G. P. SRIVASTAVA et al. : Microwave Faraday Effect in n-Type Silicon 285

phys. stat. sol. (b) 62, 285 (1972)

Subject classification: 18 and 20; 22.1.2

Department of Physics and Astrophysics, University of Delhi

Measurement of Microwave Faraday Effect in n-Type Silicon BY

G. P. SRIVASTAVA, P. C. MATHUR, and A. KUMAR

Measurements of the Faraday rotation in n-type silicon have been performed a t room and a t low (93 OK) temperatures. A crossed-waveguide technique (18.0 to 26.5 GHz), similar in principle (but in a simplified form) to the technique of Bouwknegt and Volger [4], has been used for the measurements. The rotation has been studied as a function of dc magnetic field up to 10 kOe. The effect of multiple reflections has also been studied a t low temperatures. The theory of the Faraday effect in anisotropic n-silicon, previously develop- ed by the authors [3], is modified taking into account the contribution of mixed scattering of lattice and impurities. The modified theory is compared with the experiment, and an estimation is made of the relative importance of the impurity over the lattice scattering. It is shown that impurity scattering gives an appreciable contribution to the rotation.

Bei Zimmertemperatur und niedrigeren Temperaturen (93 OK) wurden Messungen der Faraday-Rotation in n-leitendem Silizium durchgefiihrt. Eine Technik gekreuzter Hohl- leiter (18,O bis 26,5 GHz), die im Prinzip der Technik von Bouwknegt und Volger [4], jedoch in vereinfachter Form, ahnlich ist, wurde fur die Messungen benutzt. Die Rotation wurde als Funktion des konstanten Magnetfeldes bis zu 10 kOe untersucht. Desgleichen wurde der Effekt der Mehrfachreflexionen bei niedrigen Temperaturen gemessen. Die Theo- rie des Faraday-Effekts im anisotropen, n-leitenden Silizium, die von den Autoren [3] ent- wickelt wurde, wird modifiziert und beriicksichtigt den Beitrag der gemischten Streuung von Gitter und Storatellen. Die modifizierte Theorie wird mit dem Experiment verglichen und die relative Bedeutung der Storstellen- gegenuber der Gitterstreuung berechnet. Es wird gezeigt, daD Storstellenstreuung einen bedeutenden Beitrag zur Drehung liefert.

1. Introduction Donovan and Webster [l] developed the theory of Faraday effect in aniso-

tropic semiconductors and applied it to n-germanium [2]. Recently the authors [3] have extended the theory of Faraday effect to n-silicon. The expressions for high-frequency magnetoconductivity tensor elements are derived taking into account lattice scattering. The theory has been used to explain the Faraday rotation measurements a t room temperature, where the contribution of impu- rity scattering may be neglected. For low-temperature measurements, or in highly doped samples of silicon, the contribution of impurity scattering should also be included.

In the present work the theory has been modified taking into account the contribution of mixed scattering of lattice and impurities. The measurements are performed a t room and a t low (93 OK) temperatures in some silicon samples with 1 to 10 Qcm, and with their plane perpendicular to either (111) or (110) axis. A crossed-waveguide technique [4] is used for the measurements. The experi- ment is compared with the theory for different values of the parameter which measures the relative importance of two scatterings. It is found that the impu- rity scattering has an appreciable contribution to rotation.

286 G. P. SRIVASTAVA, P. C. MATHUR, and A. KUMAR

2. Theory

The expression of relaxation time z, for mixed scattering of lattice and impu- rities, is given by [5]

x3/2 (1) 7 = a

P + X Z ’

where a is the lattice scattering parameter. p is the factor which measures the relative importance of the two scatterings, and is given by [5]

where b is the impurity scattering parameter, p!, and ,u; denote mobilities of charge carriers due to lattice scattering and impurity scattering, respectively. Further the expression for mobility ,u is given by

where n is the carrier density and a the dc conductivity. Using the expression (1) and proceeding in the same way as discussed in

paper [3], the expressions for the high-frequency magnetoconductivity tensor elements can be found out, and are found to be the same as given in this paper, with the only difference that the transport integrals as given by the equation ( 1 1) of Donovan and Webster [2] should be replaced by

W

and 00

I i (4)

Using the expressions for conductivity tensor elements the Faraday rotation can be calculated following the analysis discussed by Donovan and Webster [l]. The effect of multiple reflections has also been included as discussed in another paper by Donovan and Medcalf [el.

