MICROWAVE MEASUREMENTS OF HALL MOBILITIES IN SEMICONDUCTORS by Yuiehiro Nishina A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Physics Approved: Signature was redacted for privacy. In Chargé of Major Work Signature was redacted for privacy. /Head of Major Department / Signature was redacted for privacy. Iowa State University Of Science and Technology Ames, Iowa 1960 il TABLE OF CONTENTS Page ABSTRACT vi I. INTRODUCTION 1 A. Hall Effect 1 B. Hall Effect In High Frequency Field .... 4 C. Previous Work on Microwave Hall Effect Measurement 6 D. Purpose of This Investigation 8 II. THEORY 9 A. Conductivity Tensor 9 B. Conductivity Tensor and the Hall Effect . 12 C. Principle of Measurement 15 1. Microwave degenerate cavity 15 2. Microwave power relationship 19 III. EXPERIMENTAL PROCEDURE 26 A. Microwave Measurement 26 1. Microwave cavity £6 2. Sample preparation 29 3. Temperature measurement and control . 31 4. Microwave measurement 34 B . D . C. Measurement 38 1. Cryostat and sample preparation .... 38 2. Electrical measurement 41 IV. EXPERIMENTAL RESULTS 45 A. N-type Germanium 45 1. Room temperature measurement 45 2. Low temperature measurement 52 B . P-type Germanium 57 iii Page V. DISCUSSION 65 A. Errors 65 1. Microwave measurement 65 c- D-C. measurement 68 B. Effect of Depolarization 70 C. Effect of Cyclotron Resonance 73 D - Future v/ork 76 VI. CONCLUSION 78 VII. REFERENCES 80 VIII. ACKNOWLEDGMENTS 83 IX. APPENDIX 84 iv LIST OF TABLES Page /Pjj i Table 1. Magnetic field dependence of VPII/-= i 1_ + R_| I for n-type germanium (GK-ô) at room temperature 85 Table 'c. Geometrical dependence of l^~ i i 1 — VP1 I 1 + R I Bo for n-type germanium (GN-6) at room temperature 86 Table 3. Magnetic field dependence of for n-type germanium (GK-5) 87 Table 4. Microwave Hall mobility of n-type germanium (GN-5) in low temperatures. ... 89 Taole 5. D.C. electrical measurement of n-type germanium ( GN-5-b) 90 /p7 Tacle 6. Magnetic field dependence of l~ for p-type germanium (GP-1) 91 Table 7. Microwave Hall mobility of p-type germanium (GP-1) in low temperatures. ... 93 Table 8. Magnetic field dependence of microwave Hall mobility in p-type germanium (GP-1) .... 94 Table 9. D.C. electrical measurement of p-type germanium ( GP-1-1) 95 Table 10. Magnetic field dependence of D.C. R^• 0~(B0) of p-type germanium ( GP-1-1) 96 V LIST OF FIGURES Page Fig. 1. Principle of microwave Hall mobility measurement 16 Fig. 2. Cavity construction for low temperature measurement 27 Fig. 3. Details of sample mounting on microwave cavity 30 Fig. 4. Schematic diagram of temperature measurement and control 33 Fig. 5. Schematic diagram of microwave circuit ... 35 Fig. 6. Cryostat for D.C. measurement 39 Fig. 7. Sample holder for D.C. measurement 40 Fig. 8. Notations for D.C. electrical measurement. 42 /Pc 1 Fig. 9. Magnetic field dependence of Ljz. V r j Jl + n| for n-type germanium at room temperature . 49 /pp i i Fig. 10. Geometrical dependence of . „• — V^i jl + K| ti0 for n-type germanium at room temperature . 53 IpZ Fig. 11. Magnetic field dependence of ^ for n-type germanium 4 f 55 Fig. 12. Hall mobility of n-type germanium vs. temperature 56 Fig. 13. Magnetic field dependence of for p-type germanium .4- 57 Fig. 14. Hall mobility of p-type germanium vs. temperature 61 Fig. 15. Hall mobility of p-type germanium vs. magnetic field 62 Fig. 16. Effect of cyclotron resonance on microwave power relationship 75 vi ABSTRACT The Hall mobilities of germanium single crystals were measured with a microwave frequency of 9000 Mc/sec over the temperature range 30°K to 300°K. A rectangular sample occu­ pied the central part of a wall of a rectangular cavity, which was doubly degenerate in the TE^01 mode and in the TE011 mode at a single resonance microwave frequency. The external mag­ netic field and the microwave field associated with one of the two modes gave rise to the other mode of oscillation, owing to excitation by the microwave Hall field. The theo­ retical analysis was verified by measurements on an n-type sample having a room temperature resistivity of 0.40 ohm cm. The measured Hall mobility at microwave frequencies (with a size correction) was compared with the D.C. Hall mobility between 30°K and 300°K. The maximum discrepancy was 15%. The estimated experimental error in the microwave measurement was 16/&. The magnetic field dependence of the microwave