Reflectivity measurements of coupled collective cyclotron excitation-longitudinal optical phonon modes in polar semiconductors

Solid State Communications, Vol. 6, pp. 721

-

725, 1968. Pergamon Press. Printed in Great Britain

REFLECTWITY MEASUREMENTS OF COUPLED COLLECTWE CYCLOTRON EXCITATION-LONGITUDINAL OPTICAL PHONON MODES IN POLAR SEMICONDUCTORS E.D. Palik, B.W. Henvis, J.R. Stevenson* Naval Research Laboratory, Washington, D. C. 20390, U. S. A. and S. Iwasa Department of Physics, Massachusetts Institute of Technology Cambridge, Massachusetts, 02139, U.S. A. (Received 25 June 1968) The magnetoreflection of n-type InAs has been measured In the far infrared region near the optical phonon frequency for both the Voigt and Faraday geometries. Such measurements illustrate the different possible types of cyclotron, p].asmon, and phonon interactions which can occur.

RECENTLY, we measured resonant absorption by coupled collective cyclotron excitation-longitudinal- optical phonon modes in thin-film samples of InSb In the Voigt geometry. ~ 2 The magnetic field dependence of the frequencies of these modes was well accounted for by a classical coupled oscillator model. The single particle cyclotron excitations couple to the free carrier plasmon excitations and to the LO phonons via a macroscopic longitudinal electric field arising from an electric polarization due to the influence of the Lorentz force on the free carriers. These coupled oscillators have transverse as well as longitudinal character, the coupling to EM radiation being through the transverse nature of the mode.

=

2

(n

1

+

=

1+

e

L

1kjj

2 2 +

-

w,

c” ~

-

2

2 -~

(w+j’y)2

cii

=

(nil +lkj 2

= c~.

ir’ ~

-

-

________

ui2 + i ~

ir w)

D

(H=0)

1) where the definitions

For the Voigt geometry (direction of propagation perpendicular to external magnetic field) two linearly polarized EM waves will propagate in the medium. The behavior of these waves depends on the dielectric constants C.L and ~ for polarizations perpendicular and parallel to the external magnetic field H. The dielectric constants are

D

r =

1 -

w~-iyu~+

L

(ui2 -w2-iFw)

(~iy)~ =

-

ui~~

L

-4

uj~

w~= 4r~ne2/m’c~ and u, ~ eH/rn*c hold; e, n, and m’ are the charge, density and effective mass of the free carrlerss respectively; c~is the high frequency dielectric constant; and F are the damping frequencies of free carriers and

_________________

‘Permanent address: Department of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A.

~‘

721

REFLECTWITY MEASUREMENTS

722

200

100

~ ~O(ii) 40

30

,J ~

0.3

25

I

I

I

0.2

n~ 2 2*lO”Cm’

~

Vol. 6, No. 10

~X/ ~

1

~

~0~2

—

02

~

07

-60

FARADAY

rcp

I

0 © 0

00

W

200 (cm’)

300

0

400

-

(c) Faraday geometry right circularly polarized radiation (cyclotron inactive), -

phonons; uL and (UT are the longitudinal and transverse optical vibration frequencies. The coupled mode frequencies ±are given by + 4)2

-

4(~W~.+w~w~)J2

-p

(‘f) for the lossless case ~

=

F

80

Reflectivity vs. H for the 1-InAs sample at several frequencies for unpolarized radiation. The experimental dashed curves were normalized to the solid calculated curves at H = 0.

(b) Faraday geometry left circularly polarized radiation (cyclotron active).

+4 ±r[(w~+

60

FIG. 2

(a) Calculated bulk reflectivity R~ for the 1-InAs sample in Voigt geometry for several magnetic fields.

+ w~’

40 H ~kG)

FIG. I

2w~=

20

=

0.

