, which would tend to wash out the desired potential

profile. In this context, t is mo re desirable to apply method

(2) because of the lower diffusion coefficients in alloys.

forms known as polytypes, in which one sees a kind of one-dimensional

* Some crystals such as hexagonal SIChave a number of different structural

type of crystal. The potential amplitude and resulting energy gap are probably

superlattice structure. The periods range from 15 to 53A,depending on the

too small to demonstrate any of the effects described here.

62

L. ESAKI AND

R. TSU

In the alternating impurity system (Fig. la), the peak-

to-peak amplitude of the periodic p otential can be hosen,

in principle, as any value up to that of the energy gap,

whereas, in the alternating alloy com position system (Fig.

lb ) , t hevalue would be limited to abo uthalf of the differ-

ence between the energy gaps of the twomaterials involved.

It would not be possible to ob tain the desired superlattice

structure (with a 100-A period and a 0.1-eV potential

amplitude) with relatively light doping because of space

charge effects.

We considered two potential functions, sinusoidal wave

and

a

periodic square wave, as illustrated in Fig.

1,

a n d

two typ ical values of effective m ass, 0.025mo an d

0.07m0,

which are applicable to the InAs-based alloy an d GaAs-

based alloy systems, respectively. The Ge-GeSi system is

also attractive, particularly from an experimental point

of

view. W e would choose relatively pur e elemental or binary

compound semiconductors as the narrower gap semicon-

ductor, corresponding o th e alleys in the otential profile,

to ob tain a favorable electron scattering time.

Band

structure

We are concerned with the energy bands in a one-dimen-

sional uperlattice represented by a periodicpotential

V(x)

=

V(x

f nd

with a period d typically 10 to 20 times

greater than the lattice constant

a

in the host crystal. The

usual Brillouin zone will be subdivided into minizones as

shown in Fig. 2a. Since we ar e interested only i n t he first

minizone, because of low carrier concentrations we may

assume that the E-k relation in the directions parallel

to

the superlattice planes is parabolic as usual. For the sinu-

soidalpotential V(x)

=

Vl [cos (27rxld) - 11, the wave

equ ation in the irection of the superlattice, denoted by x,

has the fo rm of Mathieu's equation and has been studied

in great detail by Slater.3

Th e reduced energy an d amplitude of the perturbing

periodic potential are defined by

=

q(k,) = (E, - Vl)/Eo

and y = V,/Eo, respectively, where

VI

is the am plitude of

the periodic superlattice potential and Eo

=

ktk:/2m; here

kd = 7r/d and m is the effective mass. In Fig. 2b we plot

r ]

vs. y for the sinusoidalpotential, using a parameter

/3 = k,/kd. Th e allowed solutions of th e wave equation

are represented by the non-shaded regions, while the for-

bidden solutions fall n the shaded regions. For the case of

y = 0.5, 7 vs. is plotted in Fig. 2a.

We also calculated the energy-momentum relation for

the periodic square-wel l p~tent ia l ,~sing the expression

given by Smith.5 A co mparison o f the E-vs.-k curves

for the two otentials is shown in Fig. 3, where d

=

loo

VI =

O.leV, and m

=

0.025mo (Fig. 3a) a nd

o.07m0

(Fig. 3b). The zeros of the energy scales in Fig. 3 are

set at the bot tom of the conduction band in the narrower

energy-gap material.Therefore th e first ban d is moved

up by anamo unt slightly less than

Vl.

In he case of

IBM

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alternating impurities, the energy gap is narrowed slightly

by the introduction of the superlattice structure.

The energy band width El =

E ?r/d)

- E(0) decreases

as increases from 0.667 to 1.85; i.e., electrons are pro-

gressively more localized in he direction of the super-

lattice as the periodic perturbing potential increases. At

y

=

4, El is only one-tenth the unperturbed energy band

width in the first minizone, which means that the effective

mass in the superlattice direction

is

ten times greater than

the unperturbed value. This trend leads to virtually a two-

dimensional electron gas system.6

For large y, and particularly for the periodic square-

well potential, the E-k relation can be approximated by a

sinusoidal form, E,

=

$E1(l - cos k,d); the inflection

point is located at the center, ?r/2d,of the first minizone.

In general, each carrier with a definite ffective mass

interacts with the superlattice and generates a correspond-

ing set of energy bands. Therefore, whenever more than

one effective mass

is

involved, the otal population of

carriers is redistributed among the respective bands.