3. Experimental Sct-up and Measurements

The experimental technique applied to measure the Faraday rotation is a crossed-waveguide method, and is a simplified and modified form of the bridge technique used by Bouwknegt and Volger [4]. The experimental technique is shown in Fig. 1 . The microwave generated by a 20V10 OK1 klystron is divided into two arms of the bridge using a tuned magic “T”. The microwave in one arm is allowed to propagate through the square coupling hole in the waveguide coupler, which can be placed in two orthogonal positions. I n the parallel posi- tion of the two waveguides in the coupler the maximum coupling of the power occurs, whereas in perpendicular position no power is coupled except when the

Measurement of Microwave Faraday Effect in n-Type Silicon . 287

Fig. 1. Block diagram of the experimental sot-up

sample placed between the two waveguides of the coupler produces Faraday rotation. The other arm contains the calibrated phase shifter and attenuator (rotary vane type). The divided power is mixed with the help of a directional coupler and is finally detected by a 1 N 26 crystal rectifier. For obtaining a better balance of the bridge, the phase shifter and attenuator are introduced in both arms. The output power is read finally on a sensitive nanovoltmeter.

This technique differs from Bouwknegt and Volger [4, 71 in the sense that the power in the arm containing the sample is not modulated. The phase and ampli- tude of the two orthogonal components of the microwave electric field are measured in the following way :

Let the amplitude of the wave in the arm containing the sample be B, and that of the wave in the other arm be A . Adjusting the phase of the wave in the arm A, maximum power is detected on the nanovoltmeter. Maximum electric field is given by

Em,, = A $. B . ( 5 )

Next adjusting the phase shifter to another position, minimum output is detect- ed on the meter ; hence

Emin = A - B . (6 )

With the quadratic behaviour of the crystal rectifier, i.e., U = c E2, one has the relation

(7 1 provided the amplitude in the arm A is kept constant. Balancing the bridge, the phase of the wave in arm B is measured in usual way. The phase and ampli- tude of the two orthogonal components of electric vector ( E , and Ey), transmit- ting the sample in the presence of magnetic field, are measured successively by mounting the waveguide coupler in parallel and crossed positions. From the known values of amplitude and phase of the two components, the rotation and ellipticity are calculated from the following relations of Bouwknegt [7] :

(Urn,, - Umin) = 2 c A B N B,

A' (1 + tan2 0) tan 8 (1 - 0 ' 2 )

' tanpl =

288 G. P. SRIVASTAVA, P. C. MATHUR, and A. KUMAR

sample

n-Si I

n-Si 111

____-__

n-Si I1

where y is the phase difference between two components, @ denotes the ampli- tude ratio of two components (tan @ = ampl. EJampl. Ev), 0 is the Faraday rotation, and A' is the ellipticity. A graphical inversion procedure is used for measuring 8 and A' from the known values of y and @.

conductivity 1 carrier density orientation (n-l ( ~ m - ~ )

0.40at 93 "K 1 <111;

0.91 at 300 "K I <110) 1 5.0 x lOI5 0.85 a t 93 O K 1 <111> 1.2 x 1015

4. Analysis and Results

The rotation for the samples whose specifications are shown in Table 1, is theoretically computed in the following way : The high-frequency magnetocon-

T a b l e 1 Specifica.tions of the samples

ductivity tensor components are calculated using equation (8) of the earlier paper (Srivastava et al. [3]). The values of various parameters used in the cal- culations are :

m, = 0.19 mo , m3 = 0.98 m, , K = 5.16, and F = 11.50. (9)

The factor a occurring in the calculations is obtained from the expression [3]

4 N e 2 a 3 n m* (To = -i,z-- , (10)

where 0, is the dc conductivity, N is the carrier density, and m* = 0.26 m,. The value of carrier density for the samples is taken from the work of Irvin [8]. It has been assumed that all the impurities remain ionized up to liquid N2 tem- perature (77 OK). Therefore the value of carrier density a t 93 OK remains the same as a t room temperature.

O"0 2 4 6 8 1 0 fiefdfk0e) A field (k0el

I<'ig. 2. Variation of Warsday rotation with magnetic field a t 93 "K for the sample n-Si I(z = 0.075 cm,

Big. 3. Variation of Faraday rotation with magnetic field at 93 O H for the sample n-Si 11 (z = 0.050 cm,

I = 20.62 GHz) Y = 20.62 G H p )

Measurement of Microwave Faraday Effect in n-Type Silicon

n-Si I n-Si I1 n-Si 111

289

I

0.500 0.526 0.118

Big. 4. Variation of Faraday rotation with the thick- ness of the sample for n-Si I and n-Si 11, and at 93 OK

( u = 20.62 GHz, H = 10 kOe)

Fig. 5. Variation of Faraday rotation with magnetic field for the sample n-Si 111, and at 300 OK (2 =

= 0.040 cm, Y = 18.5 GHz)

While using equation (8) of paper [3], the angle qj occurring in the expression is put equal t o 57.3" for the magnetic induction along the (1 11) axis, and y = = 90" for the (110) axis. Using the calculated value of conductivity tensor elements, the Faraday rotation is calculated from equation (41') of Donovan and Webster [l]. The effect of multiple reflections has also been included using equations (13) to (15) of Donovan and Medcalf [6].

