125

Cornel HaŃiegan

The Ben Daniel-Duke Model Applied to Semiconductor Hetero-

structure - Part 1

We investigate the semiconductor heterostructure with the Ben Daniel-Duke model applied for the lowest conduction states Ga As-Ga (1) as and for the heavy levels at 0=⊥k in any heterostructures (1). In a quantic level we obtained the familiar staircase density of states(2). In (3) we calculated the incrgy position of the interface state in a single HgTe-CdTe heterojonction. We also obtained the existence of the interface state relies only on the relative position of the I’8 edges of HgTe and CdTe, their actual energy position, as well as their behavior at 0≠⊥k .

Keywords: Ben Daniel-Duke model, heterostructure, quantum well, Ben Daniel-Duke quantum well

1.The Ben Daniel-Duke model

This model of heterostructure energy levels is the simplest one and works qualitatively for the lowest conduction states GaAs-Ga(A1)As heterostructures with GaAs layer thickness larger than ~100Åand for the heavy levels at k┴=0 in any heterostructure. It amounts to assuming that the heterostructure envelope function is build from host quantum states which belong to a single parabolic band. For simplicity we shall take an isotropic band and for definiteness, a conduction ban. The effective masses in the A(B) layers will be denoted by mA

(mB). Each of

the levels is twice degenerate (Kramers degeneracy). Equation (1’) simplifies considerably as all the host bands are remote for the conduction edge under consideration. Thus, we can write

( )∑=

⊥⊥⊥

−+∂∂−

∂∂−++

8

1 002

2

0

2

0

22)(

220m

zlmmA

m mplm

k

zmpl

m

i

zmm

kzV

hhhh δε

ANALELE UNIVERSIT ĂłII

“EFTIMIE MURGU” RE ŞIłA

ANUL XVI, NR. 1, 2009, ISSN 1453 - 7397

126

∑=

+

∂∂+

∂∂−

∂∂

∂∂−

yxzlm

zlm

zzlm

kMzzM

ki

zMz ,

22 11

2

1

2 ααααα

hh

81;1

2 ,,

2

≤≤=

+ ∑=

lkM

k lmyx lm

εχχββα

αβαh

(1’)

( ) ( ) ( ) ( ) ( )zzz

k

zzzzVSS εχχ

µµε =

+

∂∂

∂∂−+ ⊥

2

1

2

222hh

(1)

( ) ( ) 110

1 −−− += zzssMmzµ (2)

( )

=layerBantoscorrespondzifm

layerAantoscorrespondzifmz

B

Aµ (3)

( )

=layerBatoscorrespondzifV

layerAantoscorrespondzifzV

ss

0 (4)

and Vs, is the algebraic energy shift of the S band edge when going from A to the B material. The boundary conditions at the A-B interfaces are also very simple. They are such that:

( ) ( ) dz

d

zandz

χµ

χ 1 (5)

are both continuous. It should be noticed that the effective mass mismatch contributes to the total confining barrier by a term which is k┴-dependent and, like VS(z) exhibits step-like variations. This extra term is however small in most instances(e.g. conduction states in GaAs-Ga1-xAlxAs, Ga0.47In0.53As-InP quantum wells) although it leads to a decreasing effective barrier height with increasing k┴ if mB>mA>0.

It is also interesting to notice that the effective mass mismatch leads to a discontinuity in the derivate of the envelope function at the interfaces. In the extreme cases where mA and mB are of the opposite signs, this discontinuity causes a cusp at the interfaces of the envelope function. The latter situation occurs in HgTe-CdTe heterostructure (but only at k┴=0)

2. The Ben Daniel-Duke quantum wells(mAmB>0).

The k┴ -dependent potential energy ( ) ( )z

kzVs µ2

22⊥+

h is even with respect to

the middle of the A layer. Thus as in chapter I, one can look for bound states solution in the following forms:

127

( ) ( )

( )( ) ( )zz

Lz

LzkBz

LzzkAz

eveneven

AABeven

AAeven

χχ

χ

χ

=−

≥

−−=

≤=

22exp

2cos

(6)

or:

( ) ( )

( )( ) ( )zz

Lz

LzkBz

LzzkAz

oddodd

AABodd

AAodd

χχ

χ

χ

−=−

≥

−−=

≤=

22exp

2sin

(7)

with:

BB

Bs

AA

As m

k

m

kV

m

k

m

k

2222

22222222⊥⊥ +−=+=−

hhhhεε (8)

Figure 1. – Dispersion relations versus the real and imaginary wave vectors in A and B layers stacked to form a BAB quantum well. The three dashed lines, drawn for three energies in the heterostructures and for k┴=0, show which wave vectors in each kind of layers participate in the heterostructure state. The upper line corresponds to a delocalized quantum well state and the middle line to a quantum well bound state. No heterostructure state

128

can be associated with the lower line as the carrier effective masses are assumed to have the same sign in this particular Ben Daniel-Duke quantum well.

Equations (6-8) hold if 02 >bk , i.e. if the heterostructure state is built from

the evanescent state of the B layer (see Fig 1.)

