Helicon propagation in n-type germanium

Solid State Communications Vol. 4 pp. 665-669, 1966•

HELICON PROPAGATION

Pergamon Press Ltd. Printed in Great Britain

IN n - T Y P E G E R M A N I U M

K. Saermark, N. Falsig Pedersen, J. Lebech and H.H. Andersen

Fysisk Laboratorlum I, The Technical University of Depm~rk, Lyngby, Denmark. (Received 26 October 1966 by Ho H. Jensen)

A report on preliminary experimental resuRs on propagation of helicon waves in n-type Ge is given. Propagation in the (I, I, 1)and (1, 0, 0) - directions is considered. The experimental results show that helicon propagation in these directions in n-type Ge m a y take place under circumstances which in an essential way differ from the usual helicon assumption • << %, ~=. R is shown that the results are in good agreement with the dispersion relation given by Wallace.

In these expressions eL, c,q, • have the usual meaning while ~o = e H / m c , m being the free electron mass. Further c~L = m / m L , a T = m / m T where m L and m T are the longitudinal and transverse masses. ~ is the plasma frequency for the electrons belonging to a particular valley direction. ~

IN A SERIES of papers Wallace t-3 has given a theoretical discussion of helicon propagation in n-type germanium and silicon taking into account the actual bandstructure of these materials. Propagation along the (I, I, I) - and (I, 0, 0)- directions was considered in particular. The analysis given by Wallace may for the present purpose be s u m m a r i z e d as follows:

(I, 0, 0) - direction The d i s p e r s i o n relation is still given by (1) but (2) is r e p l a c e d by

Consider a right circular polarized wave exp [i (qx - wt) ], q, • > o, travelling along the direction of the magnetic field. The dispersion relation m a y then be written in the following way.

Ld =

(I, 1, I) - direction C

2

2 (~

--

w

2 0~

CL

=

t'(._~o ) ~

-

(I)

÷7

=

-

/

0CT(2C¢L + a : T ) ( ~ ' )

(3)

w (2CXL + aT) (~o)2

aT td

l_= r(O )

In Fig. 1 the function f (wo / ~)has been plotted for the (1,1, 1) - direction a s a function of(~o / w ) o r alternatively - for a fixed frequency - as a function of magnetic field strength H. As pointed out by Wallace = = for(wo/w) < 0 the c u r v e s may be i n t e r p r e t e d as giving the d i s p e r sion relation for left c i r c u l a r l y polarized waves travelling along the magnetic field. For a fixed

(2)

(t~aL * 5c¢T) + (8a L + aT)aT(~-q ) } 1 -

1 1 -~aT

-

F o r Germanium a L and a T may be taken equal to 0. 61 and 12.2 respectively.

~d

1

(2a T + aL)

p

where r(;--)

3

W

2 W

b,

°T (8=L *

(o)2

665

666

H E L I C O N P R O P A G A T I O N IN G E R M A N I U M

/:~) ~ (/./.,,,/

t', i 1 I

'l

alpo

w _

(%>

(4)

p

°

$Ill

II

2

>.

i ii I ', @

Vol. 4, No. 12

.

.

.

.

\

will resuR in a propagating wave. For the I direction this condition corresponds to (a) the normal helicon wave in the region to the right of point A on Fig. 1 and (b) the 'optical' helicon branch ~ in a part of the region from the point B to the point A on Fig. 1. For the H-direction the consequences of the condition (4) depend markedly on the actual value of!w/wp) 2. Thus, a propagating m o d e is possible for all values of (wo/w)to the left of the point C on Fig. I if ( w / ~ ) 2 > a, where a = 4.62 for the (I,I, I) direction and a 8.95 for the (I, 0, 0)-direction. For values of ,~/w~ )2 satisfying a>(u:/~) 2 > 0 only parts of these regions will allow a propagating mode. In particular, for ~ ~>~, only the region immediately to the left of the point C on Fig. I will allow a propagating mode, however,

\

0 6 e {tOOL p =5.,2

o, FIG. 1 Solid curve: Dispersion relation after Wallace ~ for helicon waves in the (I, I, I) - direction in Ge. Broken curve: Dispersion relation in the usual helicon-approximation. Broken vertical lines: Asymptotes corresponding to cyclotron frequencies.

: :'""=:2.

o., %~

propagation direction w e shall refer to the direction of the magnetic field corresponding to(~o /~) > 0 as the I- direction and the opposite direction where(~o/~) < 0 as being the H direction.

