Microwave Spectroscopy in Semimetallic Bismuth–Antimony Alloys

W. BRAUNE e t al.: Microwave Spectroscopy in Sernimetallie Bi-Sb Alloys 95

phys. stat. sol. (b) 89, 95 (1978)

Subject classification: 13.1 and 19; 21.7

Sektion Physik der Humboldt- Universitn‘t zu Berlin. Bereich Tieltemperatur-Pestkvrperphysik

Microwave Spectroscopy in Semimetallic Bismuth-Antimony Alloys

BY W. BRAUNE, G. KUKA, H.4. GOLLNEST, and K. HERRMANN

Results are given of systematic microwave absorption measurements a t 35 and 70 GHz in semimetallic BiSb alloys. The great niiniber of experimentR permits to made representative statements on the dependence of the cyclotron masses and the relaxation times of the charge carriers from the Sb concentration. Measuremsnts of the electron-hole hybrid-resonance field strengths vs. the antimony concentration (z at?; of Sb) show that the band parameters of the holes at T do not substantially vary for z < 6. Es werden Resultate von systsrnatischen Mikrowellenabsorptions~nsssungeii bei 35 und 70 GHz

an halbinetallischen BiSb-Legierungen dargelegt. Die Vielzahl der Experimente gestattet, repra- sentative Aimsagen iiber die Abhangigkeiten der Zyklotronmassen und der Relaxationszeiten der Ladiingstragzr von der Sb-Konzentration zii machen. Messungen der Elektron-Loch-Hybrid- resonanzfeldstarken in AbhLngigkeit von der Antimonkonzentration (z At?& Sb) ergeben, daD die Bandparameter der Locher in T fiir .c < G sich nirht nesentlich andern.

1. Introduction The objective of this paper was to carry out systeniatic cyclotron-resonance iiieas-

urenients of electrons in seniimetallic BiSh alloys [ 11. Moreover, in the work reported here i t was possible t.0 obtain paramelers of the hole band at the T point of t h e Bril- louin zone HS functions of the Sb concentration. Due to the great number of samples investiguted in these experiments, representative statements are made on the vuria- tion of the relaxation times of t,he charge carriers which occur in alloying and doping.

The methods used by 11s for det,ermining the Sb concentration, such as polaro- graphy, microprobe measuremcnt.s, and neutron activation analysis in combinat,ion with conductivity nieasiirements showed that BiSb alloys show a semiinetallic character of conduction from z = 0 to 5.7 ( x at% Sb) [ 21 and subsequently a semiconducting character u p to J = 22 [3].

Starting from bismuth, the overlapping energy E,, = 15’; + Ek between the band extrema a t L and T decreases with increasing ant,iniony concentration, becoming zero a t about 6 ato/,. The associated variation of the band shape and the decrease of the Fermi energy with decreasing Sb concentration have an influence on t.he cyclotron l l lt lS8 1 / l c .

Moreover, it can he checked by experiment whether in the semimetallic alloy range the charge carrier concentrat.ion N , of the electrons coincides with the charge carrier concentration N , of the holes. Thus it can be estimated to which degree the random doping causes a change of the cyclotron mass.

The results of microwave absorption nieasurement.s performed by us on BiS b saiiiples with 0 5 z < 6 a.re presented in the following sections.

First the theoretical basis of the description of microwave spect,ra is outlined, considering aspects of both the microwave ahsorpt.ion of the exactly compensated ( N , = Nl,) plasma and the non-compensated plasma under the conditions of the normal skin effect. Sect.ion 3 describes the measoring apparatiis as well as the preparation and characterization of the samj)lcs used. Section 1 presents the results and their discussion.

96 w. BHAUNE, G. KUKA, H.-J. GOLLNEST, alld R. HERRMANK

2. Microwave Absorption In seminietals, electron- and hole-charge carriers are present., the densities of

which, N , and A',,, exactly compensate each other ( N , = N,, = N ) . Generally the charge-carrier densities of the semimetals are less than those of the metals by orders of magnitude (N(Bi) = 3 x lo1' em+). Cyclotron resonance spectra of Bi have been explained on the basis of the occurrence of both the normal and the anomalous skin effect [4 to 61. But actually the spectra can be explained most satisfactorily on the basis of assuming the weakly anomalous skin effect [7 to 91.

