Zero-point plasmon motion in the extreme quantum limit in InSb

.Solid State Communications, Printed in Great Britain.

ZERO-POINT

Vol. 47, No. 6, pp. 509-513,

1983.

0038-1098/83 S3.00 + .OO Pergamon Press Ltd.

PLASMON MOTION IN THE EXTREME QUANTUM LIMIT IN InSb R.E. Harper* and D.P. Tunstall

Department

of Physics, University of St Andrews, St Andrews, Scotland (Received

21 February

1983 by R.A. C’owley)

Further experiments at helium temperatures are described on the anomalous NMR relaxation peak, first reported by Bridges and Clark, in the extreme quantum limit in doped InSb, extending th;previously reported density range of the phenomenon up to 5 x 10 me3 of donor electrons. New features of the anomaly, in particular an isotope effect, are reported. The large density range over which the effect has now been observed, taken with the new features, render many of the previous theories of the peak untenable. Based on the zero-point plasmon motion of the electron gas, we put forward an interpretation which appears to explain the various aspects bf the phenomenon.’ InSb, WlIEN DOPED with donors to densities greater than 10” m-‘, forms a dilute metallic electron gas at low tempcraturcs; such a system shows for example strong Shubnikov-dc Ilaas oscillations in clcctrical conductivity when subjected to moderate magnetic fields [I]. Many other properties of the material show oscillatory bchaviour; in particular the nuclear resonance spin-lattice relaxation rate 1/T, exhibits strong oscillation in a certain range of magnetic field [2]. (This reference wilt hcrcaftcr bc refcrrcd to as BC.) BC investigated scvcral transmutation-doped InSb crystals, and found rather good cxperimcnt-theory agrecmcnt for the positions of the peaks in l/T1 as the external magnetic field B. was varied; these were the peaks associated with Landau levels crossing the Fermi surface. As the field was incrcascd further, beyond the last Landau peak, when all the conduction electrons lie in a single Landau sub-level, the O+, (i.e. the Landau level were n = 0 in the orbital energy expression E = (n + f)h w, with o, the cyclotron resonance frequency, and all the electron spins have a sub-quantum number of + $), then the theoretical expectation of l/T, tending to zero contrasted sharply with the strong experimental peak that appeared for all three isotopes In”‘, Sb’*’ and Sb123. The peak, known as the A-peak, occurred at a value of Bo, for a given nD, below the value BFO required to cause the metal to non-metal transition known as magnetic freeze-out [3]. This range of field, below freeze-out but with all the conduction electrons lying in the lowest sub-Landau level, is known as the extreme quantum limit. * Now at the Department of Electrical and Electronic Engineering, University of Surrey, Guildford, Surrey. 509

The value of the external magnetic field B,” at which the peak occurred varied as the donor density nD was changed, appeared to follow a Be a ng’ dependcncc. The range of density over which the effect was observed, 0.0765 x t O** < lr,, < 0.33 x 10” me3, was timitcd by the range of external fields available for the experiments. At the time of the work several groups advanced thcorctical intcrprctations. The ideas put forward involved ptasmon excitation [4] and D- states [5 J, whilst BC themselves thought that some magnetic field-modulated hopping process of the electron spins might be the cause. Since the early experimental and theoretical work above, there has been a further programme to measure the Knight shift [6] in the same samples and magnetic field range; as with earlier Hall effect measurements no evidence was found for any significant localization effects around the position of the A-peak. There has also been some considerable theoretical work on the possibility that the extreme quantum limit in such semiconductor systems could be fertile ground for the observation of a magnetic-field induced Wigner crystallisation or chalge density waves [7: 8). This letter reports further experimental work on the position of the A-peak in mainly transmutationdoped samples of InSb, ranging in density up to n, = 4.9 x IO** rne3, involving magnetic fields up to 10 T. The details of the experimental technique closely match the BC experiment, and will be reported in more detail later. A range of relaxation times at different fields was obtained by the method of field-cycling and absolute values of T1 were frequently inferred from the initial recovery of the NMR signal after saturation, using a comparator NaCl signal. Measurements were taken at

ZERO-POINT PLASMON MOTION IN InSb

510 K)-

0 0

“m” / .

I-

+/

/

0,

/ /

/

O/ /

/

.’

