Hall mobility of hot electrons in CdS

Solid State Communications, Vol. 17, pp. 957—960, 1975.

Pergarnon Press.

Printed in Great Britain

HALL MOBILITY OF HOT ELECTRONS IN CdS Y. lye and K. Kajita Department of Physics, Faculty of Science, University of Tokyo Bunkyo-ku, Tokyo, Japan (Received 13 June 1975 by W. Sasaki)

Photo-Hall effect of.hot electrons in a pure single crystal of CdS was observed, for the first time. The measurement was carried out for electric fields up to 4,300Vfan in magnetic fields up to 40 kOe at 4.2 K. The saturation of the drift velocity Vd of electrons due to optical phonon emission was observed. The saturated value of 1~dis found to be about 1.7 x iO~cm/sec. IN STUDYING the behaviour of electrons at high electric fields, ionic crystals such as Ag-halides and CdS are believed to be suitable materials because they have highly insulating character at low temperatures with deep lying impurity levels for electrons. Thus, extremely high electric fields can be applied without causing the heating of the specimen or the impact ionization of impurity. Nonlineartransport phenomena in ionic crystals at high electric fields have been investigated by a number of experimentalists.13 Severalimportant problemson the hot electron effects in ionic crystals, however, stifi remain unsolved. For example, we have little knowledge on the reason why the current saturation due to optical phonon emission has not been observed even at extremely high electric flelds.12 In such fields, the kinetic energy of electrons is considered to be comparable to the optical phonon energy. In relation to this problem, it is necessary to carry out the Hall effect measurements which are essential in distinguishing the effect caused by a change in the mobility of electrons from that caused by a change in the trapping lifetime of electrons. In the Hall effect measurements in ionic crystals using Redfield’s arrangement of electrodes, the electric field was usually limited to 100—200 V/cm,”4 while, in the pulse photoconductivity measurements with blocking electrodes, the electric fields up to several thousand V/cm were applied.1 In the present study, we have suceeded to observe the Hall effect and magnetoresistance over a wide range of electric fields up to

4300 V/cm and of magnetic fields up to 40 kOe in a pure single crystal of CdS. An arrangement of electrodes used in the photoHall signal measurement is ifiustrated in Fig. 1. This modification of electrodes of Redfield’s arrangement5 enabled us the application of higher electric fields to the crystal. Apulsed electric field is applied between the lower electrode and a pair of the upper electrodes. Carriers are created by a light pulse of about 1 psec duration generated by a xenon flash lamp. If the electrode arrangement is perfectly symmetric and the magnetic field is absent, the amount of charge induced on each of the two upper electrodes by the carrier motion in the specimen should be identical. Hence, the output of the differential signal is zero. When the magnetic field is applied in the z-direction, a differential signal proportional to the transient Hall signal appears between the two upper electrodes. A sample of CdS pure single crystal used in the present study is grown by the Piper method.6 All the experiments were made at 4.2 K. At this temperature, the magnetoresistance mobility of the electrons in this sample was about 31,000 cm2/V sec. It is known that the holes make little contribution to the electric conduction.7 In the,present experiment, the crystal was placed so that the c-axis is always parallel to the y-direction.

957

958

HALL MOBILITY OF HOT ELECTRONS IN CdS ne2

L~t Ele ctrodes

Quartz

C~

L

F,eld

rator

Q

Recorder

FIG. 1. Schemetic diagram of the electrode arrangement and the detection system used in the photo-Hall signal measurement. _______________

~

I

1

e

2 2~

r~HE

(2)

2100 3300

ioo 230

Q,,

(3)

M(H) ~ H2

(4)

—

Q,, (H)] /Q~ (H) is the

magnetoresistance. Small anisotropy in the electron scattering relaxation time reported by Fujita eta!.7 is neglected in the following analysis. Figure 2 is the magnetic field dependences of Q~,

E (Vk~)

i

~H

hold. Here, M(H) = [Q~ (0)

CdS 4.2 K

/

pulse, m tronic where ncharge, is the the effective Cnumber the light of mass velocity, electrons of an w electron, released eH/m*C by r the athe light scattering the electron, e the cyclotron relaxationtime frequency, Tt theoftrapping lifetime of electhe electron, and H the magnetic field. The bracket () indicates the statistical average over the distribution function of electrons. In the low magnetic field region, it is well known that the relations,

Differentia Amplifier

mt

Vol. 17, No.8

in the inset. At H = 0, nonlinear relation between Q~ and E becomes apparent at electric fields above 30 V/cm. It should be noted that Q,, increases monotonically with increasing electric field without any indication of saturation up to the highest fields. At at several electric fields with the data of Q~at H = 0 low electric fields where electrons are “cool”, the —H curves rise linearly with increasing H at low H,

,,,~43oo

73

10

CdS

PV 23906

4.2 K

46

29 15

I

~cE

~ /

-

1

H(kB)

or

too— -~

10 ~0.5

FIG. 2. The magnetic field dependences of the photosignals Q~at several electric fields. The electric field dependence of the photo-signal Q,, at H = 0 is shown in the inset. In the transient condition, the photo-signal Q~ and the Hall signal Q, in the present configuration in the limit of the low electric field E are expressed as follows, ___2 I r ________

ne m * ~l + w2r2) r~E

0.1

.

E (V/cm)

FIG. 3. The electric field dependences of the photosignals Q~andPH Q,,Eat(c/JI)(Q~ low magnetic field limit, and the Hall mobility Relative values of Q~and Q, are nonnalized at /Q~,). low electric fields.