With the assumption of the normal skin effect and of elliptic Fernii surfaces, Smith et al. [4] were able to explain the essential absorption curve in agreement with the experiment. 2.1 Microzvave absorption it& the exactly compensated local semimetallicplasnza(N, =Nh)

Under the conditions of the normal skin effect the absorption R(a), B ) of a semi- infinite metal sample (yz-plane) is given by

where c denotes t,he velocity of light, o/2n is the measuring frequency, and Q I / X, E 1 ) y, B 1 1 x . are coniponents of the dielectric tensor which take into account the contribution of the displacement current.

I n high magnetic fields the absorption increases in proportion to the magnetic field strength B, which indirectly indicates the weakly attenuated propagation of magnetoplasma waves. For sufficiently long relaxation times z (oz > l), the increase of the function R(B) occurring in this case does not depend on the frequency, but only on tthe charge carrier concentration N and the effective masses mi and Mi of the electrons and holes, respectively (m,i, M i effective masses in the crystallographic axis system, see for instance [4]).

I n Section 4.3 results of absorption measureinents will be presented, which were obtained in the configuration Q 1 1 C3, B 1 1 C,, E / / C,.

I n this configuration, in high fields one obtains

( E , dielectric constant) . 4 8 R = = ___ V/'3N/E~

For short relaxation times conies frequency-dependent.

therefore the frequency dependence must be taken into account for frequencies o / 2 x 70 GHz in (2).

At the so-called hybrid resonances ( q --f 00, E~~ = 0 for ot + 00) absorption takes place, a t these resonances the radio-frequency field has only a longitudinal component as the field is generally elliptically polarized in the metal.

The charge carrier groups oscillate in opposite direction, such that no radio-frequency current is flowing ( j z =1 0). For the above configuration, the field strength of the electron-hole hybrid resonance is obtained in good approximation from

< lo), however, the absorption in high fields be-

For the investigated alloys, generally the relaxation times are relatively short,

where ??/,,h is the cyclotron mass of the holes, m, is the mass of the free electron. Above this field strength of hybrid resonance the weakly attenuated propagation of magnetoplasma waves occurs. From (3) it follows that this field is essentially determined by mch.

Microwave Spectroscopy in Semimetallic Bismuth-Antimony Alloys 97

2.2 Microwave absorption it& the non-compensated local sentimetallic plasma (Ne f Nh)

the question arises whether even accidental dopings with foreign atonis can have an appreciable effect on the deviation from the relation N , = N,.

In high fields, taking into account the difference in the charge carrier concentration of electrons and holes, (1) is given in good approximation by

Especially for antimony concentrations around x = 5 ( N = 3 x 10lg

(& component of t.he so-called lattice dielectric tensor). According to (4), for N , $- N , a cut-off or a change of the dispersion hehaviour q ( B ) occurs a t high magnetic fields B. A t higher measuring frequencies t,he contribution of non-compensation is lower.

The quantisation of t.he energy levels in the magnetic field and hence the dependence of the charge carrier concentration is neglected in (4). For Sb concentrat,ions of about 5 atyo the quant,um liinit is reached for B = 3 kG in the field direction discussed here. Taking into account these effects in the derivation of (4), this dispersion relation gets more coniplicat.ed and conclusions regarding a cut-off cannot be drawn from i t in such a simple way. We will therefore discuss another method to clear the influence of doping (see 4.2).

3. Experimental

3.1. Spectroitieter

The surface impedance R and the derivative of the surface impedance with respect to the iiiagnetic field dRldB has been measured as a function of the applied magnetic field B. The changes in surface impedance of the sample were measured as changes in the loss in min-wave cavities with f,, = 35 and = 70 GHz, respectively. The sample was mounted as end wall in t,he cavity with a possibility of shielding off part of the sample with a circular metal iris of different size.

The 35 GHz cavity was operated in the TE,, mode and the 70 GHz cavity was operated in the TE,, mode. Both generators were stabilized to the resonance frequencies of the cavities nsing reflector niodulation of the klystrons.