The index of refraction n and extinction coefficient k have been used in standard thinfilm formulas including multiple reflections to analyse the transmission data. These thin-film calculations have been useful in studying the line half-width and intensity of the coupled modes. ~ Thin film reflectance and bulk reflectivity were also calculated. In Fig. I (a) is shown the magneto-reflectivity R.1. of the 1-I.nAs sample. Rn is just R.L (H = 0). PrevIous magneto-plasma 5 have usually been done with samples of such high carrier density reflection measurements that the condition up 25 wL held. Then the plasma reflection edge was split at u~, so that one minimum moved to higher frequency and one moved

Vol. 6, No. 10

REFLECTIVITY MEASUREMENTS

723

TABLE 1 Parameters for two n-type InAs samples Sample

n 3)

m’/m

V (cnf1)

5p

(cm’~)

(cm

Ca,

WT

-

)

1 (cm” )

(U

(cm

1-InAs

(cm 2.2 x 10’~’

0.03

30

2

11.8

236

219

243

2-InAs

2.9xI0~

0.027

20

2

11.8

90

219

243

to lower frequency. The present calculation mdicates that the minimum which moves to lower frequency stops at ~L• The other “plasma reflection edge” below ui~’also splits, the higher frequency minimum moving toward but stopping at ~T while the lower frequency minimum simply moves to lower frequency. The reflectivity minima are associated with unities in the real part of the dielectric constant CR. At the coupled mode frequencies uj as well as ~T, the real dielectric constant tends to infinity. The dielectric constant is zero just to the low frequency side of where it is unity. This is near the top of the steep plasma reflection edges. For the lossless case R would reach unity at this point, A zero in the real dielectric constant at H = 0 locates the resonant frequency of a longitudinal oscillator with which transverse EM radiation cannot interact. Thus in a normal mcidence, thin film experiment, no resonant absorption of radiation occurs at 5R = 0 although at oblique Incidence a resonance is observable. ~ As Fig. 1(a) indicates, there are two such oscillators or longitudinal modes when H = 0. These are the coupled longitudinal plasmon-phonon modes discussed by Varga6 and measured by Mooradian and Wright. With magnetic field, there are four zeroes just to the low frequency side of the four unities corresponding to four reflection minima. Resonances occur at u±in the longitudinal as well as the transverse aspect of the modes. Examination of the equations of motion Indicate that no longitudinal resonances of the charge system occur at the four zeroes in ~R~’ In FIg. 1(b) and (c) is shown the reflectivity spectrum for the Faraday geometry (direction of propagation parallel to H). This was calculated from the dielectric constant

r c~

,r

=

L

2~

1+

1

-

4-w2-iflu

w(w;w~-lv)J

(3)

for the left (Lcp, cyclotron active) and right (rcp, cyclotron inactive) circular polarizations. In this case, no macroscopic electric field arises from the Lorentz force to couple the single particle, cyclotron excitations to the plasma excitations and the LO phonons as was the case for the Voigt geometry. Thus, there are only two Infinities in ~R now located at w. and wT. At H = 0 there are again the two coupled longitudinal plasmon-phonon modes5 at = 0. When H ~ 0, the coupled plasmonphonon modes continue to oscillate at w~ even though eRA r ~ 0 at ~ Also no longitudinal modes of the charge system occur at ~ r = 0. These points may be termed dielectric anomalies. The plasma reflection edges shift for Lcp and rep, since the unities and zeroes In eR~r shift. We have measured the magnetoreflection of two n-type samples of InAs characterized in Table 1. The experiment was performed in the far infrared with a conventional grating monochromnator optical system In a dry box. Because considerable water vapor absorption persisted even with drying, the experiment was done by field sweeping at fixed frequencies. The sample was glued to a cold finger of a liquid helium optical dewar and was substantially above 4. 2°K. Unpolarized radiation was usedin the experiment, the ratio of to changing somewhat with wavelength because of polarization of the optical systern. Some results are shown in FIg. 2 for the 1-InAs sample. The calculated reflectivity contaming both R 11 and R~.contribution is plotted against H for several frequencies along with the ~.