By matching χ(z) and ( )dz

dz

χµ 1− at the omterface 2

ALz = (or

equivanlently 2

ALz −= ), one obtains the implicit equations whoseroots are the

bound solutions of the Ben Daniel-Duke quantum well problem. These are:

statesevenfork

k

m

mA

B

A

A

BA 0sincos =− ϕϕ (9)

statesoddfork

k

m

mA

A

B

B

AA 0sincos =+ ϕϕ (10)

AAA Lk2

1=ϕ (11)

A comparison with the results obtained in chapter I shows equations (9-11) are the same as those of text book quantum wells except that the wave vectors kA,kB have been replaced by kA/mA and kB/mB. This is a direct consequence of the

matching conditions of ( )dz

dz

χµ 1− at the interfaces. In all other instances

however, many of the results obtained in chapter I can be applied to equations (9-11). In particular, the number of states bound by the well (at 0=⊥k ) is equal to

+=2

1

222

,2

int1 AA LV

mN

πh (12)

In figure 2 are shown the variations of the confinement energies E1,E2,E3 at 0=⊥k of a Ben Daniel-Duke quantum well (LA=100Å, Vs=0.3eV, mA=0.07m0)

with the mass ratio A

B

m

m . It can be seen that all the E1’s decrease with increasing

A

B

m

m and tend to values ∞

+1pE which are such that:

( ) ,.....2,1,01 ==∞+ ppLEk ApA π (13)

129

Figure 2.- Evolution of the confinement energies E1,E2,E3 with the mass

ratioA

B

m

min a Ben Daniel-Duke quntum well L=100Å; Vs=0.3eV;mA=0.07m0.

This equation resembles the bound states equation in a quantum well with

an infinite barrier height. It expresses however quite a different physical situation. In a quantum well with infinite Vs, the envelope function vanishes at the interface. Besides, according to equation (13) N is infinite. If Vs diverges, kB also diverges. The leads either to 0cos =Aϕ (Eq. (9)) or 0sin =Aϕ (Eq.(10)). The ground

state solution, which is nodeless, fulfils ( ) π=∞=sAA VkL , the associated

envelope function having a finite slope at 2

ALz ±= . On the other hand, when Vs

is kept fixed but A

B

m

m increases to infinity, N remains unchanged. Moreover, if the

ground states envelope function is to nodeless, it has to be a cosine in the well. It should barely penetrate the barrier (since kB increases) and, in addition, should

have a derivate at 2

ALz ±= whose modulus becomes smaller and smaller to

comply with the continuity of ( )dz

dz

χµ 1− at the interface. Thus at infinite A

B

m

m

ratio the only possible wave function is constant in the well and zero in the barrier, so that both the envelope function and its derivate are also zero in the barrier. The only ( )zk Acos function which is constant corresponds to 0=Ak and thus to

130

01 =E … This is why when A

B

m

m diverges and Vs is fixed the even states of the

well fulfil 0sin =Aϕ (and not 0cos =Aϕ ), i.e. admit 01 =E as an acceptable

solution. Symmetrically, the odd states fulfil 0cos =Aϕ and finally the series of

levels (Eq.(13)) is recovered. The ground state envelope function of a quantum well with either infinite Vs or finite Vs but infinite mB is shown in figure 6 in order to depict the differences between the two physical situations.

When the effective masses mA and mB are not widely different, as for instance in GaAs-Ga(Al)As heterostructures, the in-plane dispersions of the subbands attached to the 0=⊥k bound states of a quantum well are nearly

parabolic in k┴:

( ) ( )n

nn m

kEkE

20

22⊥

⊥ +=h

(14)

The in-plane mass mn should, in principle, be obtained by numerically solving equations (6-11). However when k┴ is small enough, an approximate scheme can

be designed in the following way. The term ( )z

k

µ2

22⊥h in equation (1) is formally

rewritten:

( ) ( )

−+= ⊥⊥⊥

nn mz

k

m

k

z

k 11

222

222222

µµhhh

(15)

Figure 3.- Ground state envelope functions for a quantum well with infinite sV

(dashed line) or for a quantum well with finite sV but infinite a

b

m

m(solid line).

131

whose eigenstates are in the form given by equation(14). The first order corrections to these eigenstates are given by:

nE∆ =2

22⊥kh

−+−

nnb

Bnb

A mEP

mEP

m

1)(

1](1[

1 (17)

where:

∫∞

==2

22 )(2)(

AL

Bnnb k

BdzzXEP (18)

is the integrated probability of finding the electron in the barriers while in the nE

state. The first order energy shift will vanish if:

).(1

)](1[11

nbB

nbAn

EPm

EPmm

+−= (19)

References

1. Duggan G., The Journal of Vacuum Science and Technology. B3 (1985) 1224.

2. Guldner Y., Bastard G., Vieren J.P., Voos M., Faurie J.P., Million A., Physics Review, B51 (1983) 907.

3. Yia-Chung Chang, Schulman J.N., Bastard G., Guldner Y., Voods M., Physics Review, B31 (1985) 2557.

4. Lin Liu Y.R., Sham L.J., Physics Review, B32 (1985) 5561. 5. Voisin P. Two dimensional Systems, Heterostructures and Superlattices

edited by G. Bauer, F. Kuchar and H. Heinrich, Springer Series in Solid State Sci. 53 ,Springer Verlag, Berlin, 1984, p. 192.

6. Bir G.L., Pikus G.E., Symmetry and Strain-induced Effects in Semiconductors, Wiley, New York, 1974.

7. Marzin J.Y., Heterojunctions and Semiconductor Superlattices edited by G. Allan, G. Bastard, N. Boccara, M. Lannoo and M. Voods, Springer Verlag, Berlin, 1986, p. 161.

Address:

• Dr. Cornel HaŃiegan Resita, Al Breazova, nr. 2/A, ap. 19, 320067, Resita,