Journal ArticleIn recent communications1'2 Dousmanis et al.  suggest that the re-entrant energy contours of  the heavy-hole bands in Ge and Si3 could contribute  a negative resistivity component to the overall  resistivity of these materials, possibly related to a nonequilibrium density of carriers in these re-entrant states

Mattis, Daniel C.

Stevenson, M. J.

English

Journal ArticleIn recent communications1\u272 Dousmanis et al.  suggest that the re-entrant energy contours of  the heavy-hole bands in Ge and Si3 could contribute  a negative resistivity component to the overall  resistivity of these materials, possibly related to a nonequilibrium density of carriers in these re-entrant states

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THEORY OF NEGATIVE-MASS CYCLOTRON RESONANCE D. C. MaUis and M. J. Stevenson Research Laboratory, International Business Machines Corporation, Poughkeepsie, New York (Received May 19, 1959) In recent communicationsl , 2 Dousmanis et al. suggest that the re-entrant energy contours of the heavy-hole bands in Ge and Si3 could contri-bute a negative resistivity component to the over-all resistivity of these materials, possibly re-lated to a nonequilibrium density of carriers in these re-entrant states. Such a phenomenon could be observed under cyclotron resonance conditions (Hdc along [100] axis, electric field transverse to Hdc) as an emission spectrum cen-tered about the cyclotron frequency of these states, or as a reduction in the strength of the background absorption at that frequency. We have examined this idea in some detail, and find that except under very restrictive conditions, these states will in general add a normal, posi-18 tive resistivity contribution to the material in-dependently of their population. Therefore, in-disputable experimental evidence of any emissive properties at the cyclotron frequency of the car-riers in these "dimple" states will be shown to reveal more about the nature of conduction in the valence band of Ge and Si than merely the exist-ence of these re-entrant energy contours, or de-viations from thermal equilibrium, or a combi-nation of these. As an introduction to the analYSis consider the expression for the integrated absorption coeffi-cient A, calculated for a hypothetical system of carriers of constant, isotropic effective mass m *. Under crossed dc magnetic and rf electric fields, the carriers will be stimulated by the VOLUME 3, NUMBER 1 PHYSICAL REVIEW LETTERS JULY 1, 1959 latter to make transitions between Landau levels (denoted by quantum number n). We obtain A = (7Te2 /2m *)~ (n + 1) (f(n, k) - fen + 1, k)] = (7Te2 /2m*)~f(n, k) =7Te2N/2m*. (1) The sum in Eq. (1) is over all quantum numbers, only n changing in a transition. The result we obtain is a sum rule independent of the magnetic field, namely, that the integrated absorption de-pends only upon the total density of carriers, N. Were it possible to isolate carriers in a region of opposite band curvature where the energy is given by E =Eo - pl/2m*, such that over the whole range where carriers exist the effective mass -m* is constant, isotropic, and negative, the in-tegrated absorption coefficient would be A = (7Te2 /21 m* I)~ (n+ l)(f(n+ 1, k) -fen, k)] (2) This system of negative masses is emissive, just as the system of positive masses was ab-sorptive. Now, it might be argued that the sign of the mass determines whether a system of par-ticles is emissive or absorptive. We believe this point of view to be unsound. Our arguments, to be developed below, will indicate that in the case of distributions which can be characterized by a function of energy alone, emission will occur only if there is a maximum energy cutoff on the states of interest (such as Eo in the case of nega-tive masses discussed above) combined with an unbalanced "inverted" distribution. These argu-ments do not hold when the distribution is aniso-tropic, as is the case for the NEMAG (negative-effective mass amplifiers and generators) sug-gested by Kromer.4 However, in experiments where holes are optically excited, thermal scat-tering of the holes along a surface of constant energy is presumably rapid enough that the po-pulation of states along such a contour may be taken to be constant. We idealize the experiments of Dousmanis et al. by means of the following assumptions: 1. Holes are excited up to the optical phonon energy Eopt. (Those created at higher energies will rapidly emit optical phonons and fall to this energy or below.) 