Temperature Dependence of Intrinsic Carrier Concentration in InSb: Direct Determinationby Helicon InterferometryK. K. Chen and J. K. Furdyna Citation: Journal of Applied Physics 43, 1825 (1972); doi: 10.1063/1.1661403 View online: http://dx.doi.org/10.1063/1.1661403 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/43/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Temperature dependence of photoluminescence from InNAsSb layers: The role of localized and free carrieremission in determination of temperature dependence of energy gap Appl. Phys. Lett. 102, 122109 (2013); 10.1063/1.4798590 Effects of diffusion current on galvanomagnetic properties in thin intrinsic InSb at room temperature J. Appl. Phys. 45, 3530 (1974); 10.1063/1.1663814 Intrinsic Concentration and HeavyHole Mass in InSb J. Appl. Phys. 41, 1804 (1970); 10.1063/1.1659107 HeliconWave Propagation in InSb J. Appl. Phys. 38, 4888 (1967); 10.1063/1.1709238 Distribution Coefficients and Carrier Mobilities in InSb J. Appl. Phys. 30, 559 (1959); 10.1063/1.1702405

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LASER -HEATING STUDIES 1825

VII. STRESS GENERATION WITH BLOWOFF

To further elucidate phenomena accompanying rapid energy deposition in materials, the following experi­ment was carried out. The dashed curve in Fig. 5 shows the results of a measurement of the stress wave when the vaporization of the carbon causes surface erosion at the front of the sample. These two curves (the elastic wave and the wave generated when blowoff occurred) were obtained at the same fluence on the same sample in subsequent irradiations. The physical mechanism for this seemingly paradoxical situation is the following: On the first irradiation the PMMA matrix cracks and the vaporized carbon is dispersed. The cracking of the ma­trix weakens the matrix and the dispersion of the carbon shortens the absorption depth. These two effects are additive and the result is that on the second and subse­quent irradiations material removal from the front sur­face results. This is the physical mechanism for im­pulse generation and detailed studies of the shape of curves such as this allow insight to be gained in the pro­cesses occurring dudng this dynamic process. For the case of the elastic wave (solid curve, Fig. 5), the in­tegral

£:'" Pdt=O

in zero. For the case with material blowoff, this inte­gral is not zero.

VIII. SUMMARY

A simple hydrodynamic theory for the prediction of

stress generation in composite materials due to sudden energy deposition has been presented. Several laser heating experiments of carbon-fiber-PMMA composites were carried out; the results are in accord with the theory if it is assumed that about one-half of the energy initially deposited in the carbon is rapidly transferred to the PMMA matrix. A plausible mechanism for such transfer is given.

*Work supported by the U. S. Atomic Energy Commission. IN. C. Anderholm, Appl. Phys. Letters 16, 113 (1970). 2E.D. Jones, Appl. Phys. Letters 18,33 (1970). 3W.B. Gauster, J. Mech. Phys. Solids 19,137 (1971). 4C. Mark Percival, J. Appl. Phys. 38, 5313 (1967). 5J.C. Bushnell and D.J. McCloskey, J. Appl. Phys. 39, 5541 (1968).

6W.B. GausterandD.H. Habing, Phys. Rev. Letters 18, 1058 (1967).

7Bernard Budiansky, J. Compos. Mater. 4,285 (1970). 8A. S. Vuylsteke, J. Appl. Phys. 34, 1615 (1963). ~.J. Rundle, J. Appl. Phys. 39, 5338 (1968).

l'M.H. KeyandD.A. Preston, J. Phys. E3, 932 (1970). l1Transoptic powder, Buehler Ltd., Evanston, Ill. 60204. 12R. L. Lachance (private communication). 13R.A. Graham, F. W. Neilson, and W.B. Benedick, J. Appl.

Phys. 36, 1775 (1965). 14R. E. Hutchinson (private communication). 15Hadron, Inc., Westbury, N. Y. 11590. 16Dennis B. Hayes and Lynn W. Kennedy (private communica­

tion) • 17C. Kittel, Intr04uction to Solid State Physics (Wiley, New

York, 1966), p. 183.

