Cyclotron FIR emission from hot electrons in GaAsGaAlAs heterostructures

Solid-State ElectronicsVol. 37. Nos 4~, pp. 1213-1216. 1994 Pergamon

0038-1101(93)E0048"6

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

CYCLOTRON FIR EMISSION FROM HOT ELECTRONS IN GaAs-GaA1As HETEROSTRUCTURES W. ZAWADZKII']", C. CHAUBET2, D. DUR2, W. KNAP2 and A. RAYMOND2 qnstitut ffir Halbleiterphysik, Kepler Universit/it, 4040 Linz, Austria and 2Groupe d'Etudes des Semiconducteurs, Universite Montpellier lI, 34095 Montpellier, France Abstract--We study far infrared emission from GaAs-GaAIAs heterostructures, induced by electric pulses in the presence of a magnetic field and a hydrostatic pressure. Cyclotron masses are measured as functions of 2D electron density in the strong electron heating regime at pressures P = 0 and P = 7 kbar and the detection energy of 4.43 meV. The results are described by an effective two-level k. p theory, which takes consistently into account the effect of band's nonparabolicity in GaAs on electric and magnetic quantization. It is shown that the observed emission spectrum is due to eight transitions between Landau levels (populated up to the optic phonon energy), since under the strong heating conditions the 2D electron gas is nondegenerate. This is independently confirmed by magnetotransport measurements, Very good theoretical description of emission experiments at pressures P --- 0 and P = 7 is achieved with the use of bulk GaAs parameters. Theoretical estimations of the heating conditions in crossed magnetic and electric fields indicate that the electric field in our GaAs--GaAIAs structures is highly inhomogeneous.

i. INTRODUCTION Cyclotron masses of 2D electrons in G a A s - G a A 1 A s heterostructures have been subject of intense experimental and theoretical studies in the last years. Although bulk G a A s is not a typical narrow-gap material, its conduction band is markedly nonparabolic, which leads to a well observable increase of the mass at accessible electron energies. In the present paper we are interested in the dependence of the cyclotron mass in G a A s - G a A I A s heterostructures on the 2D electron density Ns (cf. Refs[2-12]). The influence of the band nonparabolicity on the electric and magnetic quantization in heterostructures has been treated by Zawadzki[13,14] (cf. also Ref.[3]). Other, less consistent schemes have been used in the literature. A frequent procedure is to neglect the band nonparabolicity in the electric and magnetic quantization and then to calculate an energy-dependent effective mass from the bulk twolevel k . p formula:

m*(E)=m~(l + 2 ~), in which m* is the band-edge mass value and E 8 is the energy gap. One then introduces ex post the electric and magnetic quantization by breaking the total electron energy E into an "electric" part (related to the electric quantization in the well) and the "transverse" part (related to the magnetic quantization or free motion parallel to interface). This scheme was used for example by Hopkins et a/.[5] and Warburton et al.[12]. Another frequently used procedure intro?Permanent address: Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland.

duces a position-dependent mass m*(z), resulting from the fact that in 2D systems the electron energy E, as counted from the band edge, is often a function of the electron position z. One calculates then the "average" or the "'most probable" electron position z0 and the corresponding mass m*(z0). This scheme was used for example by Chaubet et al.[! I} in the description of the cyclotron mass as a function of hydrostatic pressure. Thiele et a/.[6] used m*(z) concept in the multi-level k . p scheme[l 5]. Batke el al.[9], describing the magnetic field dependence of the cyclotron mass, used the above mentioned theory of Zawadzki. Finally, Lassnig[4] employed an incomplete five-level k - p model in a selfconsistent calculation, which included a penetration of the electron wavefunction into the GaAIAs region. 2. EXPERIMENTAL PROCEDURE

The investigated samples were G a A s - G a A I A s heterojunctions grown by M O C V D technique (the undoped G a A s layer was n-type residual). We used a liquid medium pressure cell to produce hydrostatic pressures up to l0 kbar at room temperature. The pressure was always applied at room temperature and then the cell was slowly cooled down. The cell had a saphire window for the propagation of far-infrared (FIR) radiation and a plug of fit-through connections. These allow one to heat the electron gas by electric pulses to obtain the F I R emission and to measure P~x and p.~,. magnetoresistance components. The studied sample (emitter) was placed in one coil and the detector in the other. The F I R radiation was guided to the detector by a copper light pipe. A magnetically tunable G a A s photoconductive detector was used. Magnetic field of the emitter was tuned

1213

W. ZAWADZKIet al.

1214

well described by the five-level k - p model[l]. The calculated dispersion relation for the conduction band E(k) can be well approximated by the two-level formula: 2m~'=E

a.u.

t 2

[

a

3

B (Tesla) Fig. 1. Cyclotron-resonance emission peak obtained from GaAs43aAIAs heterostructure (sample B, P = 0). and that of the detector was kept constant. An example of the CR emission spectrum is shown in Fig. 1. Due to heating of 2DEG, necessary to obtain the CR emission, the Q u a n t u m Hall Effect disappeared. In this regime, the 2D density of electrons N s was determined from the components of the resistivity tensor:

1 PxyB Ns= 2 + " " e Pxx P':,.

1+

,

(2)

where E is the electron energy (counted from the conduction band edge), m~' is the band-edge effective mass, and E* is an effective energy gap, obtained from a fit to the real E(k) dispersion. The value of E* = 0.98 eV has been calculated, as compared to the real gap value E(F~6)- E ( F [ ) = Eg = 1.52 eV. Using the semiclassical procedure we generalize eqn (2) for the presence of magnetic and electric fields. Thus, we replace hk by p + eA (where A is the vector potential of magnetic field B) and E by E - U, where U is the potential energy due to electric field. This results in a differential equation for the envelope function ~P associated with the conduction band: [2--~n, (P + cA):

' ] ~=0. -E--~(E-U)(E~+E-U)

(3)

We take the potential U = U(z). The magnetic field B is also parallel to the z direction and we describe it by the Landau gauge A = [ - B y , 0, 0]. The solutions of eqn (3) take then the form:

qt = e i k ~ x c k . ( ~ ) f(z )'

(4)

(1)

The transport experiments were performed by the a.c. method, in order to measure the resistivity components in the same electrical conditions as those used for the heating of the 2D electrons in the FIR emission experiment. Typical results are shown in Fig. 2. Average electric field in the sample was around 50V/cm. However, it seems that the local fields responsible for the CR emission were approximately ten times higher. At low temperatures, the effect of pressure is to diminish the density of 2DEG in the channel[ll]. In order to change N s at low temperatures one illuminated the sample by a red light emitting diode. This transfers the electrons from the DX centres to the conduction band. The system was sufficiently sensitive, allowing one to control Ns in this way.

where L = (h/eB) ~:2 is the magnetic radius, ~b. is the harmonic oscillator function, yo=kxL 2, and n = 0, 1, 2,... is the Landau quantum number. The

-8

(b)

(a) a

0.50 --

6 b

xx

x

0.25 2

3. T H E O R Y

The theory employs a two-level k. p model of the band structure, generalized for the presence of electric and magnetic fields. However, GaAs is not truly a narrow-gap material and the two-level model is not directly applicable to the description of its conduction band. For this reason we use the following approximate procedure. The conduction band of GaAs is

0

1

2 B (Tesla)

3

0

Fig. 2. Magnetoresistance components p,., and p,, measured on GaAs-GaAIAs heterojunction in the conditions of FIR emission (a.c. method). Due to the heating of 2DEG by electric pulses the quantum Hall effect has disappeared.

1215

Cyclotron FIR emission from hot electrons in GaAs-GaAIAs heterostructures x, y motion is then quantized into the Landau levels. After some manipulations one obtains:

GaAIAs-GaAs 0.074

/ f

I

2

1

-

ei

-

v)

×(E~ + E + E ± - U ) ] f ( z ) = O ,

(5)

0.072

where E . is the transverse energy: *

1'~-]1/2 0.070

and toc = eB/m~. If spin is included, the Landau energy under the square root should be modified by the Pauli spin term with the spin g-value taken at the conduction band edge. For E + E . - U <~ E * , eqn (5) reduces to the standard one-band effective mass approximation. We assume that the potential in G a A s - G a A I A s heterojunction is triangular: U = eFz, with the infinite barrier at z - - 0 , so that f(z = 0 ) = 0. One can then use the W K B approximation to quantize the motion along the z direction. The result is[18,19]:

(a + b)alab l/: + (b - a) 2 In

Ibll2 _ ail:l

=(~,2--~,,/ E:

/ / P=0

0.068

1-*0

~Ro = 4.43 meV

/

I

2

I 4 Ns(1011

I 6

I 8

I 10

cm-2)

Fig. 3. Electron CR mass vs 2D electron density, determined from FIR emission on GaAs-GaAIAs heterostructure (sample B, P = 0). Short solid lines are theoretical, calculated for transitions between Landau levels l ~0, 2-,I, etc. Long solid line is theoretical weighted average over all the eight transitions (see text).

4eFhn (i + 43-), (7)

where a = E - Ez and b = E* + E + E±. The quantum number i = 0, 1, 2 .... numeretes electric subbands. In the samples investigated we deal with the lowest electric subband. In the triangular well approximation the average electric field F is: F(Ns) = 4he (Na + ½Ns),

(8)

K0

where r0 is the static dielectric constant and Nd is the depletion density. Once the electron energies are calculated from eqns (6) and (7), one can define the C R mass m* by the relation h e B / r n * = E n + l - E , . The mass m* is always larger than m* because of the band nonparaholicity. 4. RESULTS AND DISCUSSION