In the case of cyclotron resonance measurements a usual magnetic field modulation method was used. In the case of absorption nieasurements (cf. 4.3) the sample was coupled relatively strongly to the cavity. In [lo] it was investigated to which degree the &-value of the resonator can be rcduced without losing required stability. This had to be taken into account, because according t,o ( 2 ) in high magnetic fields the absorption increases linearly with increasing field. Moreover an increase of absorption with increasing is observed in the alloys as compared to Bi which is due to the decrease of charge carrier concentration. All measurements were carried out in the Voigt configuration (q 1 B ) . The static magnetic field was calibrated by means of a nuclear resonance technique with an error of < 1%. The temperature could be chosen between 1.4 and 4.2 K. i phssica (h ) R9jl

98 W. BRAUNE, G . KUKA, H.-J. GOLLNEST, and R. HERRMANN

3.2 Meastcritig samples

The BiSb crystals were grown from 99.9999yo pure Bi and Sb, respectively, by means of the zone-levelling technique. From the bar-shaped inaterial the measuring samples were obtained by cleaving in liquid nitrogen.

The ant,iinony concentration x of the measuring samples was determined from microprobe measurements. For x Ss 1 the stated value of the absolute Sb concentration has an error of about 20%. For higher x values a much more precise absolute determination is possible. I n this case the error is about 5% of the measured value.

4. Results and Discussion 4.1 Relaxation times of the carriers

It is known that the relaxation times zh and z, of the holes and electrons, respectively, can be determined from the cyclotron resonance spectra. On the basis of a great number of mcasurenients we can state the following about the dependence of the relaxation time on the antimony concentration x: In the range 0 < x 0.5, z, decreases by almost one order of magnitude with respect to the Bi value. At higher Sb concentrations up to a = 6, z, shows only minor variations. Typical values for semimetallic BiSb samples were Z, ==. ( 1 to 2) x lo-" s. In good Bi samples it can be found from cyclotron resonance measurements a t around 35 and 70 GHz and tempera- tures of liquid He that uzh > WZ, [ll]. This situation changes with increasing teni- perattire and by alloying Bi with Sb, respectively. The relaxation time of the holes decreases stronger than that of the electrons. For Sb concentrations x 2 1 we could not detect cyclotron resonance of the holes in the frequency range used by us. Moreover it should be noted that in the semimetallic BiSb alloys z,, can be estimated from the slope of the absorption curve in high fields (see 4.3).

4.2 C@otroim masses of the electrons

Fig. 1 a and b show the dependence of the light cyclotron masses on x for B 1 1 C, and B 1 I C, and that of the heavy cyclotron mass for B 1 I C, in the semimetallic alloy range with concentration up to 6 atyo, as referred to the corresponding cyclotron masses of pure Bi (m,, = 0.0082m0 for B I ( C,, m,, = 0.0094m0 for B ( 1 C, and rnt; = = O.138m0 for B / ! C, [l, 91). The anisotropy of t,he masses was taken into account for their experimental evaluation.

I n ordinary configuration ( E 1 1 B ) we found a better possibility of measuring the mass anisotropy and of determining the cyclotron masses in coniparison to the extra- ordinary configuration ( E 1 B ) . This fact is attributed to the masking of the light electron fundamental resonance by magnetoplasma effects a t E J- B.

Fig. 2 shows a typical spectrum. The relative high resolution and the presence of non-local effects even for Sb concentrations of about :c = 5.4 are illustrated.