Transport properties

We used a simplified path integration method7 to obtain a

relation between the applied field F i n the direction of the

superlattice and the average drift velocity u . The equations

of motion are

- h k z =

eF

and

u

=

h dE,/dk,; 1 )

the velocity increment in a time interval dt is

du, = eFh-2 (d2E, / ak~)

t.

2)

The average drift velocity, taking into account the scat-

tering time T is written as

ud

=

lm

xp (- t / r du

=

eFh-' lmd'E, /dk;) exp (- t / r t . 31

Using the sinusoidal E-k approximation, we obtain

Od

g ( l ) [ h k d / m ( o ) ] 4)

and

dl

=

E / 1

+

T212>

5 )

where = eFr/hkd

=

k , / k d ; the effective mass

m(0)

is

determined by the curvature

of E ( k )

and

is

equal to

2h2/E1d2. he function g 4) (shown as the dashed curve

in Fig. 4 has maximum at =

l/?r

and thereafter

decreases, corresponding to a decreasing average drift

velocity, which results in a negative differential conduct-

ance because the current is proportional to

vd.

At high

fields the current isproportional to ( F T ) - ~ . he value

=

l / r corresponds to eFrd/h = 1 This threshold

(j = ,

9 ;

-

_

, x:R

4   -----

Figure

2 Reduced energy asunction of (a) wave

vector k , or reduced wave vector

p

for y = 0.5 and (b)

reduced amplitude y of the sinusoidal superlattice potential;

=

E ,

-

VI)

/ E o ,

y

=

V J E 0

and p

k, /ka.

Figure

3

Electron energy E, as a function of wave vector

k ,

in the direction of the superlattice. Solid curves are for

the sinusoidal potential, long-dash curves are for theperiodic

square-well potential, and the short-dash curves are for the

unperturbed cases. The arrows refer to the points of inflec-

tion. Values

of

theparameters are

VI

= 0.1 eV and

a)

rn =

0.025rno, y

=

0.667; (b)

rn =

0.07rno, y

= 1 85

/6

I

/'

.

-

2d

d

2d

7

7 317

,(

.*

6

JANUARY 1970

SUPERLATTICE

NEGATIVE

DIFFERENTIAL

CONDUCTIVIT

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I

Figure 4 Drift velocitymplitudes

as

functions of the

reduced parameter =

2 e F ~ d / h k , / k d :

(a) sinusoidal

potential; (b) periodic quare-wellpotential for

k , / k d =

0.5; and

(c)

periodic quare-well potential for

k i / k d

=

0.82.

Here

ki

is the wave vector at the inflectionpoint of

the E-k curve.Arrowsndicatehe eaks of the drift

velocity functions.

condition can be achieved with a n electric field strength

F = lo3

V/cm and

a

scattering time

T =

0.67psec.

For

small

y,

when

E k)

is not a sinusoidal function,

the

E-k

relation was approximated by sections of two

parabolas of opposite curvature, joined at the inflection

point

Ei , k i ) .

or the average drift velocity in this case

we obtained

and

Th e functionf(() is plotted in Fig. 4 for

k i /kd

=

0.82 (y

=

0.667) and 0.50 (y

=

1.85). These curves also indicate th e

existence of negative conductance, but the hreshold value,

2

0.4

or

eFrd/A

2

1.26

for the top curve, is slightly

greater tha n or he sinusoidal

E-k

relation. Since the

inflection point is shifting tow ard the minizone boun dary,

it is understandable that higher fields or longer scattering

times are required to obtain negative conductance.

As

the a pplied voltage is increased, howe ver, effects such

as Zener tunneling, avalanching and impact ionization set

in; eventually the negative conductance would be offset

by these effects. The possibility of Zener tunneling

to

the

64

second minizone when electronseach the first minizone

L. ESAKI AND R.

TSU

boundary could be a dominant factor in thecase of small

amplitude of the periodic potential. If,however, the am pli-

tude VI is of the orde r of 0.leV an d the applied field is

of the order of

lo3

V/cm, the tunneling probability can

be kept negligibly small. If the electron scattering time is

sufficiently long, electron s will undergo rf oscillation du e

to th e reflection at the minizone boundaries, the so-called

Bloch oscillation. This occur s fo r

eFrd/A >

27r, which

is several times the threshold value for negative conduct-

ance. Th e frequency of the Bloch oscillator is

eFd/h = 250

GHz for

F

= 103V/cm an d

d

=

l O O A .

The scattering

time then should be greater than 4 psec.

