The electronic properties of the miniband and the effect of external electric fields in superlattices

Super/attices and Microstructures, Vol. 7, No. 2, 1990

147

THE ELECTRONIC PROPERTIES OF THE MINIBAND AND THE EFFECT OF EXTERNAL ELECTRIC FIELDS IN SUPERLATTICES C.Y. Pong and R.F. Gallup Department of Physics University of California Davis, CA 95616 L. Esaki and L.L. Chang IBM Thomas J. Watson Research Center Yorktown Heights, NY 10598 (Received 30 January 1990)

The properties of the lowest-energy miniband in a square-well model superlatrice and the electronic properties of this superlattice under an external electric field are studied by simple variational calculations within the concept of the supercell. The calculated width of the miniband (2.4meV) agrees well with the value measured for a GaAs/AIAs superlattice having similar parameters (2meV). For weak fields, the results suggest coherent electron conduction over a few superlattice periods between scattering events, a conclusion which is consistent with experimental results. For strong fields, the formation of Stark ladders is illustrated.

1. Introduction The electronic properties of the minibands and the influence of external electric fields in superlattice semiconductors have recently attracted more attention! 1-4] Schneider et al [1} performed time-resolved photoconduction experiments and found that, within a limited range oil small electric field magnitudes, the electron travels over several superlattice periods between scattering events. Other experimental results [2] obtained using large electric fields provide convincing evidence for the existence of Stark ladders. Theoretically, Ferreira and Bastard [5] used the envelope function approach and atomic wave functions to examine the behavior of an electron moving in a superlattice under the influence of an electric field as a function of time. h~ this paper, we report the results of simple variational calculations using plane waves as basis functions and periodic boundary conditions to study the steady state properties of the lowest energy miniband and the behavior of an electron under several different external eleclric fields in a square-well superlattice. From

0 7 4 9 - 6 0 3 6 / 9 0 / 0 2 0 1 4 7 + 05 S02.00/0

our calculations, the width of the lowest energy miniband with no electric field was found to be 2.4meV, This result agrees well with the measured value of 2meV for a GaAs/AIAs superlattice with superlattice parameters which are close to our model's values! 1l and with the 2.3meV value calculated from the Kronig-Penney model! 6] Our results also indicate that for electric fields which produce potential differences between adjacent superlattice wells which are small compared to the bandwidth of the lowest miniband, the electron can move coherently over several superlattice periods between scattering events in response to the electric field. As the field strength increases, however, Stark ladders prevail. The present approach provides new insight into the problem of superlattices under electric fields. Since the basic ideas involved in the calculations are very simple, it is possible that this method can be applied to the study of other systems, especially those with smaller barrier heights than the one for GaAs/A1As. The remainder of this paper will be organized as follows: In section 2, the model and method will be © 1 990 Academic Press Limited

Superlattices and Microstructures, Vol. 7, No. 2, 1990

149

well or barrier is set equal to the value of the potential calculated for the center of the well or barrier (Fig. l b dashed lines).

~l-'a ....

i . . . .

I- ~ '

~1 ......

~. . . . .

-~

'-

.g B. Method W i t h the periodic structure, we applied the variational method and used plane waves as basis functions to solve the Schrodinger equation with the model potential given in Fig. lb. For the case of a superlattice with no external electric field, a supercell containing one superlattice period was used. W i t h this supercell, only about 180 plane waves are needed to give a convergence of the order of 0.0t meV for the energy eigenvalues of the wavefunctions. This conclusion is obtained by comparing the results of the 180 plane-wave case with a calculation using 400 plane waves.

i

g

\ I

3

4

5

x {lc~) Fig. 2a. The superlattice charge density for the lowest energy state at k = 0.

With an applied electric field, the supercell which was used is eight times larger than the unit cell of the superlattice. By using the same cutoff kinetic energy as for the 180 plane-wave case, matrices with sizes of about 1400 x 1400 were diagonalized.

i

r ~r

~J

i ;3

i

L t

I

r

i ,

[

I!

[

>,

2~

3. Results and Discussions In order to provide a proper reference, we shall first present the results for the lowest energy miniband without an electric field. These are followed by the results first for a weak field case and then for a strong field case.

,/¸ ',,

/'\,

I: / \

i

In Fig. 3, the one dimensional energy-momentum relation for the lowest miniband is plotted. The energy is expressed in meV while the m o m e n t u m is in units of 2%/a. We calculated the width of the lowest miniband ~.o be 2.4meV. This result agrees very well with the mea-

/ ,

\ '

I

~: I:

\,~

,/

(1

The charge densities of the states in the lowest energy miniband at k = 0 and k = 0.5(27r/a) are given in Figs. 2a and 2b, T h e dashed lines in these figures indicate the positions of the barriers. A total of four superlattice periods are plotted. Since the phase factors of the Bloch states cancel when the absolute value squared of the wavefunctions is calculated, only the periodic parts of the Bloeh functions are plotted. T h e two charge densities are quite similar. The maxima of the charge densities are near the center of the wells as expected.

