Non-linear transport properties of miniband conduction in the presence of crossed electric and magnetic fields: A semi-classical approach

~

Solid-State Electronics Vol. 37, Nos 4--6,pp. 1015--1019,1994

Pergamon

Copyright ~ 1994ElsevierScienceLtd Printed in Great Britain.All rights reserved 0038-1101/94 $6.00+ 0.00

NON-LINEAR TRANSPORT PROPERTIES OF M I N I B A N D CONDUCTION IN THE PRESENCE OF CROSSED ELECTRIC A N D MAGNETIC FIELDS: A SEMI-CLASSICAL APPROACH F. ARISTONE 1"2, J. F. PALMIER 3, A. SIBILLE4, D. K. MAUDE l, J. C. PORTAL 1"2and F. MOLLOT 5 ~CNRS-SNCI, B.P. 166-X, 38042 Grenoble Cedex, :CNRS-LPS-INSA, Complexe Scientifique de Rangueil, 31077 Toulouse Cedex, 3CNET, 196 Av. Henri Ravera, 92220 Bagneux Cedex, 4ENSTA-Department d'Electronique, 32 Boulevard Victor, 75015, Paris and 5IEMN-DHS/UMR-USTL, 59655 Villeneuve d'Ascq, France

Abstract--In this paper we compare a series of perpendicular transport measurements for short period GaAs-AIAs superlattices in presence of a magnetic field applied parallel to the layers with the semi-classical calculations of the miniband conduction properties. The theoretical approach takes into account the local solution of the Boltzmann equation in the presence of scattering mechanisms such as phonons and interface fluctuations. The Poisson and drift-diffusion equations have been self-consistently solved to obtain the current density vs the applied electricalfieldcharacteristics, which is directly compared with the experimental measurements. A good agreement is found at 300 K for all values of the electric and magnetic fields. At low temperatures a pronounced disagreement is observed when the electric field F is greater than the critical value F, that determines the negative differential resistance regime in the current-voltage characteristics. We discuss the physical origin of this difference.

The concept of superlattices (SLs) was introduced some years ago by L. Esaki and R. Tsu[I] as a new class of semiconductor material. SLs are formed by the alternate growth of electronically different materials (typically two) along one direction, with individual widths small enough to give significant quantum effects, i.e. of the order of 100/~. This new one-dimensional man-made crystal presents a Brillouin zone smaller than that of the bulk materials due to the larger period of the artificial lattice. The energy spectrum along the growth direction is composed of minibands and minigaps. The first miniband width for conduction electrons is typically of the order of a few meV up to 200 meV, in comparison to the usual ~ l e V for the conduction band in pure materials. These minibands occur because of the non-zero tunneling probability between two consecutive wells. The energy-momentum dispersion of these minibands should allow the observation of the negative differential velocity (NDV) regime for electrons moving in the presence of a sufficiently large applied electrical field. The NDV regime is a direct consequence of the negative effective mass experienced by the carriers. This new type of material with a miniband width controllable only by external (growth) parameters has attracted much interest due to the potential applications for the elaboration of devices such as ultrahigh frequency (/> 10 GHz) oscillators. Very recently Hadjazi et a/.[2] have obtained a 60 GHz oscillator with a GaAs-AIAs SL structure. SLs have been studied either by electrical transport measurements,

such as thermal-saturation of miniband transport[3], tunnelling between two-coupled SLs[4], under hydrostatic pressure[5] and also using optical techniques such as photo-luminescence[6] and electro-reflectance[7]. The existence of the NDV regime in SLs as a bulk material effect has been demonstrated by Sibille et a/.[8] in undoped samples. Studying n-type doped samples they also demonstrated the existence of the negative differential resistance (NDR) in short period SLs[9]. Palmier et al.[10] have calculated in detail the effects of a crossed electric and magnetic fields on the velocity vs the electrical field relation-- V (F )--by solving the Boltzmann transport equation in the presence of a limited number of scattering mechanisms. As a starting point we take the local V ( F ) relation and calculate the current density as a function of the applied bias voltage solving a self-consistent system of the Poisson and drift~liffusion equations, taking into account the conservation of the carriers. These results are directly compared to the experimental curves, obtained for different values of the applied magnetic field and over a range of temperature. The samples studied consist of a series of periodic SLs of GaAs-AIAs hetero-layers grown by molecular-beam-epitaxy. They are lightly Si-doped, i.e. n -~ 2 x 1016cm-~ and were grown on a highly doped substrate ( ~ 2 x 10t~cm-3). The doping concentration was graded on either side of the active region to avoid abrupt