Discussion

In obtaining the solutions fo r the average drift velocity,

Eqs. 4 and

6 ,

we made twoassumptions; namely, that

k, =

eFf /A and tha t T is time-independent. The form er

assumption implies that k ,

= 0

a t

t

=

0,

which is justifiable

in relatively lightly do ped semiconductors.' The atte r

assumption is

a

reasonable approxim ation in the case in

which the allowed ban d width is ma de narrow.

In

the two

examples, m

=

0.025m0

and o.07m0, the smallermass

case equires an electron temperature close

to

1000 K,

whereas the larger mass case requiresonly

100 K,

t o

reach the negative differential conductivity region. This

electron temperature, which is also a function f the ampli-

tud e and the profile of the periodic potential, could be

kept very low with proper design of the structure. In such

cases the specimen could be operated with low electric

fields at cryogenic temperatures; ower emperatures are

helpful in obtaining onger cattering times.

Using th e Heisenberg uncertainty principle we estimated

the values of electron scattering time

T

and the mean free

path I that are required for these quantum mechanical

effects. For

AE

=

O.lE,

=

0.003

eV and

Ak

=

O.lkd

=

3

X

lo5

cm-', the inequalities are I2

330A

and

r 2 0.22

psec. This indicates that the mean free path should be at

least three times as long as the superlattice spacing. The

scattering time here is abo ut one-third

of

that previously

estimated for obtaining anegative differential conductance.

The scattering time

is

an impor tant factor in the effects

described and more-detailed calculations are being made

to verify the model. If the superlattice were perfect, the

scattering ime would be infinite, as is the case with an

idealcrystal lattice. Small deviations fro m he perfect

periodic potential, even when the long-range ord er is pre-

served, act as localized scattering centers. If the super-

lattice structure is prepared by

a

periodicvariation of

alloy composition, there will be unavoidable an dom varia-

tions in the magnitude of the thereby introduced super-

lattice potential maxima (at =

d/2,3d/2, . ).

However,

the probability density of conductionelectrons in he

superlattice structur e of the sinusoidal potential indicates

that electrons in the conduction ba nd would be bunched

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in the potential minima at

x

=

0

, 2 d , . Therefore,

small variations in themagnitude of the potential maxima,

where the electron probabilitydistribution has minima,

would have little effect.

If the deviation fro m an ideal superlattice becomes so

large that the long-range order is no longer preserved, a

disordered system will be obtained, w hich might be called

a

disordered one-dimensional superlattice. Even this struc-

ture, however, may provide

a

testing ground for mathe-

matical models used in he study of one-dimensional

disordered systems.' Correlation between theory an d exper-

iment on a disordered superlattice would lead to better

understanding of a three-dimensional disordered system,

i.e., an am orpho us substance. A lthough it may be a for-

midable task

to

construct the proposed superlattice, we

believe th at efforts directed

to

this end will ope n new areas

of investigation in the field of semiconductor physics.

Acknowledgments

The autho rs are pleased to acknowledge

W.

E. Howard

and R. Ludeke for theircontributions and P. J. Stiles

for stimulating discussions. Th e Bloch oscillator described

here is an outgrowth of discussions with W. E. Howard.

J A N U A R Y

1970

References

1 . L. Esaki, Phys . Rev . 109,603 1958).

2.R.N. Thomas and M. H. Francombe, Ap p l .Pkys .

Letters 13, 70 1968). Vapor rowth at ultra-high

vacuum may be suitable for this purpose.

3. J C . Slater, Phys .Rev . 87, 807 1952).

4.R. de

L.

Kronigand W. J Penney, Proc .Roy . SOC

A130,499 1930).

5. R. A. Smith,

Wave Mechanics of CrystallineSolids,

John

Wileyand Sons, Inc., New York 1961, Chap.

4.

6.

F.

Stern nd

W. E.

Howard,

Pkys .Rev .

163, 816

1967).

7. R. G. Chambers, Proc.Phys.SOC. London)A65, 458

1952);A. B. Pippard,

TheDynamics

of

Conduction

Electrons, Gordon and Breach Science Publishers,

Inc.,

New York 1965.

8. P.A. Lebwohl and R. Tsu showed thathe use of

the path ntegration method gives exact results under

these conditions private communication).

9. See,

for

instance, E. H. Lieb and D. C . Mattis, Mathe-

maticalPhysics in OneDimension,

Academic Press,

Inc.,

New

York

1966, and J.

Hori, Spectral Properties

of

DisorderedChains and Lattices, Pergamon Press,

Oxford 1968.

Received September 3,

1969

6

SUPERLATTICE

NEGATIVE DIFFERENTI AL CONDUCTIVI