0 1 1 3 4 5 6 x/at%/ -

"L,, 0 1 2 3 4 5 6

XldZi -- Fig. 1. Dependence of a) light cyclotron mass ( B 1 1 C,, B [ I C2) and b) heavy cyclotron mass

( B ( [ C, I( E ) of the electrons on the antimony concentration

Microwave Spectroscopy in Scrnimetallic Bismuth-Ant,imony Alloys 99

Fig. 2. Typical spectrum in the trigonal plane (q 1 1 C3), w/2n = 34.4 GHz, T = 1.4 K, B 1 1 C, 1 1 E, z = 5.4. The upper part of the figure shows the anisotropy of the cyclotron-resonance field strength of the heavy-mass electrons

?$ '%

in the neighbourhood of the C,-axis

0 50 I00 600 800 I000 I200 BIG) --

Looking a t Fig. l a , one notices sonie scattering of the experinient,al values in the low concentration range. Considering the relative precise measurements in this range, such a relatively great scattering seems not to be explicably in a simple way. Never- theless, no linear dependence is obtained starting from x = 0. This behaviour was pointed out previously [ 11. I n the present paper some experimental mass values from [ 13 have been correct,ed due to new nieasurenients and lineshape calculations [9], respectively.

For the dependence of the light cyclotron masses on the antimony concentration (2 at%), a linear relationship is found for 2 < x < 6, which can be approximated by the following expression:

Within the liniits of measuring error the dependence of the heavy cyclotron masses B 1 1 C2 on the Sb concentration is linear, too:

The dependence of the small ext,rerne cross-sections of the electron Fermi surfaces S, on the Sb concentration is given by [12]

s* = 1 - 0.22 + 0 . 0 0 8 5 ~ ~ (2 < G ) . S2(0)

( 7 )

Our quantum oscillation measurements of a sample with x = 4.5 and the result for x = 2.35 [13] suggest a linear dependence of the great extreme cross-sections of the electron Fermi surfaces S, on the Sb concentration:

For Sb concentrat,ions of about x = 0 the energy overlap between the conduction band a t the L point and t,he valence band a t the T point of the Brillouin zone disap- pears. Consequently in this alloy range the Fermi energy tends toward zero or t,oward a finite value, respectively, which is determined by the accidental doping. If the doping concentrations occurring in the seiniconductive alloy range [ 151 are taken as a starting point, then the accidental doping effect,s a variat,ion of the Fermi energy by 1 meV. For x = 5 the variation of the Fermi energy by 1 meV corresponds to a variation of the cyclotron mass by about 10%. In this case it should, however, be noted that this variation of Fernii energy causes a non-compensation ( N , + N h ) of 7.

100 W. BRAUNE, G. KLJKA, H.-J. GOLLNEST, and R.. HERRMANN

Fig. 3. (a) Experimental absorption spectrum R(B) of a semimetallic sample with z = 5.4. B 1 1 C2; E 1 1 C , ; q 1 1 C3; T = 1.4 K ; u/2n : 70 GHz. (b) Absorption curve R(B) as calculated according to (1) for the configuration of fields stated in (a), ut, = 5 ; W t h = 2; N , = AT, = 3x 1016 ~ m - ~ . The effective masses of the electrons, mi, amount to 50% of the bismuth value [4]. LE 0 5 la t5 EfkGi - M The values of the effective masses of the holes, Mi, re- main unchanged in the semimet,allic alloy range. (c) Ab-

sorption curve R(B) as calculated according to (4) for the configuration of fields stated in (a) and (b); ~ ' t , = 5; W t h = 2; N , - Nh = 2.6 X 1016 ~ m - ~

the charge carrier densities. This is caused by the considerably higher density of states of the hole band a t T. According to (4) this non-compensation should manifest itself experimentally in the propagation of magnetoplasma waves.

For samples with x-values of about, 5 a t y , the following estimation holds:

Eg = 4 meV , E$ = 2.5 me17 , N, = N, , = :3 x 10l6 . (9)

Increasing the Fcrnii energy of the electrons by 1 meP, one obtains

E$ = 5 meV , Ek = 1.5 meV , N, = 3.6 x 10l6 C I H - ~ , N,, = 1 x 1O1o cm-3 . (10)

For N , - N , =: 2.6 x 10'6 ~ l n - ~ , from (4) one would obtain a cut-off field strength of B < 20 kG a t o/2n = 35 to 70 GHz. Our investigations for field strengths up to 15 kG as well as other investigations for magnetic field strengths up to 60 kG do not show a cut-off of the propagation of magnetoplasma waves [14]. Because of t.he influence of the quantum effects this question requires a more detailed investigation. To clarify the influence of doping the dependence of the dielectric anomaly on the antimony concentration was also investigated experimentally and theoretically. For a non-compensation corresponding with (lo), the dielectric anomaly shifts by about 50% towards higher fields in comparison with its value for exact compensation (Fig. 3 curve b and c). The experimental spectra do not show such a field shift of the dielectric anomaly (Fig. 3 curve a).