+ ~1

For the (I, 0, 0)-direction the curves will be qualitatively similar. However, in this case there will be only one asymptote for the I-direction. F o r fixed values of ~ and q f o r the helicon wave and a given value of ~ it is s e e n f r o m equation (1)that the corresponding value(s) of the magnetic field or (~o / ® ) m a y be found by i n t e r s e c t i n g the c u r v e s on Fig. 1 with a s t r a i g h t line p a r a l l e l to the ('~o /'~) - axis. Thus, since q >= o, only values of (~'o / ~ ) s u c h that

|

I

i

I

i

I

I

I~

FIG. 2

Experimental values of the p l a s m a freque n c y ( % / w ) ~ s a function_of l / T , determined by m e a n s of Rayleigh interference p a t t e r n s . The s t r a i g h t line is a least s q u a r e s fit to the g e o m e t r i c a l r e s o n a n c e points. it will be strongly damped due to the n e a r n e s s to the cyclotron r e s o n a n c e . This damping effect may a l s o modify the shape of the d i s p e r s i o n

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HELICON PROPAGATION IN GERMANIUM

667

coil allowed the t e m p e r a t u r e of the sample to be v a r i e d in the r a n g e 4 - 25°K. The s a m p l e s had a r o o m t e m p e r a t u r e specific r e s i s t i v i t y of 2 ohmcm. The following m e a s u r e m e n t s were p e r f o r m ed: (a) F o r a number of t e m p e r a t u r e s in the range 10-25°K the t r a n s m i s s i o n through the sample was observed a s a function of magnetic field, both for the 1-direction and for the IIdirection. The m e a s u r e m e n t s w e r e made both for the (1, l , 1) - and the (1, 0, 0) - direction.

Imovm & 3 2 1 0

(b) For a number of t e m p e r a t u r e s in the range 10-25°K R a y l e i g h - t n t e r f e r e n c e patterns were obtained for both directions of magnetic fields and for both c r y s t a l l o g r a p h i c directions.

y os

(c) For the (1, 1, 1) - direction the t r a n s m i s s i o n through the sample was observed as a function of t e m p e r a t u r e for a number of fixed values of the magnetic field.

,I0i O,Cf

An analysis of the r e c o r d e r t r a c e s gave the following results.

p-4

V ~, .

. ~,"..

FIG. 3 Wave number q as a function of magnetic field in the region C-B of Fig. 1. The i n s e r t shows for magnetic field in the H - d i r e c t i o n t h r e e r e c o r d e r t r a c e s of the t r a n s m i s s i o n of helicon waves through a (1, 1, 1 ) - direction G e - s a m p l e . The t r a c e s e-~ a r e taken at 11.2 and 11.7 and 12.2 K, respectively. For t r a c e g the amplification was a factor of 3 lower than for the c u r v e s e and f. The base line is c o m m o n to the t h r e e traces. curve in this region. In the region between the points C and B on Fig. 1 propagating modes will also be possible for suitable values of (~ / w~), examples of which may be seen on Fig. 3 which shows some r e c o r d e r t r a c e s of the t r a n s m i s s i o n through the sample as a function of magnetic field. As mentioned in Ref. 3 for (~/w p) >> 1, this region gives r i s e to the usual c y c l o t r o n resonance. T h e experimental s e t - u p used was s i m i l a r in principle to the microwave i n t e r f e r o m e t e r d e s c r i b e d by Furdyna " a n d by Flietner and Kliefoth. ~ The fixed frequency of the i n t e r f e r o m e t e r was 43.9 GHz. The sample - lengths 3 or 4 m s , diameter 3 mm - was mounted in an i r i s i n s e r t e d in a c i r c u l a r waveguide . A heating

(1) The direct t r a n s m i s s i o n curves show peaks corresponding to the well-known g e o m e t r i c a l r e s o n a n c e s given by the condition p . ½k = d where d is the length of the sample. E x p r e s s e d in t e r m s of the wavenumber this condition b e c o m e s q

= P ~

*

(5)

F r o m the position of the peaks in the t r a n s m i s s i o n curves it was possible to a s s i g n a p-value to each of the peaks. By means of these p-values and the equations (1) - (3) the value of ( ~ / ~ ) ~ was calculated as a function of the sample t e m p e r a t u r e m e a s u r e d by means of the carbon r e s i s t o r . The; r e s u l t s a r e plotted in Fig. 2 where the logarithm is shown against 1 / T . F r o m this figure it may be seen that all of the points, i . e . for both c r y s t a l lographic directions and for both directions of the magnetic field, lie r e a s o n a b l y well on a straight line. The solid line shown has been dete r m i n e d by a least s q u a r e s fit to the points belonging to the p-values 7, 5 (I-direction) and 4, 2 (H-direction). The slope of the straight line c o r r e s p o n d s to an impurity ionization energy of 0. 0100 eV. It may be noted that t h e r e is a tendency for the points belonging to p = 6 and p = 3 (not shown on Fig. 2) to s c a t t e r somewhat m o r e than the r e s t of the points although they a r e in g e n e r a l a g r e e m e n t with the straight line shown on Fig. 2.