Temperature Dependence of Intrinsic Carrier Concentration in InSb:

INTRODUCTION

Direct Determination by Helicon Interferometry

K. K. Chen and J. K. Furdyna Department of Physics, Purdue University, Lafayette, Indiana 47907

(Received 29 October 1971)

The phase velocity of helicon waves depends on the carrier concentration. In intrinsic semi­conductors sharp changes in helicon phase can therefore be produced by varying the temper­ature of the material. Basic features of this effect are analyzed for intrinsic semiconduc­tors of InSb and HgTe type. Helicon velocity in these materials is determined almost entire­ly by the electronic plasma alone and, within a limited field and temperature region, is described in a simple analytic form. Experiments performed at 35 GHz on intrinsic InSb between 200 and 350 OK are discussed. Interferograms obtained by sweeping the sample temperature at constant magnetic field provide a fast and accurate method of directly re­cording the temperature dependence of intrinsic carrier concentration nj • The agreement between observed and calculated values of nj is very satisfactory over the investigated temperature range.

Low-frequency electromagnetic waves can, under cer­tain conditions, propagate in conducting solids in the presence of a strong external magnetic field. In un­compensated conductors the propagating waves are cir­cularl:y polarized in the sense of the cyclotron motion of mobile charge carriers and are referred to as heli­con waves. Helicon dispersion in metals and extrinsic

high-mobility semiconductors provides a method for determining the carrier concentration. 1 The method is both highly accurate and does not require the use of electrical contacts. In the case of semiconductors, with which this paper concerns itself, microwave frequen­cies are particularly well suited for helicon transmis­sion experiments.

Dispersion of helicon waves in an extrinsic semiconduc-

J. Appl. Phys., Vol. 43, No.4, April 1972

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1826 K. K. CHEN AND J. K. FURDYNA

w a: ::::> !;i

;;-500 'E

u ".

Q x

200 c z

100 Q ~

50 g: z

FIG. 1. "Mapping" of the field and temperature range where conditions

a: w Q. ~ 200 w

20 u ---~~~~==------------------1 w 10 ~ 5 u

(4) and (5) are satisfied for intrinsic InSb. The boundary curves were cal­culated using published parameter values. The shaded area indicates conditions where Eq. (9) is expected to hold to within 5%. The vertical arrows show the range of the tempera­ture sweep experiments presented in Figs. 6 and 7.

t-

, , , , , ,

2 ~ I z

a: t­z ....

1000~'~~~~~--~~--~4~0~~5~0~~6~0----~70~--~80~--~9no---Tr100 MAGNETIC FIELD (kG)-

tor is conveniently described by the Drude free-carrier model. The helicon wavelength ;\ is determined by the phase constant a= 21T/;\.= (fJ.oewn/B)1/2, where fJ.o is the permeability of vacuum, e the electronic charge, w the frequency, n the charge carrier concentration, and B the external dc magnetic field. The helicon wavelength ;\ will therefore vary with either w or B. In experiments involving propagation in slabs, the variation in ;\ mani­fests itself as Fabry-Perot resonances1• 2 or, in more lossy samples, as interference of the transmitted heli­con wave with a reference signal. 3 Carrier concentra­tion n can be determined from either effect.

Helicon waves can also propagate in intrinsic semicon­ductors, as long as the mobility of one type of carrier greatly exceeds that of the other (as, e. g. , in intrin­sic InSb or HgTe). 4 It is readily shown that the presence of the low-mobility carrier significantly modifies heli­con damping. In a limited range of magnetic field, how­ever, the transmitted phase of the waves is, to a good approximation, still described by the phase constant a'" (fJ.oewn/B)l/2, where now n represents the concentra­tion of the high-mobility carriers only (e. g. , electrons in InSb).

In intrinsic materials, carrier concentration changes sharply with temperature. Then, under conditions when the above form of a applies, helicon wavelength can be varied continuously by sweeping the temperature at a constant magnetic field, resulting in interference phenomena similar to those observed as a function of B or w. This provides a quick and accurate method of directly recording the temperature dependence of the intrinsic concentration. Although the magnetic field is held constant during the temperature sweep, its pres­ence is of course necessary to render the material transparent.