Figure 3 shows the CR effective mass measured with the use of Landau emission on sample B under zero pressure. Short solid lines have been calculated using the above theory with the GaAs parameters: m ~ ' = 0 . 0 6 6 0 m 0 , E * = 0 . 9 8 e V , x 0 = l l . 9 1 , Nde¢= 3 x 10 I° c m - : . The free electron density Ns influences the C R mass via band's nonparabolicity in two ways. First, with increasing N s the electric field in the junction becomes stronger. As a consequence, the electric subband energy E0 becomes higher and the corresponding mass increases. Secondly, if the electron gas is degenerate, the density N s determines the position of the Fermi energy EF, which in turn determines

which Landau levels are involved in absorption or emission processes. With increasing N s the Fermi energy moves toward higher numbers n and the CR mass increases. If a transition from n + l ~ n to n + 2 ~ n + 1 processes is abrupt, the mass increase should have a jump-like character, whereas for a given CR transition the mass should increase continuously due to the F(Ns) dependence. Thus a given short line in Fig. 3 corresponds to a specific n + 1 -~n transition between the Landau levels, when the Fermi energy is located between these levels. The experimental data in Fig. 3 exhibit a completely different behaviour than that described above. The reason is that, in order to induce the emission, we heat the 2D gas by the electric pulses. For our detection energy of hto = 4.43 meV and htOL = 36.2 meV, we populate the Landau levels up to n = 8, so that eight C R transitions participate simultaneously in the observed emission. The above reasoning is corroborated by the width of the CR emission peak as shown in Fig. 1. The half-width of the peak is AB ~ 0.6T, and the maximum occurs at B0= 2.8T. This corresponds to Am*/m* ~0,21, i.e. to the mass spectrum covered by all the CR transitions shown in Fig. 3. Thus in our emission experiment we observe the C R mass averaged over all the transitions. In order to calculate this mass average one should account for the fact that the probability of the CR transition between n and (n + I) Landau levels is proportional to (n + I) (of. eg. Ref.[16]). Thus, assuming that the population factors are the same for

1216

W. ZAWADZKIet al.

all the levels, we calculate a weighted mass average, using the weights of (n + 1) for the corresponding C R masses. The result is shown in Fig. 3 by the solid line. If one keeps in mind that there exists some uncertainty concerning the average electric field in the junction, the agreement between the theory and the experiment should be considered very good, both as far as the absolute mass values and the slope of m*(Ns) dependence are concerned. Similar results for sample A, investigated under hydrostatic pressure of P = 7 kbar, are shown in Fig. 4. The theory can be used directly for this case, if one specifies the band-edge mass rn~" and the effective gap E* for the sample under pressure. We use the value of d m * / d p = 4.2 x 10 -4 k b a r - l (cf. Warburton et al.[12]), which is close to 4.09 × 10-4m0kbar -1, determined by Wasilewski and Stradling[17]. As to the dependence of the energy gap on pressure, we take d E J d p = !1.5 meV/kbar, measured by Zallen and Paul[18]. We assume that dEg/dp ~ d E * / d p , which seems a reasonable approximation. Using the above parameters of G a A s under pressure we obtain a very good theoretical description of the data shown in Fig. 4. It is worth mentioning that the non-weighted mass averages are distinctly lower, occuring between the mass values for 4--*3 and 5 ~ 4 transitions. Such mass averages give poorer theoretical fits, in particular for the results obtained under pressure (Fig. 4). Thus, in some sense our fits verify experimentally (although not with a sufficient precision) that the probability of the C R transition between (n + l) and n Landau states is proportional to the factor (n + 1). This property is very difficult to verify for single C R lines, as it would require a comparison of the absolute values of the calculated and observed C R absorption (or emission). The most uncertain parameter in our description is the average electric field F in the junction, as given by GaAIAs-GaAs 0.076

J

~ 0.074

~

~

"E J 0.072 1

0.070

/ / ---*0 I 2

P = 7 kbar ~to = 4.43 meV I

I

t

I

4

6

8

10

N s (1011 cm -2)

Fig. 4. The same as in Fig. 4 for sample A (P = 7 kbar).

eqn (8). This field determines the energy of the electric subband through eqn (7) and, hence, the mass value at the subband edge. We take the depletion density Nd = 3 × 101° cm -z, in the range of typical values for GaAs~3aA1As technology. The value of Na fixes the effective mass in the limit of Ns = 0. The good agreement between experimental and theoretical C R masses confirms in a sense the validity of the average field estimation, as given by eqn (8). Our preliminary estimations of the heating conditions for 2 D E G in crossed electric and magnetic fields show that it is possible to induce a sizable C R emission at our magnetic fields by the electric field of about 500 V/cm and to equally distribute the electron population over the Landau levels up to the optic phonon energy at the field of about 1000 V/cm. This means that the electric field distribution in our sample is strongly inhomogeneous. We estimate that the observed emission comes from the high-field regions of 10/~m in size.

Acknowledgement--We thank Dr J. P. Andre of Laboratoire d'Electronique et de Physique Appliquee, Philips, for providing the samples.

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