Consequently the doping effect can be neglected in these experiments. If the dependence of the band-edge cyclotron masses as measured by Oelgart and Herrmann [15] in the semiconducting range (EF/E, < 1) is extrapolated into the semimetallic range, then the measured cyclotron masses of about x =r 5 appear relatively high. Since the doping effect is negligible, the ratio EFIE, should assume values of about 2 and one obtains from measured the S21ni,, ratios for x = 5 an energy gap of 5 meV.

This conclusion on the basis of a simple dispersion relation of the carriers includes a relatively great uncertainty of t,his value.

4.3 Absorption nieaszirements

Direct absorption spectra R ( B ) were measured on seven samples (z = 0.4; 1.5; 2.3; 2.9; 3.7; 4.5; 5.4). Fig. 3 curve a shows a typical spectrum as obtained from a sample with x = 5.4. The parameters of the hole band can be determined by fitting the calculated spectra to the measured ones using (3) and (2). In this way i t is concluded from the nieasurenients that within a measuring accuracy of 10% the effective masses of the holes in the semirnetallic alloys up to x = 5.4 do not differ from those of pure Bi. This result is in accordance with [16, 171. For wzh < 10 the increase of absorption R in high magnetic fields strongly depends on the relaxation time z,, of the holes. From this it was possible to estimate the relaxation time of the holes.

Microwave Spectroscopy in Semimetallic Bismuth-Antimony Alloys 101

Fig. 3 curve c shows a calculated absorption spectrum for the set of parameters (10) stat,ed in Section 4.2 neglecting quantum effects. The effect of non-compensation on the spectra is clearly observed. As mentioned above, this effect could not, however, be detected by experiment.

References [l] R. HERRYANN, W. BRAUNE, and G. KUKA, phys. stat. sol. (b) 68, 233 (1975). [2] R. KUHL, W. KRAAK, H. HAEFNER, and R. HERRMANN, phys. stat. sol. (b) 7 i , K109 (1976). [3] G. OELGART, G. SCHNEIDER, W. KRAAK, and R. HERRMANN, phys. stat. sol. (b) 74, K75 (1976). [4] G. E. SMITE, L. C. HEBEL, and S. G. BCCHSBAUM, Phys. Rev. 129, 154 (1963). [5] V. Y. EDELMAN and J. M. TCHEREMICIN, Zh. eksper. theor. Fiz. - Pisma 11, 373 (1970). [6] H. D. DREW, Phys. Rev. B 5, 360 (1972). [7] A. AKAHANE, J. Phys. SOC. Japan 31, 990 (1971). [8] J. NAKAHARA, H. KAWAMURA. and I. SAWADA, Phys. Rev. B 3, 3155 (1971). [9] G. KUKA, W. BRAUNE, and R. HERRRIANN, phys. stat. sol. (b) 76, 633 (1976). 101 J. DUSEK, Czech. J. Phys. B 11, 528 (1961). 111 H.-J. GOLLNEST, G. KUKA, W. BRAuNE, and R. HERRnfANN, to be published. 121 I?. BRODE, Diploma work (unpublished). 131 W. BRAUNE, G. KUKA, S.HESS, H.-U. MULLER, and T. JUNG, phys. stat. sol. (b) 79,510 (1977). 141 W. BRAUNE, J. LEBECH, and K. SAERMARK, to be published. 151 G. OELGART and R. HERRMANN, phys. stat. sol. (b) 58, 181 (1973); 61, 137 (1974). 161 R. HERRMANN and P. GOY, phys. stat. sol. (b) 80, 207 (1977). 171 B. N. BRANDT and S. M. CHUDINOV, Zh. eksper. theor. Fiz. 59, 1491 (1970).

(Received May 8, 1978)

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Microwave Spectroscopy in Semimetallic Bismuth–Antimony Alloys