THEORETICAL DETAILS

We consider plane electromagnetic waves propagating in a conducting medium along an external dc magnetic field B taken along the z direction, B = B,i. We seek

J. Appl. Phys., Vol. 43, No.4, April 1972

solutions in the harmonic form

(1)

where E is the wave amplitude, q[=a+i13J is the com­plex propagation constant, and a[ = 21T /:\ J and 13 are its real and imaginary parts, which determine the phase and the damping of the propagating wave, respectively. The wave equation yields two solutions (normal modes) for this geometry5:

q!=(a"+i13,,)2=(w/c)2K,,, (2)

where the subscripts (+) and (- ) refer to two opposite circular polarizations, c is the speed of light, and K +

DIRECTIONAL

~==~C~O~U~PL~E~R~====~~~====~ -i<l-F DETECTOR

FREQUENCY METER

ATTENUATOR

ATTENUATOR

PHASE SHIFTER

KLYSTRON

~~========~==~ DIRECTIONAL COUPLER

SAMPLE ASSEMBLY

ROTATING JOINT

CIRCULAR POLARIZER

CIRCULAR WAVEGUIDE

DEWAR

FIG. 2. Microwave bridge arrangement used for measuring helicon dispersion at 35 GHz. Signal transmitted through the sample beats with the larger reference signal from the by-pass arm, producing a well-defined interference pattern as a function of B or T.

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CARRIER CONCENTRATION IN InSb 1827

..J <I Z l.? Vi 0: g til >­w o

o

InSb 35.0 GHz d ~1.02 mm T~ 321°K

20

Direct Transmission "-".

40 60 MAGNETIC FIELD (kG)

/ I

I I

I I

I I

\ I

Y 1\

I \ I \

I \ I \

I \ I \

\

I

/ /

/

80

FIG. 3. Typical set of interferograms obtained for InSb by sweeping B at constant temperature in the intrinsic range. The overlapping curves are obtained with four consecutive phase settings of the microwave bridge, 900 apart. The truncation of the envelope arises from the presence of holes which, how­ever, do not affect the transmitted phase perceptibly.

and K_ are the complex dielectric constants appropriate for each normal mode, determined by the properties of the medium according to a chosen model.

For a system consisting of ne electrons and nh holes per cm3

, K" can be written in terms of the classical Drude model in the standard form as 5

= K (1 + i w~e + i W~h ) , wlle-i(w±wee ) Wllh-i(w'Fw eh)'

(3)

Here K, is the dielectric constant of the lattice, II is the collision frequency, wp[= (ne2Im*EoK,)1/2] is the plasma frequency, and we[ = eEl m *] is the cyclotron frequency, with e the magnitude of the electronic charge, m * the effective mass, and % the permittivity of free space; the subscripts e and h refer to electrons and holes, and single and double primes indicate real and imaginary parts, respectively. The expressions are in mks units.

At microwave frequencies and in moderate magnetic fields, intrinsic lnSb is described by the following in­equalities:

w!,lw»wee»w, lie (electrons),

W!iW»lIh»W, weh (holes).

(4)

(5)

Inequality in the form of Eq. (4) is often referred to as the helicon limit. Owing to their large effective mass and high scattering rate IIh in the intrinsic temperature range, the holes generally do not satisfy the helicon limit condition below about 100 kG. The region of tem­peratures and fields where conditions (4) and (5) are fulfilled is mapped in Fig. 1 for the case of intrinsic lnSb.

When inequalities (4) and (5) apply, it is readily shown that the real part of the dielectric constant becomes

K' "''I' K,W:e (1 _ W~h) '" 'F~ »/(" "wwce II~ w~ ,,'

(6)

6~--~-----r----~----.----'-----r-----r----'

f 5 z (f)

[54 <.!l lJJ I­z .... 3 >-a: <[ a: ~2 a: <[

InSb (nA"'8.5XIOI4cm-3)

d = 1.02 mm

T=321°K

FIG. 4. Helicon dispersion analysis by a standard" integer" plot. Equal increments of phase obtained from Fig. 3 are plotted vs B_l/2. The slope yields carrier concentration according to Eq. (10).