+ BC 0 This work

/+-

/ I 1.0

0.1 no

Fig. 1. A log-log plot of the value at 1.5 K of the magnetic field B.A. in Tesla, at which the anomalous peak in l/T, occurs for all three nuclear species, as a function of doping density 11~. BC data is included. The x-axis is marked in units of 10” rnV3, and the y-axis in Tesla. The dashed line represents an n$* fit to the low density data. 4.2 K and below. Annealing of samples to remove radiation damage followed the recipe of Clark and Isaacson [9]. One sample, of density 4.9 x IO** me3, was melt-grown, tellurium-doped. Two samples, of densities 5.7 x IO*’ nib3 and 3.3 x 10** rnd3 , were doped at the PLUTO reactor in an environment having a thermal to fast neutron ratio of 800 : I, whilst the others were doped at the reactor at S.U.R.R.C., East Kilbridc, whcrc the thcrrnal : fast neutron ratio was close to I : I. The consistency of the observations of 0,” over samples involving such a wide range of preparation tcclmiqucs adds support to the hypothesis that the A-peak cffcct is an electronic phcrlomcrron. not associated

with

dcfcct

and/or

impurity

structures.

(Ilowcvcr the size of the peak in the melt-grown sample was somewhat less than expected.) WC concentrate in this brief report on the low temperature (I .5 K) data that we have obtained, Fig. 1, which cxtcnds the BC data, also included in Fig. I, so that the range of density over which the effect appears now extends frorn 0.077 up to 4.9 x 10” m-‘. It should be emphasized that this peak in I/T1 as a function of B. is substantially bigger in magnitude than the oscillations in l/T, that occur below the quantum limit at lower values of B. as 3 result of Landau level oscillations. In our work the amplitude of the l/T, peak appears to scale approxiniatcly linearly in tID (although the rneltgrown sample dcviatcs somewhat). Furthermore, the value of B,” shows little or no temperature dependence within the at densities above IO** III- 3, for tcmperaturcs range I .5 to 4.2 K, a bchaviour rather different from that reported by BC for their samples. A further new feature of the results presented here emerges when one considers the relative amplitudes of the I/T, peaks at 06’ for the different isotopes in the samples. Sb’*’ has a bigger gyromagnetic ratio than Sb’*-‘. by a11rmt a factor of two, so for any magnetic

Vol. 47, No.

relaxation mechanism, all other things being equal, one would expect (1 IT,) for Sb”’ to be substantially greatel than for Sb’23 . Whilst ail the low-density data presented by BC indicates that, after background subtraction, the Sb’23 peak is bigger than the Sb’*’ peak at the A-position, a careful check on our 3.3 x lO22 rnw3 sample produced the result that at 4.2 K, (l/T1) for Sb’*’ was about three times as big as (1 /TI) for Sb’23 at the A-position, a result for the relative magnitudes that correlates rather well with the behaviour for the two isotopes in the lower region of field where standard metallic relaxation and Landau level oscillations in (l/T,) are observed [2]. We believe that this change of relative amplitude of l/T, at the A-peak position for the two Sb isotopes is driven by the change of electron density. We turn now to consideration of the importance of the new data in adjudicating between the various hypotheses as to the mechanism underlying the effect. If we consider first the localized D- hypothesis [5] in the light of the further knowledge of D- states that has crnerged in the past decade, one would have to discount this is a viable explanation, if only because the system is still strongly metallic at the n-peak position. at the time the D- states arc supposedly swept past the Fermi surfact by the magnetic field. The plasmon excitation mcchanisrn [4j is afflicted by its incapacity to explain the dcpcndencc of the amplitude of the l/T1 A-peak on tcrnperaturc in the helium range. Neither of the above two mechanisms has any capability of explaining the variation in the difference in the relative magnitudes of the Sb’*’ and Sb’*’ l/T,‘s as the density is varied. The electron hopping hypothesis [2] requires some localization of electrons and/or defects to act as paramagnetic centrcs of relaxation. Our observation of substantially the same A-peak effect in samples produced in both the PLUTO and the SURRC reactors, for which defect concentrations presumably differ sharply, even after annealing, and the lack of observation of evidence for any localization effect in transport studies around the A-peak field, lead us to discount hopping and/or paramagnetic impurity/defect species as possible explanations. We are led therefore to consider alternative hypotheses; one such possibility for the n-peak is to associate it with a field-driven phase transition in the electron gas, either of the Wigner crystallisation type [7] or involving a charge-density wave [8]. The conditions of our experiments arc quite close to those predicted to be necessary for such transitions, with magnetic fields close to IO T and temperatures of around I K. However, we only use the high fields at large donor densities; at the 10 T end of the range, the field is of course deep into freeze-out for the low density samples. A brief analysis