1015

F. ARISTONEet al.

I016

hetero-junctions between the SL and the contacts. Standard contact techniques with typical mesa's areas of 2800/~m 2 were used. In this work we present results obtained for three samples of miniband widths equal to 4, 9 and 16 meV. These samples are nominally of 22/8, 19/7 and 22/5 mono-layers of GaAs/AIAs respectively. The active region is about 1/~m, which means typically a total of 125 periods in each sample. The miniband widths have been calculated using a KP model, taking into account a constant averaged mass of GaAs and AlAs. Identical results have also been obtained using a transfer matrix approach. In Fig. l(a) a schematic representation of the sample structure is shown. The miniband width, the applied electric field and the orientation of the magnetic field are represented. Only the F conduction band for the GaAs and for the AlAs is shown. Transport properties of electrons moving along the growth direction of a SL were first investigated by Shik[l 1]. He also considered the presence of a magnetic field applied perpendicular to current. However, in his approach only low values of B for which the energy quantisation by the magnetic field was negligible have been taken into account. Recently Palmier et al.[10] have presented a detailed numerical solution of the Boltzmann equation under the same conditions

and they obtained the local V(F)--velocity vs the applied electrical field-characteristics for different values of the magnetic field intensity. In their work the temperature Tand the interface fluctuation A that describes the interface roughness, are additional parameters in the formulation of the problem. We have considered the solution of the Boltzmann equation in presence of polar optical and transversal phonons together with the interface roughness scattering mechanisms, as described in Ref. [10]. To calculate the J ( V ) - - c u r r e n t density vs the applied bias voltage-curve in order to compare it with experimental results we have used the Poisson equation: ~3F ~3z-

q (n Er

Nd)

(1)

and the drift-diffusion equation:

J.=q{nVD(F)+ D.(F) dn J

where n is the density of negative carriers, N~ is the doping concentration of donors, er is the static dielectric constant and D, is the diffusion constant. The carriers conservation equation in the absence of any generation and recombination mechanism (dark conditions of measurements) and in steady-state condition is given by: 1 ~J --= O.

(a)

(3)

q c3:

V

(2)

In order to apply suitable boundary conditions we simply write eqn (!) in terms of the electrostatic potential 4~, given by F = -0~b/dz, and from eqns (2) and (3) we obtain:

0

Z

~r qnVD(F)+qD"(F)o:

/A

/A

a

a

b Period

/A a

b

: d=a+b

(b)

n÷ GaAs

Fig. 1. Schematic SL device and the experimental set-up. (a) Potential profile for a periodic SL of materials A and B. A is the miniband width of accessible energy for electrons. (b) SL device in presence of the applied electric and magnetic fields.

=0.

(4)

Assuming a known starting expression for the n (z) distribution and the F(z) in eqns (!) and (4) respectively, we rewrite (I) and (4) in terms of difference equations using the finite-element method (PI)[12]. The non-linear decoupled system is solved by the Newton-Raphson algorithm. The iterative procedure is repeated until a stable and convergent solution for n (z) and F(z) is simultaneously obtained. We note that the present model may not be applied when ~VD/~F < 0 nor when NdL exceeds a certain critical value[13], where L is the length of the (active) SL region. This procedure has been validated for pure GaAs samples, whose VD(F) law is well known. In our experiments we have measured the current-voltage characteristics for a series of samples in presence of a magnetic field applied parallel to the SL's interfaces, i.e. perpendicular to the current direction. In Table 1 we present the most important parameters of our samples.

1017

Perpendicular transport methods Table I. Main parameters of the three samples studied in this paper Sample A (meV) Widths(,~) No. of periods SI 4 62/23 118 $2 9 54/20 135 $3 16 62/14 131 The minibands widths have been calculated with a Kronig-Penney model using a constant averaged mass and also using the transfer matrix technique. The period of these SLs have been obtained from X-ray diffraction. The electrical contacts were formed using standard lithographic techniques and had an area of 2800#m:. The active region of the samples were doped at 2 x 101~cm 3.

than Ft. However, in the low temperatures regime the experimental results in the N D R region diverge

l l l l | l l l l | l l l !