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HALL MOBILITY OF HOT ELECTRONS IN CdS

have a peak at about 3 kOe and fall at higher fields, The magnetoresistance M(H) is proportional to H2 below 2 kOe. At high electric fields where electrons are “hot”, there also exists a magnetic field region where Q,~is proportional to H. It turned out that, in the magnetic field region where Q,, ~ H, M(H) is proportional to H2. The relations, Q~,, ~ H and M(H) ~ H2, observed at high Eand low H are formally identical with those observed at low E and low H. [Equation (3) and (4)1. We, thus, may deal with the data in this region assuming that electrons have a distribution and a mobility which are not affected

959

scatteringmechanism with the rise of the electron temperature. At 4.2 K, the piezoelectric potential scattering is dominant in the low electric field region. Above 100 V/cm, the deformation potential scattering becomesimportant. Our main interest in the present study is in the region (d), where the kinetic energy of electrons is estimated to be comparable to the optical phonon energy of the crystal. _________________________________

tO CdS PV 2 3906

appreciably by the magnetic field.*

4.2 K

Closed circles in Fig. 3 represent the electric field dependence of Q, at a weak magnetic field in the2 regionover where relations ~ Hregion. and M(H) H hold the the whole electricQ,,field Open circles are Q~at H = 0. As can be seen from the figure, the entire range of the electric field in the present study can be divided into four regions according to the characteristic behaviours of the two quantities Q~ and Q~: (a) E~30 V/cm Both Q~and Q,, increase linearly with E, i.e. show the ohmic behaviour; (b) 30 V/cm~E~ 100 V/cm Q~and Q~show superlinear dependence on E. Q~,rises up more steeply than Q~. —

—

(c) 100 V/cm~E~500 V/cm Q~is approximately proportional to the square root of E. Q~,tends to a constant value independent of E. —

(d) 500 V/cm E Q~gradually rises up again, while Q,, remains constant. ~

rn6

___________________________________ 10 100 1000 E (V/cm)

FIG. 4. The electric field dependence of the drift velocity Vd PH (E).E with the data of the photosignal ~ for comparison. The value of Q~is normalized to Vd at low electric fields. The dashed line indicates the value of the drift velocity of an electron whose kinetic energy of the dirft motion along the electric field is equal to the optical phonon energy.

—

The experimental results of Q~below 500 V/cm

and Q~,below 100 V/cm are in good agreement with the corresponding data for the same sample reported by Fujita eta!.7’8 and by Shiga and Onuki.4 The behaviours of Q~and Q~in these regions have been interpreted by them in terms of the change in the ______________

tmAs has been pointed out in the reference 3, hot electrons in the transverse magnetic fields reduce their kinetic energy with increasingH. The experimental results of M(H) and Q.~in CdS suggests that the kinetic energy of hot electrons in CdS remains unchanged at low H and decreases abruptly at intermediate magnetic fields. This phenomenon will be discussed elsewhere.

To investigate the mechanism of the scattering which plays a significant role in the region (d), we calculate the Hall mobility PH (E) (c/I-f)(Q~/Q~) from the data shown in Fig. 4. Triangles in Fig. 3 represent the calculated Hall mobility. The Hall mobility PH (E) is constant at electric fields below 30 V/cm [region (a)] and increases slightly in the region (b).4 In the region (c), PH (E) decreases as E~2 and falls more steeply in the region (d). We attribute this steep fall O~PH(E) in the region (d) to the optical phonon emission by electrons. To see this point, we plot the quantity PH (E) E in Fig. 4, and compare it with the data of Q~. The quantity PH (E). E is identical with the drift velocity of electrons, apart from the small difference between the Hall mobility and the drift mobility. As can be seen in Fig. 4, PH (E). E agrees fairly well with Q~ below 500 V/cm.

960

HALL MOBILITY OF HOT ELECTRONS IN CdS

Above 500 V/cm, the PH (E) E curve shows saturation. The kinetic energy of an electron due to the drift along the electric field at the saturation is estimated to be approximately 0.4 x 1u~,where h~,,,is the qptical phonon energy of the crystal. This suggests that the average drift velocity of an electron is limited by the optical phonon emissioil process. Thus, the rise m Q~ in the region (d) should be attributed to the Change in the trapping lifetime, whose microscopic mechanism is presently unknown.

Vol. 17, No.8

In conclusion, these experimental results suggest that the kinetic energy of electrons at high electric fields in CdS is limited by the optical phonon emission process, while photo-signal increases monotonically up to extremely high electric fields. We are grateful to Prof. K. Kobayashi and Dr. M. Onuki for kindly providing the sample. We thank Profs. K. Kobayashi, T. Masumi and Mr. S. Komiyama for valuable discussions and critical reading of this manuscript. On of the authors (K K.) is obliged to Sakkokai Foundation for financial support. Acknowledgements

—

REFERENCES 1. 2. 3.

MASUMI T.,Phys. Rev. 159,761 (1967). NAKAZAWA F. & KANZAKJ H., J. Phys. Soc. Japan 20,468 (1965). KAJITA K. & MASUMI T.,Proc. 12th mt. Conf. Phys. Semicond. Stuttgart, p. 844 (1974).

4. 5.

ONUKI M. & SHIGA K., Proc. fin. Conf. Phys. Semicond. Kyoto, 1966, J. Phys, Soc. Japan 21 Suppl., 427 (1966). KOBAYASHI K. & BROWN F.C.,Phys. Rev. 113, 507 (1959).

6.

ONUKIM.&KUBOS.,J.Phys.Soc.Japan 17,1516(1962).

7.

FUJITA H., KOBAYASHI K. & TAKANO K.,J. Phys. Soc. Japan 21,2569(1966).

8.

FUJITA H., KOBAYASHI K., KAWAI T. & SHIGA. K., J. Phys. Soc. Japan 20, 109 (1965).