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Vol. 47, No. 6

PLASMON MOTION IN InSb

511

Zawadzki [ 121. Briefly we think of our conduction electrons in the extreme quantum limit as occupying a Fermi cylinder in k-space, of radius kl = ((2m*/h’) x and of length 2 k, where - fg*PBf))1’2, (f% k, = { 2n2 hnD/eBt}. k, and kl are then evaluated using experimental vahies of Bf and nD, m* = 0.0137 m, and g’ = 49.7, and a value of k at the top of the distribution, k = { kz + k:)“2, obtained. This is the value that determines how far up the (non-parabolic) band we are, and is now inserted into the expression for A [Ill .

P /

I I2

I

I

I

3

4

5

I

6

I

7

I

8

a$ expt.

Fig. 2. A plot of the theoretical value of the magnetic field Bf at which equation (1) in the text is satisfied against the experimental values of the field at the peak of the NMR relaxation rate I/T,. The dashed line corresponds to perfect one to one correspondence between the two axes. Both axes are marked in Tesla. of the theoretical papers above has led us to conclude that freeze-out occurs at a lower magnetic field in InSb than any other electronic phase transition. Close obscrvation around the n-peak fields of the transport propcrtics of our samples has in any case produced no cvidcnce of any effect in the resistivity; WCconclude that Wigncr crystallisation and/or CDW formation do not provide a basis for understanding the n-peak in InSb. It is our belief. however, that zero-point plasmon motion of the electron gas system can provide a natural explanation for the ,4-pcnk. Gunther cl ul. [4] have, of course, already connected the A-peak field with that at which the electron spin resonance frequency is equal to the plasmon frequency op. We illustrate this remarkable equality in Fig. 2 where the enlarged density range of the data now available makes the agreement, with no adjustable parameters, seem completely non-coincidental. In Fig 2 we plot the experimental values of ,!I,” against theoretical values of the same B,” obtained as solutions of the equation

i.e. where, in the latter equation, ?rD is in units of 102’ 111~~ and m* in units of 9.1 x lo-” kg, and where E = 15.7 [IO]; g’ and m* have been obtained by a procedure that accounts for the non-parabolicity of the InSb conduction band as B. and tIg are varied. This procedure for calculating g* and m* is bused on the work of Zawadzki and Szymanska [I I ] and of

where eg = 0.235 eV is the energy gap in InSb. From this value of A, following the recipes outlined [ 11, 121 values of eF, g*, and m* can be obtained. Since it is necessary in evaluating the importance of Fig. 2 to realize how far into non-parabolic regions of the band our experiments go we tabulate the data obtained by this procedure for our samples in Table I. The final column documents how far, weighted by their importance in equation (I), our “top-of-thedistribution” paramctersg* and m’ deviate from their “bottom-of-the-band” values. In the context provided by Table I, it can be seen that the data prcscntcd in Fig. 2 covers quite a considcrablc range whcrc the clcctron gas occupies nonparabolic regions, whcrc large changes in g* and tn* occur; yet the equality between the plasmon and ESR frequencies persists. We take this as definite evidence for the existence of a plasmon process for the driving mechanism for the A-peak. It is however clear that the first-order process, as explored already [4], does not provide an answer. WC propose a second order resonant process involving the modulation of the electron-nuclear dipolar coupling by the zero-point plasmon motion. This motion provides a fluctuating magnetic (or indeed electric, since ESR can be excited here by electric fields) field at the electron positions, and of the right frequency to induce electron spin resonance. Electron spin flips cannot be produced in first-order by this process because both the electron spin and the plasmon systems, at the A-peak magnetic field B,f and at the low temperaturcs of our experiments, are in their ground states, i.e. energy conserving transitions. where. say, an electron spin flips up, absorbing a plasmon. are forbidden. No such problem is encountered. however. when resonant time-dcpcndent perturbation theory is used in second-order, and the process can be just as strong as the typical first-order process. With this relaxation mechanism, the process causing nuclear relaxation around the A-peak is thus a double

512

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Vol. 47, No. 6

PLASMON MOTION IN InSb

Table 1. Numerical data, used in evaluating equation (I), illustrating the non-parabolic effects in these experiments

nD (x 10” ms3) 0

0.765 (BC) 1.2 2.1 (BC) 3.34 (BC) 5.7 21 33 49

m*(m0)

eF (K)

