Energy relaxation of warm holes in p-type tellurium

Solid State Communications,

Vol. 7, pp. 1149—1152, 1969.

Pergamon Press.

Printed in Great Britain

ENERGY RELAXATION OF WARM HOLES IN p-TYPE TELLURIUM* H. Kahiert, K. Hess and K. Seêger Ludwig Boltzmann-Institut für Festkorperphysik, Wien and Institut Physik der Universitit Wien, Austria

für

Angewandte

(Received 19 June 1969 by G. Liebfried)

Measurements of the warm hole energy relaxation time of Te in the temperature range between 150°Kand 190°K are reported. Experimental data were obtained using a microwave harmonic mixing method. These energy relaxation times are compared with calculated values assuming polar optical scattering as being the dominant energy loss mechanism in this temperature range. Furthermore, deviations from Ohm’s law applying weak and medium d.c. fields to Te samples have been investigated and will be discussed.

SEVERAL authors have reported energy relaxation times experimentally obtained with the covalent 3 In a more recent semiconductors Ge and Si.’ study4 the temperature dependence of the energy relaxation time ‘~ in the warm carrier range in these materials has been determined using 5 It is the purpose harmonic mixing of microwaves. of this paper to present data of ~ in trigonal Te. For this material one has to expect the influence of polar optical scattering on the carrier transport as has been shown by various authors.58 Data will be compared with theoretical values which are calculated under the assumption, that scattering by polar optical modes is the important energy loss mechanism in p-type Te in the temperature range under consideration.

other arm. Both waves are mixed in the semiconductor sample and produce a 1 kHz modulated d.c. signal Emjx, which is picked up by a resonance amplifier. Emjx is given by the expression5 Emix

=

(3/4)f3(1+4w2

~~)“~2E~E 2 cos2(qS+~1i) (1)

with tan2~

2 1&~

‘ri)

(2)

~/(1+3w

The coefficient ~3 describes the deviations from Ohm’s law due to the relation9 ~ (1+~E~) (3) U=a

where o is the conductivity and E the field strength. The relation between ~ and the observable phase shift ~fris shown in Fig. 2.

EXPERIMENTAL A simplified block diagram of the microwave apparatus is shown in Fig. 1. The 1 kHz modulated fundamental wave cos (~t ~ at X-band frequency 9.375 GHz is fed via a phase shifter into one of the arms of a single-ridged double-Tee junction, the second harmonic cos 2wt at K-band frequency 18.75 GHz to the ____________

In order to calibrate the system, samples of p-type Ge and n-type Si with energy relaxation times and corresponding vt-values (labelled known from previous measurements,4 were mounted in the waveguide. The position of the ~

phase shifter where the mixing ~signal vanishes i.e. where according to (1) + = 77/4 is called ~ ~ The quantity çS is the phase

* Dedicated

to Prof. Dr. Ing. Heinrich Gobrecht at the occasion of his 60th birthday.

difference between the fundamental and the 1149

1150

ENERGY RELAXATION OF WARM HOLES IN p-TYPE TELLURIUM

I~~~ic 1 kHz Reson E1Gener~dfl Amplifier

waveguide were cooled. This proved that the phase shift in the waveguide system remained constant.

______

t.~.(

X Band

~JMatthed

~ ;—~--iTermin /,~. 4,.’~

K tron 1 kHz rmd

Sample Refere’nceSarrçle

Calib.Phase

Shift~ FIG. 1. Simplified block diagram of the micro-

wave apparatus

3~

i I

/

0° -

1J2 2

I

1c1~ 2

5

The dependence of the d.c. conductivity on the electric field strength was also measured using a d.c. pulse bridge method.’° Data have been taken within the first 100 nsec of the 1 ~isec pulse in order to avoid a change of the conductivity due to the onset of acousto-electric effects. 11—13 The samples were cut from a tellurium single crystal with a hole concentration of 8.2 x i0~ 3 and a Hall mobility of 4300cm2/Vsec at cm 77°K, by means of an etch-soaked sewingthread. Samples used for microwave measure-

//

//

15°

Vol. 7, No. 16

ments were filaments of 1.7—2.2 cm length and with diameters ranging from 0.08 down to 0.02 cm.