The theoretical results based upon the above analysis are shown in Fig. 2 to 5. The experimental results are also shown in the same figures for comparison. Fig. 2 to 4 are the results a t 93 O K in silicon samples with magnetic induction along the (1 11) axis. Fig. 5 gives the results a t room temperature for a silicon sample with magnetic induction along the (110) axis. The effect of multiple reflections inside the sample and a t 93 "K are shown in Fig. 4. The values of determined from Fig. 2 and 3 are used for computation of the multiple reflection effect. From the experimental known values of B, the mobilities due to lattice and impurities scatterings are calculated from the expressions (2) and (3), and are shown in Table 2.

Table 2 Experimental values obtained for mobilities due to lattice and impurity scattering at the temperatures indicated in Table 1

' lattice mobility I (m2V-1s-1 )

sample impurity mobility

(m2 V-ls- l ) ~ . ~ . ~ _ _ _

5.00 2.81 3.53

5. Discussion

An examination of Fig. 2 and 3 discloses that the measurements at low temper- ature (93 OK) in silicon samples (40 and 85 Q-lm-l) are in agreement with theory choosing the value of parameter /3 to be 0.6 and 1.1, respectively. This shows that the scattering by ionized impurities has appreciable contribution 19 phyaica (b) 62/1

290 G. P. SRIVASTAVA et al. : Microwave Faraday Effect in n-Type Silicon

a t low temperature, and this contribution increases with doping. The figures also show that the value of the parameter has an appreciably large effect on rotation, and in no case this parameter may be omitted. I n highly doped samples (Fig. 5) with the resistivity 1.1 Qcm (at room temperature) this contribution is appreciable even a t room temperature, where the scattering by ionized im- purities is usually neglected. I n the samples with lesser doping (40 L2-lm-l) and a t 93 OK, the rotation tends to a saturation value for higher magnetic fields (Fig. 2) . However, the approach to the saturation value with magnetic field and finally inversion ( p B = 1) of rotation is less predominant in n-silicon compared with n-germanium. I n n-germanium the inversion of rotation is attained a t low temperature, with the magnetic field even less than 10 kOe. Contrary to this, the inversion is not attained in n-silicon up to 10 kOe, even in the sample with the least doping (Fig. 2). The possible reasons for this may be the relatively small value of mobility in n-silicon compared with n-germanium, and the higher value of effective mass in n-silicon due to which t,he cyclotron resonance condition is not attained a t lower fields.

From Table 2 it is evident that the mobility due to lattice scattering is nearly the same for all the samples with different dopings and a t the same temperature. The mobility of lattice scattering decreases with temperature. The mobility due to impurity scattering decreases with the increase of carrier density. The behaviour of lattice and impurity mobility shown in Table 2 agrees with the experimental findings of earlier workers [9]. This also confirms the value of B obtained experimentally, and thus i t may be concluded that the impurity scattering has an appreciable contribution.

Acknowledgement

The authors are indebted to Prof. F. C. Auluck for his interest in this work.

References

[l] B. DONOVAN and J. WEBSTER, Proc. Phys. SOC. 79, 46 (1962); 79, 1081 (1962). [2] B. DONOVAN and J. WEBSTER, Proc. Phys. SOC. 81, 90 (1963). [3] G. P. SRIVASTAVA, P. C . MATHUR, and A. KUMAR, phys. stat. sol. (b) 46, 757 (1971). [4] A. BOUWKNEGT and J. VOWER, Physica (Uhecht) 30, 113 (1964). [5] A. C. BEER and R. K. WILLARDSON, Phys. Rev. 110, 1286 (1958). [6] B. DONOVAN and T. MEDCALF, Brit. J. appl. Phys. 15, 1139 (1964). [7] A. BOUWKNEOT, Thesis, Rihksuniversiteit Antwerpen, 1965. [8] J. C. IRVIN, Bell Syst. tech. J. 41, 387 (1962). [9] G. C. PEARSON and J. BARDEEN, Phys. Rev. 75, 865 (1949).

(Received February 1, 1972)