0.0137 0.0140 0.0141 0.0143 0.0144 0.0147 0.0161 0.0169 0.0178

30 38 54 73 103 238 314 401

electron flip-flop, with a nuclear transition accompanying either the upward or the downward electron transition. Energy conservation is accommodated by some minor rearrangement of the electron dipolar environment, to take up the miniscule nuclear Zeeman energy. The whole process is driven by the resonant zero-point plasmon motion, via the electron-nuclear dipolar interaction. Such a hypothesis seems to be potentially capable of explaining the further new fcaturc, noted above, that of the isotope effect. At low donor densities, the range of X--values prcscnt in the electron gas at the n-peak position is small, so the effective mass and R* values are well-dcfincd over the whole gas. The plasmon zero-point motion and the ESR frequencies are thercfore also wclldefined. If WCintcrprct the NMR relaxation via the above process as being due to the appearance within the sample of fluctuating magnetic fields at nuclear sites at frcqucncies around zero frequency (the NMR frequencies are very close to zero-frequency on the frequency scale of the ESR and plasmon motion) as the ESR and plasman frequencies coincide, then the frequency distribution of these fields around zero frequency will be sharp for the low density samples. If this spectral density peak is sharp on the scale of the NMR frequencies, typically - 100 MHz, of our experiments, then because the Sb’23 frequency is substantially lower than that of Sb’*‘. Sb’23 relaxation would proceed faster than for Sb’*l. At higher densities the range of /c-values is much larger, and the ESR and plasmon resonance necessarily more diffuse. The zero-frequency peak of fluctuating magnetic fields associated with the proposed relaxation mechanism will then be much broader and the normal dominance in relaxation rate for the higher 7 nucleus, Sb’*’ , would be re-established. The proposed second-order relaxation mechanism can therefore account for both the trend of the data over the whole density range studied (Fig. 1) and the relative change of amplitude of the A-peak between the

g*

0

-49.7 -48.4 -48.0 -47.3 -46.8 -45.7 -40.8 -38.5 -36.1

1 .ooo 1.016 1.021 1.028 1.036 1.050 1.124 1.162 1.208

two antimony isotopes as the density is varied. The absence of observable effects on transport properties at Be is naturally explicable via this second-order process. The details clearly need more quantification; a more profound analysis would require consideration of the temperature variation of BOAat low density, the temperature variation of (l/T,) at Bt, the density depcndence of (l/T,) at I]$ and a host of other minor details. The lack of precise agrcemcnt between experiment and theory in Fig. 2 at high density is still rather puzzling. We conclude therefore that we have established a prima fucie cast that the origin of the anomalous NMR peak in l/T, in the extreme quantum limit in InSb is the interaction of the zero-point plasmon motion with the clcctron and nuclear spins via their electron-nuclear dipolar interaction. Acknowledgements

- The authors would like to thank Professor W.G. Clark for many useful discussions and technical help. MS Harper would like to thank the SERC for a research studentship for the duration of this project, which was supported by SERC equipment and NATO grants.

REFERENCES

5.

H.P.R. Frederikse 8~ W.R. Hosler, Phys. Rev. 108, 1136 (1957). F. Bridges & W.G. Clark, Phys. Rev. 182,463 (1969).J.L. Robert, A. Raymond, R.L. Aulombard & C. Bousquct, Phil. Mag. B42, 1003 ( 1980). L. Gunther. M. Revzen & A. Ron, Physics 3, 115 (1967). G. Bcnford & N. Rostoker, Phys. Rev. 182,375

6.

(1969). G. Miranda & W.G. Clark, Phys. Rev. B9.495

7.

(1974). W.G. Kleppmann

1. 2. 3. 4.

8.

& RJ. Elliot, J. Phys. C. Solid State Physics 8,2729 (1975). Y. Kuramoto, J. Phys. Sot. Japan 44, 1572 (1978).

Vol. 47, No. 6 9. 10.

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PLASMON MOTION IN InSb

W.G. Clark & R.A. Isaacson, J. Appl. Phys. 33, 2284 (1967). E. Burstein, Lartice Dynamics (Edited by R.F. Wallis), p. 3 15. Pergamon Press, London (1965).

11. 12.

W. Zawadzki & W. Szymanska, 3. Phys. Chem. Solids32, 1151 (1971). W. Zawadti, Phys. Left. 4, 190 (1963).

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