,

i/ i 1

o

5

2

A. The Lowest-Energy Miniband The model of the GaAs/AIAs superlattice discussed in section 2 is used for this calculation. The growth direction is defined to be the x-direction. The lattice constant, a, is 140.0/~.

j

x 00oh) Fig. 2b. The superlattice charge density for the lowest energy state at k = 0.5(2~r)/a.

sured value of 2meV given in Ref. 1 and with the value of 2.3meV obtained from the Kronig-Penney model. B. The Weak Field Case Let us first define the strength of the electric field F, W h e n F =(2.4meV)/ea (where e is the charge of the electron), the potential difference between adjacent wells in the supercell is equal to the energy width of the field-free miniband (2.4meV). This value of F is referred to as the critical field. We consider values of F such that eFa is less than (greater than) 2.4meV to be low (high) strength. Next, as discussed in section 2, the supercell concept was used for the cases of the GaAs/A1As superlattice under the influence of the electric fields. The period of

Super/attices and Microstructures, Vo/. 7, No. 2, 1990

150

3'

i

j

i

,

i , r

i I I

i i

I r

i

i

r

~

,

'

M i

,

g 2

ILl

o 1'

0

A 2

3

4

5

x (~ooA)

Fig. 4. The charge distribution for the weak field i

0 0

i

t

i

0.25

J

,

i

cage.

I

0.5

k (2~/a) Fig. 3. The energy-momentum relation for the lowest energy miniband of the field-free superlattice.

the supercell is eight times the lattice constant of the superlattice. However, we will use charge densities from HALF of the supercell for our discussions, We would also like to emphasize that the values of the energies of the states are not affected by the symmetric arrangement of the electric field within one supercell except that the energies are doubly degenerate as compared to the ones obtained using the asymmetric field configuration. In addition~ because the potential is symmetric in space, the wavefunctions have definite parity. The odd parity states will be used in the following discussion. In Fig. 4, we present the charge density of the lowest band for an F value of 0.3meV/ea. The prominent feature exhibited in Fig. 4 is the nonuniform distribution of the electronic charge density as compared to the results given in Fig. 2a. With the application of a weak electric field, the charge density accumulates in the low potential region. Notice, however, that the charge distribution is not localized solely at the lowest potential well. Instead, the wavefunction is delocalized over 4 periods of the superlattice. We interpret this result as suggesting that an electron can travel coherently over several lattice constants between scattering events when the superlattice is subjected to a weak electric field. (Notice, however, that no particular scattering mechanism is analyzed or suggested in our calculations.) After each scattering event, an electron experiences a new period of the electric field, all previous history is lost in the scattering event.

g

g g O I T t

,{ 1

2

3

4

5

x (1ovA)

Fig. 5. The charge distribution for the critical field case.

C. The Critical Field Case In Fig. 5, we have plotted the charge distribution of the lowest energy state for the critical electric field value of F =(2.4meV)/ea. This charge distribution indicates that a qualitative interpretation of the critical field is the value of the field at which the lowest energy electronic state begins to be localized solely within the lowest energy potential well. D. The Strong Field Case The results to be discussed are for an electric field F =19.6meV/ea, which is definitely larger than the critical field value of F =(2.4meV)/ea. In Fig. 6, we have plotted the charge distributions of the lowest few states. The lowest energy state is completely confined in the region of the lowest potential.

Superlattices and Microstructures, Vol. 7, No. 2, 1990 H~ i

/' ',,

S',

3 LT i!

i v >,

?l

~i

/

'~

~

/' ""

?

!

/ ?'

i

¢

il

ri

', I

I

'

) 4

I

I

'

'L[

i

i

'

I

'

i

6

/I i ~!

i/

4!

Ii ] ]

fl

i !,~

,,

151

electric fields, however, our results show that the electronic states correspond to Stark ladders. Since the basic ideas involved in this calculation are very simple and the boundary conditions of the supercell can be treated easily, it is possible that this method can be applied to the study of other systems. Acknowledgment-CYF would like to thank the IBM Corporation for summer faculty support in 1989, and H. Ohno for several helpful discussions. References

0

I

2

3

4

5

x (~ooh)

Fig. 6. The Stark ladders produced in response to a strong electric field. The next higher energy state is localized in the well with the, second lowest potential and so on. The energy difference between two states is just equal to the potential difference due to the electric field. Consequently, Stark ladders are obtained from our calculations. 4. Summary We have used a simple variational method with plane waves as basis functions and a square-well superlattice model to study both the lowest-energy miniband structure of a GaAs/A1As superlattice and the electronic properties of this superlattice under the influence of several external electric fields having a wide range of magnitudes. For the cases with an electric field, we used the concept of the supercell which is similar to that used in the treatment of amorphous materials. We showed that for weak electric fields, the electronic states extend over several lattice constants of the superlattice, This result illustrates the experimental results which indicate that an electron can travel coherently for a few laltice constants between scattering events. For strong

1. H. Schneider, H.T. Grahn, and K. yon Klitzing, presented at the 4th International Conference on Modulated Semiconductor Structures, Ann Arbor, July (1989); H. Schneider, K. Von Klitzing, K. Ploog, Superlattices and Microstructures, 5, 383 (1989). 2. E.E. Mendez, F. Argullo-Rueda and J.M. Hong, Physical Review Letters 60, 2426 (1988). 3. P. Voisin, J. Bleuse, C. Bouche, S. Gaillard, C. Alibert, and A. Regreny, Physical Review Letters 61, 1639 (1988). 4. J. Bleuse, P. Voisin, M. Allovon, and M. Quillec, Applied Physics Letters 53, 2632 (1988). 5. R. Ferreira, and G. Bastard, Physical Review B38~ 8406 (1988). 6. C. Kittel, "Introduction to Solid State Physics", 2nd Edition, J. Wiley and Sons, New York, p. 281. 7. J.S. Nelson, C.Y. Fong, and I.P. Batra, Applied Physics Letters 50, 1595 (1987). 8. L.H. Yang, C.Y. Fong, and C.S. Nichols, Proceedings of the Materials Research Symposium 118, 513 (1988). 9. K.Kunc, in "Electronic Structure, Dynamics, and Quantum Structural Properties of Condensed Matter", edited by J.T. Devreese and P. Van Camp, Plenum Press, New York, p. 227(1984).