2. A negative transverse mass -mT and posi-tive longitudinal mass m L will be assumed con-stant for carriers within narrow cones of half-angle a whose axes lie along the [100] axes of Ge and Si. Beyond this cone, the heavy holes under consideration have a positive effective mass mp, whose magnitude is sufficiently differ-ent from mT that the conductivity at frequencies near wT=eH/mTc will be determined largely by the properties of the carriers within the cones. We shall consider the possibility that mT' m L> or a vary with energy. 3. steady state is described by a distribution functionf(E) dependent only on energy. The microwave fields are a small perturbation, whose effects we shall describe by the Boltzmann equa-tion. Thus, under cyclotron resonance conditions, with the magnetic field along the axis of one of the cones, we find a frequency-dependent conductivity given bY' = -Ne2 a2 IdE E3I2(df/dE)('T+ iW7'2)/[(wT7')1 + (1 + iW7')2] 2m L IdE ElIlffft) (3) The second expression follows if a"« 1. The total density of carriers within the cone is taken to be N, and the momentum relaxation time 7' obeys W T7'» 1 for a well-defined resonance. A dispersion relation valid near the axis of the cone is E=ip2(cos"8/m L - Sin28/m~ !:ipz/m L =itfi(m L cos"O -mT Sin2 8) ;,itfim L' (al « 1) (4) and was used to express the variables of Eq. (3) in terms of the energy. If wT and 7' vary slowly with energy compared withf(E), the numerator of Eq. (3) can be inte-grated by parts, and we obtain 3Ne1a l ~ 7'+iW7'2 J a(w)= 4mL L(WT7')z+(l+iW7')*j' (5) It is easily verified that Re[a(w)]>O, so that we have a direct check that under reasonable assump-19 VOLUME 3, NUMBER 1 PHYSICAL REVIEW LETTERS JULY 1, 1959 tions the conductivity , and hence the absorption for the system under consideration is positive. If all the hypotheses we stated are not valid, there will exist possibilities for these negative masses to amplify resonant radiation. We en-visage two distinct situations: the first involves a cutoff energy Ec < Eopb such that for E >Ec the shape of the energy contours has changed sufficiently that no particles' above this cutoff energy can contribute to the resonance at wT' H such is true for the energy band structure of the heavy holes in Ge or Si, emission would be ob-served at wT provided This last condition is tantamount to an inversion of the population of these low-lying energy levels, which could be achieved by a mechanism such as selective recombination of slow carriers. If the band structure does not permit such a drastic change in m T' a second situation is con-ceivable whereby the distribution function while still of cubic symmetry shows some anisotropy. H, for example, /=/1 (E) + /z(lpl), (6) those particles distributed according to /1 will absorb radiation, whereas the contribution of the carriers with negative transverse mass -m T distributed according to /a can b~ shown to be emissive, and peaked about the frequency wT' Whether emission or absorption predominates at that frequency will depend on the relative mag-nitudes of/I and/2 • In this connection we propose that light may not provide the best mechanism for obtaining a se-lective excitation of such carriers. A possible alternative excitation technique could utilize microwave impact ionization of impurity centers in p-type Ge by resonant absorption at the cyclo-tron frequency wp of the positive-mass heavy holes. With a double resonant cavity system, the changes in the rf conductivity may then be studied with a low-power probing signal at the resonance frequency of the negative-mass holes, wT' Such a technique would result in the presence of posi-tive-charge carriers only, and with proper cir-cular polarization the effects of the negative-mass holes can be separated from those of the positive-mass holes. In concluding, we express our thanks to col-leagues of this laboratory-to Mr. W. Kopka for computational work on the mM 704 computer, and to Dr. S. Koenig, Dr. R. Landauer, and Dr. E. Adams for their interest and helpful sugges-tions. ta. C. Dousmams, Phys. Rev. Letters 1, 55 (1958). 2a. C. Dousmams et al., Phys. Rev. L;tters.!., 404 (1958) • 3Dresselhaus, Kip, and Kittel, Phys. Rev. 98, 368 (1955). 4H. Kromer, Proc. Inst. Radio Engrs. 47, 397 (1959). 6For method of derivation, see, for example, D. Mattis and a. Dresselhaus, Phys. Rev. 111, 403 (1958). 'This is an idealization. In reality we would not nec-essarily expect a sharp cutoff and there would be over-lap in the spectra of states above and below E c' 

Theory of negative-mass cyclotron resonance

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Theory of negative-mass cyclotron resonance

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