The lowest-order contribution due to the presence of holes is very small and is neglected in the final form of (6). The value of K~, corresponding to the polarization, which is cyclotron-resonance-active with respect to electrons, is positive. Waves polarized in this sense, i. e. helicon waves, will therefore propagate with rela­tively little attenuation. The real part of the propaga­tion constant for the helicon polarization is then simply

..J <[ Z <.!l ii5 lJJ U Z lJJ a: lJJ lL. a: lJJ I­Z H

300

InSb - Intrinsic Range

35GHz 8=22.1 kG

d =4.32mm

, I I , I I I I I I I

\ f

\ " .J

:1 II 1\ I I I I I I I I I I

i I I I I I , I I I , I

: I I I

I I I I I I I I

I I I I I I I I

\ I I I I I I I \ I

~ : \1 ,I oJ

FIG. 5. Two interferograms observed for intrinsic InSb by sweeping T at a fixed value of B. The overlapping curves are taken at two phase settings 1800 apart. The decrease of trans­mitted amplitude with increasing temperature is a composite effect of increasing concentration and decreasing electron mobility. The sample used was n type at 77°K, with 3x1014

cm-3 donors.

J. Appl. Phys., Vol. 43, No.4, April 1972

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1828 K. K. CHEN AND J. K. FURDYNA

In Sb - Intrinsic Range x Theoretical points ~,t Experimental points at constant T - Experimental data

B=8.85kG

I~~--~~--~~~~~~~--~~--~~~~ 10 210 230 250 270 290 310 330 350 TEMPERATURE (OK) -

FIG. 6. Temperature dependence of intrinsic concentration in InSb measured by helicon interferometry. The black and white circles show values of n obtained at fixed T, as shown in Figs. 3 and 4. The solid curves represent carrier concentrations obtained from temperature sweep experiments using Eq. (9) with n(To) given by the open circles. The crosses give intrinsic concentration calculated using nl = 3. 6rs exp(- O. 26/kT) cm-6. The lower temperature sweep was obtained with the data shown in Fig. 5.

a_=~CK'2+K'r/2.., (ilge;n..r/2, (7)

correct to terms of order we"/,,,,= il"B« 1, where IJ." is the hole mobility and I K I is the magnitude of K. This expression is precisely the expression for a gas of n electrons, obtained as if the holes were not present. Helicon dispersion is thus determined almost entirely by the highly mobile electronic plasma when Eqs. (4) and (5) are satisfied. Under these conditions, the back­ground of relatively sluggish (but not motionless) holes contributes only to wave diSSipation.

The dependence of the concentration n. (henceforth written n, since ,\=n" in the intrinsic range) on temper­ature is obtained as follows: We consider helicon prop­agation through a sample of thickness d, suffiCiently large so that multiple reflections can be ignored. The transmitted phase at any temperature T is then

IT the change of phase ~4> is produced by changing the temperature from To to T while the field B is held cons tant, then

(8)

n1/2(T) _ n1/2(To)= ~ (IJ.o:W y/2[ 4>(T) - 4> (To)]. (9)

The electron concentration n(To) can be determined for some fixed temperature To by standard helicon inter­ferometry' using Eq. (7). With the value n(To) as refer­ence, neT) can then be immediately obtained for other temperatures via Eq. (9) by measuring helicon phase, using the same microwave interferometer. The shaded area in Fig. 1 indicates conditions where Eq. {9} holds to better than ~ accuracy for typical intrinsic InSb

J. Appl. Phys., Vol. 43, No.4, April 1972

parameters (see Appendix).