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F ~'• Typical J(V) characteristics for each sample are shown in Fig. 2 for different temperatures. The current increases with applied bias for applied electric fields less than the critical value Ft. At r o o m temperature we observe that the current intensity does not really decrease but remains almost constant when the applied electric field is higher than Ft. When the temperature decreases the current intensity decreases abruptly for electric fields such that F > F c, i.e. the N D R is more pronounced and the peak in the J (V) curves is clear. The increase of the current when the temperature decreases is essentially due to the decrease of the scattering relaxation time z, which occurs because phonons scattering is less important at lower temperatures. At very low temperatures ( T ~ 5 0 K ) the current decreases with decreasing temperature. A more detailed discussion of these effects will be included in a future publication. For all temperatures and miniband widths we have observed that with increasing magnetic field, Fc shifts to higher values. This effect occurs mainly because of the distortion of the electronic orbits in the presence of the magnetic field, which tends to localise the electronic movement. A small effect due to the nature of scattering mechanisms is present although it is only a second order correction. In Fig. 3 we have plotted both experimental and calculated J (V) curves. A satisfactory agreement is achieved in the non-linear but positive domain of the differential resistance for all temperatures. U n d e r N D R conditions the unity for the solution of eqns (l) and (4) cannot be asserted. This problem has been clearly explained in the framework of the G u n n effect theory. Briefly, for a certain critical value of the product nL, which depends upon d V/dF, an absolute instability appears with a high field domain near the anode. We therefore cannot interpret the N D R region in terms of our macroscopic model for two reasons. Firstly the I(V) characteristic depends on the external circuit conditions. Secondly there is the appearance of new phenomena such as resonant tunnelling in the high-field domain. Even though our model cannot be applied in the N D R region, we have obtained a similar behaviour o f theory and experiment at high temperature even for electric fields higher

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from the calculated values. The shift of the critical field Fc to higher values in the presence of an applied magnetic field is understood in terms of a competition between electric localisation, i.e. the Bragg reflection at the Brillouin zone and the magnetic localization. From our analysis we have observed that a relatively

simple classical description of the transport properties in a SL lead to a satisfactory agreement in all of the positive domain of the differential resistance. However, a better quantitative agreement between the theoretical and experimental results of the NDR regions remains to be obtained.

Perpendicular transport methods 3

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Fig. 3 (c) Fig. 3. Direct comparison between experimental results (a.2, b.2, c.2) and calculated curves (a.l, b.i, c.l) for samples Sl (a), $2 (b) and $3 (c)--see Table l--for T = 300 K. Very good agreement is obtained for all curves below the critical electric field F¢. Fair agreement is obtained in the NDR region. We note that the disagreement in the NDR region increases with decreasing temperature.

Acknowledgements--We thank S. Vuye and J. C. Esnault for sample mounting and M. Rabary for technical support. F.A. is grateful to CAPES-Brazil for a grant. We acknowledge CNET, EEC and Conseil R6gional des Midi-Pyr6n6es for financial support.

6. 7.

REFERENCES

8.

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9. 10. 11. 12. 13.

J. C. Portal and F. MoUot, Japan. J. appl. Phys. 32, 144 (1993). 8. J. Skromme, R. Bhat, M. A. Koza, S. A. Scharz, T. S. Ravi and D. M. Huang, Phys. Rev. Lett. 65, 2050 (1990). P. Voisin, J. Bleuse, C. Boucbe, S. Gaillard, C. Alibert and A. Legreny, Phys. Rev. Lett. 61, 1639 (1988). A. Sibille, J. F. Palmier, H. Wang and F. Mollot, Phys. Rev. Lett. 64, 52 (1990). A. Sibille, J. F. Palmier, H. Wang, J. C. Esnault and F. Mollot, Appl. Phys. Lett. 56, 256 (1990). J. F. Palmier, G. Etemadi, A. Sibille, M. Hadjazi, F. Mollot and R. Planel, Semicond. Sci. Technol. 7, B283 (1992). A. Ya Shik, Soviet. Phys. Semieond. 7, 187 (1973). J. F. Palmier, Resonant Tunneling in Semiconductors (Edited by L. L. Chang et al.), p. 361. Plenum Press, New York (1991). B. W. Hakki, J. appl. Phys. 38, 808 (1967).