5

I

2

S

5

-

10~

The samples for d.c. measurements were bar shaped with gold contacts alloyed to the end faces. To reduce the influence of lattice defects

FIG. 2. Relation between phase shift and energy relaxation time at the fundamental frequency

on the mobility, these samples were annealed at 350°Cfor 160 h. ~ After this treatment they

harmonic wave at the position of the sample, which is equal to the reading of the phase shifter plus a constant phase shift ç~ of the waveguide system. Now the calibrating sample is replaced by a sample with unknown energy relaxation time. The new position of the phase shifter çS~ where the mixing signal vanishes, enables one to determine the relaxation time from the relationship

showed a Hall mobility of 7700 cm 2/Vsec and a hole concentration of 3.3 x 1014 cm3 at 77°K. All Te samples were cut parallel to the c-axis. The accuracy of the microwave measurements described above is limited by the fact that rectifying effects may occur at the contacts of the sample. In this case it is not possible to detect a sharp minimum of the signal. Therefore one has to use samples long enough that microwave fringe fields do not reach the contact areas. Since the tellurium crystal was grown parallel to the c-axis with a diameter of only 1 cm, no data perpendicular to the c-axis were

=

cat

+ ~/‘cai

—

4 ~

and (2), with ~L,replaced by ç1i~ which is the phase shift of the material under investigation caused by the energy relaxation. The additional phase shift due to the finite impedance of the sample can be taken into account by reducing the cross section of the samples.4 For measurements at different temperatures and using the calibration from the measurement at one temperature it is important to know that th~stays constant. Therefore at a Ge reference sample (Fig. 1) which was always at room temperature a voltage was observed which did not change when the sample under investigation and the surrounding

obtained for this reason. Microwave measurements were also tried with annealed samples, but the value of the coefficient /3 of this material was too small to allow a reasonable accuracy with the available klystron power. RESULTS

~cai

In Fig. 3 (curve a) values of the experimentally determined energy relaxation time 7~ parallel to the c-axis are plotted vs. ternperature. At 150°K‘r~= (5.8 ±1.5) x 1012 sec and increases with temperature up to a value of

Vol. 7, No. 16

ENERGY RELAXATION OF WARM HOLES IN p-TYPE TELLURIUM

T=77°K

•~

—1

5

-2

2+ 10

1151

~

-~

x1~V~

LO

5-

FIG.4. Experimental dependence of conductivity on field strength obtained by d.c. pulses. 210_li

.f___~____°+

— -

2 ~12

b -—

—

—

I

—

1L,~J

—

—:

—

—

—

I

I

I

60

180°K

200

FIG.3. Energy relaxation time of Te c-axis vs. temperature. (a) Experimental values (b) Calculated values assuming energy loss via polar optical scattering (c) Calculated values assuming energy loss via deformation potential scattering, (7.2 ±1.5) x 10’2sec at 190°K. The quoted error in the relaxation time corresponds to a statistical error in the measurement of the phase shift of ±1°.

6 The result of the from Mendum and Dexter.’ calculation is plotted in Fig. 3 (curve b). The discrepancy between experiment and theory may be caused by the fact that a Maxwell—Boltzmann distribution at the hole temperature Th, on which the calculation is based, should not be a good approximation for the low carrier concentration in the samples used in our experiments. Furthermore it should be mentioned that this type of distribution function overestimates the energy loss especially in the case of either polar or nonpolar optical scattering.’7 This leads to the assumption that taking a more appropriate distribution function into account would yield energy relaxation times which come closer to the experimental values. For comparison, energy relaxation times