(2)

The Rayleigh i n t e r f e r e n c e patterns may

668

HELICON PROPAGATION IN GERMANIUM

a l s o , a s shown by Furdyna, 4 be utilized to d e t e r mtne(~/tv). Using methods s i m i l a r to F u r d y n a ' s we have done this f o r the d i s p e r s i o n r e l a t i o n s given by equations (1) - (3), and the resulting values of (wp /~)2 have been plotted on Fig.2. R is seen that there is a good agreement between the two ways of determining the ratio(%/~) 2. R is noted that the determination by means of the Raylelgh interference pattern does not require an assignment of a definitep-value. (3) As mentioned e a r l i e r the t r a n s m i s s i o n through the s a m p l e was o b s e r v e d as a function of t e m p e r a t u r e f o r a number of fixed values of the magnetic field. As the t e m p e r a t u r e is r a i s e d w ~ i n c r e a s e s and f o r the H - direction i t event~mlly obtains a value above which no t r a n s m i s s i o n is possible f o r magnetic fields exceeding a certain value. Below this value of ,~ transmission will be possible and the curve of transmission versus temperature will show m a x i m a when the condition (5) is fulfilled. For the H-directlon, however, only one peak was observed, although the curves corresponding to higher magnetic fields show transmission up to the temperature corresponding to the criticalvalue of % . For magnetic fields immediately to the left of the point C on Fig. 1 the damping is so high that no transmission is observed. For the I-direction two peaks were observed on the transmission versus temperature curves. By means of the appropriate q and w~ values the left hand side of equation (1) was calculated for the above mentioned peak values and plotted on Fig. I at the corresponding value of the magnetic field. Also shown on this figure is the dispersion curve corresponding to the usual helicon approximation. R is seen that the experimental points are in good agreement with the dispersion relation given by Wallace except for the point at the lowest magnetic field in the I-direction. However, as Wallace has neglected damping effects it is reasonable to assume that this deviation m a y be due to damping effects causing the f-curve to bend over.

Vol. 4, No. 12

(4) Also in the region between the points B and A on Fig. 1. minor peaks on the transmission curves were observed, and at some values of magnetic field strength also discontinuitiesin the transmission curves were observed. The discontinuities were reproducible and showed hysteresis (5) P r o p a g a t i n g modes w e r e a l s o o b s e r v e d in the region between the points C and B on Fig. 1, i . e . p a r t l y c o r r e s p o n d i n g to the H - d i r e c t i o n . Figure 3 shows s o m e e x a m p l e s of r e c o r d e r t r a c e s obtained for the H - d i r e c t i o n t a k e n at the t e m p e r a t u r e s 11.2, 11.7 and 12.2°K. This example may be c o r r e l a t e d with the c u r v e s on Fig. 3 showing q as a function o f ( u b / ~ ) . It follows f r o m Fig. 2 that at l l ° K we have a value of ( % / ~ ) ~ 0.1. F r o m Fig. 3 it is s e e n that the i n t e r f e r e n c e condition equation (5) for this value of the p l a s m a frequency m a y be satisfiedfor to neighbouring values of the magnetic field. By raising the temperature to 11.7 and 12.2°K, respectively, this possibility is eliminated and as shown on Fig. 3 the two peaks merge into one and the peak value diminishes as a consequence of the fact that the q - value moves away from the one suitable for geometrical resonance. Note, that the peak corresponding to the higher magnetic field has a somewhat larger amplitude. A somewhat similar effect m a y also be seen on some of the curves belonging to the H - direction to the left of the point C on Fig. I. Here a value of ( % / ~ ) ~ in the range 0 < ( % / w ) 2 < a m a y give rise to two peaks on the transmission curve, corresponding to the same value of the wavenumber q, when the magnetic field is raised for a constant temperature. This has been observed on some of the recorder traces. Summing up we conclude that the experimental o b s e r v a t i o n s a r e in good a g r e e m e n t with the d i s p e r s i o n r e l a t i o n given by Wallace, in spite of the fact that both damping and quantum effects have been neglected in Wallacets derivation of the d i s p e r s i o n relation.

References 1.

WALLACE P . R . , Can. J. P h y s i c s 43, 2162 (1965).

2.

WALLACE P . R . ,

Can. J. P h y s i c s 44, 442 (1966).

3.

WALLACE P . R . ,

(to be published).

4.

FURDYNA J . K . , Rev. Sci. Inst. 3_77, 462 (1966).

5.

F L I E T N E R H. and K L I E F O T H K . , PhTs. stat. sol. 14, 181 (1966).

Vol. 4, No. 12

HELICON PROPAGATION IN GERMANIUM

Ein Bertcht ~ber vorla~iflge experimentelle Resultate betreffend Ausbrettung yon Heltkon-WeUen in n-Typ-Ge ist gegeben. Ausbrettu~g in die (1, | , | ) - und (1,0, 0) - Richtungen ist untersucht worden. Die experimentelle Resultate zeigen, class HeltkonoAusbreitung in diese Richtungen in n-Typ- Ge stattflnden kann unter Umst~nden, die entschetdo end yon der gewdhlichen Helikon-Annahme ~ ~ ~ , ~c abweichen. Es tst gezeigt, class die Resultate mit der yon Wallace gegebenen D[spersionsgleichung in guter Ubereinstimmung stnd.

669