Hall mobility in a p-type sample, having a room temperature resis­ tivity of 0.77 ohm cm, was in qualitative agreement with the D.C. results obtained oy Willardson et al. 1 I. INTRODUCTION A. Hall Effect In the study of electronic structure of solids, particu­ larly in conductors and semiconductors, the Hall effect con­ tributes to our knowledge of the sign and concentration of the free charge carriers. The effect was found by E. H. Hall in 1879, and it can be explained in terms of the definition of the Hall coefficient R^ If a rectangular coordinate is designated along each axial direction of a rectangular con­ ductor, the Hall coefficient R^is expressed as Rw& ' (1'1) where Jx = trie static current density in the x direction, Bz = the static magnetic field of induction, Ey = tne static electric field induced by the Hall effect with no current in this direction, i.e. Jy = 0. All quantities are given in MKS units. Another quantity of fundamental importance is the Hall mobility, u, defined as u = F^r , (1.2) where cr is the D.C. conductivity. The physical implications of the Hall coefficient and the Hall mobility in semiconductors can be clarified if the analysis is made by means of Boltzmann1s transport equation + V-*f + ' (1*3) 1 MIc - where f is the distribution function of the carriers and ["^-£*1 is the rate of change of f due to the scattering of J° 2 f the carriers. For a steady state one has =0. As ex­ plained by Wilson (1, p. £09), when the static electric and magnetic fields are applied in a semiconductor where the existence of a relaxation time T is assumed, Equation 1.3 becomes, for the isothermal case, f - t•of __ e [E + V x B]-7 kf . (1.4) Here fQ is the value of f when there is no external force on the carriers. If f is expressed as f = f0 - "v • <f> (£ ) , (1.5) —> and if the terms of higher order in V are neglected, Equation k> 2 1.4 can be solved for spherical energy surfaces ( 6 = A_K_), provided SL. <f> ( € ) = e(mE + fx B j. (1.6) The use of Equations 1.5 with 1.6 gives the relationship be­ tween the current density and the applied field. If Maxwell-BoItzmann distribution is assumed for the carriers, one obtains, for B 1 , m CT= Ne%T> (1.7) m ' Rh - h • (1-8) U - RhCT - r! <Xy ' (1,9) • eo 3 wnere = —j= J g x^ exp x dx , (1.10) 3(W J 0 X = r£=r , (1.11) A0 1 and 3 yOÛ î (an)fc / N = / 6 ^ fo dC (1.12) " 2 7f*k3 y Vn'0 = density of carriers. Equation 1.10 Implies that <g7 is the averspe value of g over the Boltzmann distribution. If it is assumed that p r = T0(t)£ , (i.i3) Equation 1.9 reduces to i K-. T i -P I l ^ x /T"«/ 5 u = im R*• o0 U0T) NF + P)/N|) • (1-14) Hence the value of p for a scattering mechanism would be exhibited in the temperature dependence of the Hall mobility. Seitz (%) derived the temperature dependence of the mobil­ ity for lattice scattering. The analysis indicated thst the mobility should change as T In the low temper­ ature region the scattering by ionized impurity atoms be­ comes predominant. According to the Conwell and V/eisskopf formula (3), the mobility shows the T3//^ dependence. 4 In case of germanium, however, the temperature dependences of the mobilities in the lattice scattering range are T_1'for electrons and T ' for holes as they were found by Morin et al. (4) and by Prince (5), respectively. B. Hall Effect in High Frequency Field In the previous discussion all external fields applied in the solids were treated as static and the solution to the Boltzmann's Equation 1.3 was true for = 0. If the ex­ ternal field changes in a time comparable to relaxation time, the charge carriers are no longer in a steady state. Particularly, if the external fields change sinusoidally the deviation in f from f0 also has to change in the same manner. Hence one obtains -|I = jo> (f - f0) , (1.15) where co is the angular frequency of the external fields. Then the solution to Equation 1.3 can be obtained in the same manner as in the case of steady state except is replaced by ^ . When an electromagnetic wave is propagating in the Z direction along which a static magnetic field of induction B is applied, the solution to Equation 1.3 with the assumptions made in the previous discussion gives *This substitution was used conveniently in the analysis by Donovan (6). Jx " ¥ fE* <rrw> + Ey<(rrwf>i B} (1"16) y m- (- Ex<C(l + 7»^) B * Ey C + (1'17) where the higher powers of ^ ^ ^^ ^ B are neglected.