The relative change in conductivity obtained from d.c. measurements is plotted against the square of the electric field strength in Fig. 4. After an initial increase of conductivity, the deviations from Ohm’s law vanish at approximately 300 V/cm. At higher fields the conductivity lies below the Ohmic value. DISCUSSION According to a theory of Paranjape ~ the average rate of energy loss was calculated for the case of polar optical scattering in the approximation Th — T << T, Th being the carrier and T the lattice temperatures, respectively, Values of static and optical dielectric constants and Debye-temperature for the A were 6 the2-mode effective masses taken from Lucovsky et al.,

due to deformation potential scattering are also shown in Fig. 3 (curve c). Since accurate deformation potentials are not yet available,’8 a value of 5 V was used in the calculation as it was done for the estimation of the room ternperature mobility due to deformation potential scattering by Lucovsky.6 As far as the deviations from Ohm’s law (Fig. 4) are concerned one might expect, according to a theorie by Stratton,’9 a negative sign of the warm carrier coefficient /3 at 77°K. This calculation assumes that polar optical scattering alone influences the mobility. A reversal of the sign of /3 should not occur at a temperature lower than 1.080, with 0 being the Debye-temperature.

this expectation positive sign of /3 Contrary at weak to electric fields was a found in our

1152

ENERGY RELAXATION OF WARM HOLES IN p-TYPE TELLURIUM

measurements as shown in Fig. 4. This may be caused by the fact that even in the annealed samples which are used for d.c. measurements, lattice defects influence the deviations from Ohm’s law at weak electric fields, as has been 8 At higher fields the discussed by Nimtz. deviations behave as expected for the case of

Vol. 7, No. 16

polar optical scattering. Therefore we conclude that polar optical scattering is the dominant mechanism for energy loss at weak electric fields though the origin of the discrepancy between theoretical and experimental values of /3 must be sought elsewhere.

REFERENCES 1.

MORGAN T.N. and KELLY C.E., Phys. Rev. 137, A 1573 (1965).

2.

GIBSON A.E., GRANVILLE J.W. and PAIGE E.G.S., Physics Chem. Solids 19, 198 (1961).

3.

SEEGER K., Z. Phys. 172, 68 (1963).

4.

HESS K. and SEEGER K., Z. Phys. 218, 431 (1969).

5.

SCHNEIDER W. and SEEGER K., Appi. Phys. Lett. 8, 133 (1966).

6.

LUCOVSKY G., KEEZER R.C. and BURSTEIN E., Solid State Commun. 5, 439 (1967).

7.

GEIK R., GROSSE P. and RICHTER Y,I., 1st.

8.

NIMTZ G. and SEEGER K., Appi. phys. Lett. 14, 19 (1969).

9.

CONWELL E.M., High Field Transport in Semiconductors, p.24, Academic Press, New York and London (1967).

mt.

Symp. Physics of Se and Te.

10.

R~5THE.P., TSCHULENA G. and SEEGER K., Z. Phys. 212, 183 (1968).

11.

KANAI V., J. phys. Soc. Japan 14, 1118 (1959).

12.

ISHIGURO T. and TANAKA T., Jap. J. appl. Phys. 6, 864 (1967).

13.

QUENTIN G. and THUILLIER J.M., J. phys. Soc. Japan 21 Suppl., 493 (1966).

14.

SKADRON P. and JOHNSON V.A., J. appi. Phys. 37, 1912 (1966).

15.

PARANJAPE B.V., Rep. Br. elect. md. Res. Ass. L/T 285 (1953).

16.

MENDUM J.H. and DEXTER R.N., Bull. Am. Phys. Soc. Ser. 11, 9, 632 (1964).

17.

CONWELL E.M., see reference 9 p. 209.

18.

GROSSE P., Springer Tracts in Modern Physics Vol. 48, p. 96. Springer, Berlin (1968).

19.

STRATTON R., J. phys. Soc. Japan 17, 590 (1962).

Die Energie-Relaxationszeit warmer Defektelektronen in Tellur wurde im Temperaturbereich zwischen 150°Kund 190°Kgemessen. Dazu wurde die Methode der Harmonischen Mischung von Mikrowellen verwendet. Die Energie-Relaxationszeiten werden mit berechneten Werten verglichen, die man unter der Annahnie erhält, dali polare optische Streuung der dominierende Energieverlustmechanismus in diesem Temperaturbereich ist. Weiters werden die Abweichungen vom Ohmschen Gesetz bei Anlegen von schwachen und mittelstarken Gleichfeldern an Tellurproben untersucht und diskutiert.