724

REFLECTIVITY MEASUREMENTS 20 IS

H (kG) 60

40

°

so

Vol. 6, No.10

measured reflectivity normalized to the calcu]atedatH=0.

o

~‘

0.~

Since the parameters n, m’, ‘i’, r, ta,, wj’ and WL were known for each sample from

S

slightly Hall literature, andtomobility obtain we only the measurements adjusted calculated n, m’, curves andand from ofVFigs. the 2 and 3. The positions of reflection maxima and minima were the primary experimental results. These were not sensitive to V, so this parameter Is the least well determined, since we made no serious attempt to fit the magnitude of the experimentally measured reflection which Is sensitive to Y,

2

• 2~

i~j:r~a,~___,_,••___

1_,_

—

3

WL

4;’

~f~..___

.5

3

3

2-mAe

a

°°

~

-

2

o o~~—~--i----

2000

4000 H~(kG)t

6000

FIG. 3 (a) Observed and calculated posttions of reflection minima and maxima for Volgt and Faraday geometrIes, 1-InAs. (b) Same for only Voigt geometries; 2-InAs. For Voigt geometr~the solid curves are calculated positions of reflection maxima and minima; the symbol 0 experimental minima; •- experimental maxima. The dotted lines are the positions of w as obtained from equatIon (2). ~‘or the Faraday geometry, the dashed curves are calculated positions of reflection minima; the symbol ~ expertmental minima.

The positior~of the reflection maxima and minima are shown In Fig. 3, From a series of plots like Fig. 1 and from solutions of equations (1) and (3) for = 1, we determined the solid lines fitted through the experimental maxima (solid circles) and minima (open circles). The two maxima fall very near the positions of u~ indicated by the dotted lines. However, the reflectivity Is not directly Informative as to the line half width and intensity, although these are obtainable precise data. from P a Kramers-Kronlg analysis of We measured reflectivIty geometry for the 1-InAs sample in inthe the Faraday frequency region above 250 cm”’ with the use of a polyethylene wire grid polarizer and a CsI Fresnel rhomb to produce left and right circularly polarized 1 unpolarized radiaradiation. tion was used. Below The 250four cm”reflectivity minima for the Faraday geometry (two for £cp and two for rcp) nearly coincide with the four reflectivity minima for the Volgt geometry. The dashed curves in Fig. 3(a) represent the calculated positlons where ~RL = 1. The cyclotron frequency w~ for the ‘Faraday geometry, indicated in Fig. 3(a), Is manifested in the zcp spectrum of Fig. ~b) as a dip In the reflectivity.

-

The forms of e in equations (1) and (3) give a reasonable fit to the experimental transmission data3 and present reflectivity data. Both experiments have proved fruitful in the study of coupled modes.

-

References 1.

IWASA S., SAWADA Y., BURSTEIN E. and PALIK E. D., Proceedings of the International Conference on the Physics of Semiconductors, Kyoto 1966; J. Phys. Soc. Japan Suppi. 21, 742 (1966).

Vol. 6, No. 10

REFLECTIVITY MEASUREMENTS

725

2.

KAPLAN R., PALIK E.D., WALLIS R. F., Phys. Rev. Left. 18, 159 (1967).

IWASA S., BIJRSTEIN E. and SAWADA Y.,

3.

PALIK E,D,, KAPLAN R., HENVIS B.W., STEVENSON J.R., 1WASA S. and BURSTEIN E., 9th International Conference on the Physics of Semiconductors, Moscow 1968 (to be published).

4.

1WASA S., International Advanced Summer Institute on the Physics of Solids in Intense Magnetic Fields, Chania, Crete 1967. Plenum Press, New York, (in press).

5.

WRIGHT G.B. and LAX B., J. appi. Phys. Suppl. 32, 2113 (1961).

6.

VARGA B.B.,

7.

MOORADIAN A. and WRIGHT G. B., Phys. Rev. Lett. 16, 999 (1966).

8.

STIMETS R.W.,

Phys. Rev. 137, A,896 (1965).

Bull, Am. phys. Soc. 13, 725 (1968).

Die Magnetoreflexion des n-InAs wurde in dem fern-ultraroten Gebiet in der N~heder optischen phonon Frequenz sowohi fur die Voigt als auch für die Faraday Anordnung gemessen. Messungen dieser Art zeigen die verschiedenen Arten von Zyklotron, Plasmon, und Phononwechselwirkungen weiche vorkommen können.