EXPERIMENTAL RESULTS

Helicon transmission experiments in pure InSb were performed at 35 GHz in temperatures between 200 and 350 OJ(, using a simple interference bridge. 3 The experi­mental arrangement is shown in Fig. 2. Essential fea­tures of the arrangement are as follows: Circularly polarized microwaves propagate in the TEn mode along a circular waveguide, placed axially in a magnetiC field (Faraday geometry), in this case a Bitter solenoid. The sample is located at the center of the magnet, in a simple cryostat arrangement. Sample temperature is controlled by a heater wound around the sample holder. A copper-constantan thermocouple, attached to the side of the holder, is used to monitor the temperature. The transmitted helicon signal interferes with a constant reference signal from the bypass arm of the microwave bridge, thus providing a measure of the dependence of transmitted phase on either B or T. We have also used a much simpler arrangement, conSisting of a 4-in. Varian magnet with axial bore to accomodate the cylin­drical guide, and a Styrofoam sample chamber through which nitrogen gas is circulated at the desired tempera­ture, with very satisfactory results.

Helicon dispersion at a given temperature is determined from a helicon interferogram obtained by sweeping B, shown in Fig. 3 for T= 321 OJ(. The sample used in this case was p type at 77 OJ( (apprOXimately 8.5 X1014 cm-3 acceptors), and was chosen to demonstrate the domi-

InSb (nA"'8.5x 1014cm-3)

x nj (IntrinSiC)} on'" nj - t nA Theoretical

2 Experiment at constant T

- Experimental data

x

x

x

x o

o

lo'~8:;;O~--;20~O---;2;;';20;;---;:2~40;;----;::260~--~28:;;:O:----:300~'----:3;;';2:;::O---:;3~4~O TEMPERATURE (OK)_

FIG. 7. Temperature dependence of electron concentration in InSb having a slight excess of holes (nA"" 8. 5 X 1014 cm-3). The solid curve represents 7'e obtained from a temperature sweep experiment as in Fig. 6. The sample is intrinsic above 250 oK, but the electron concentration drops perceptibly below nj at lower temperatures. The theoretical values of nj and 7'e are indicated by crosses and squares, respectively.

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CARRIER CONCENTRATION IN InSb 1829

nance of electronic plasma even though nh is slightly greater than ne at the lower-temperature limits of the present experiment. The overlapping curves in the figure are traces of interference patterns obtained in consecutive sweeps at four settings of the reference phase 900 apart. The cross-over points provide a very precise means of measuring incremental changes in helicon velocity. The data are analyzed by a standard integer plot of consecutive phase increments vs B-l /2,

as shown in Fig. 4. This yields the concentration n(To) from the slope A, according to the relation3

(10)

Having established n(To), we now sweep the temperature at constant B as shown in Fig. 5. The figure shows two overlapping interference patterns corresponding to two phase settings of the microwave bridge 1800 apart, ob­tained at 22.1 kG by sweeping from 210 to 300 oK. The temperature axis of the X-Y recorder is driven by the calibrated thermocouple attached to the sample. The change in transmitted phase occurring between two tem­peratures T and To is then determined by simply count­ing interference extrema or cross-over points between T and To. Having measured n(To), we can then obtain n(T) in a continuous manner using Eq. (9). It is of prac­tical interest that the rate at which the temperature dependence of n is recorded is determined by the rate at which the sample temperature itself can be varied-in the case of Fig. 5 as little as 5 min for a 100-deg in­crement, with excellent reproducibility.

Temperature dependence of intrinsic concentration de­termined from the data of Fig. 5 and another interfero­gram extending the range to higher temperatures, is shown in Fig. 6. The open circles represent values n(To) determined by a magnetic field sweep at two fixed values of To, relative to which the phase change A<I> is measured. The full circles are additional determina­tions of n at other fixed values of T, to check the accu­racy and self-consistency of the temperature sweep data. The crosses represent theoretical values of intrinsic concentration in InSb, calculated according toB n~ = 3.6 T3 xexp(-0.26/kT) cm- B • Finally, the solid curves show experimental values n(T) obtained from temperature sweep interference patterns using Eq. (9).

Figure 7 displays similar electron concentration data obtained in a single sweep over a very wide temperature range, using the same sample as in Figs. 3 and 4. As indicated earlier, the sample was doped p type, with a nominal acceptor concentration nA =8.5 X10l4 cm-3 • It can be clearly seen from the figure that the measured electron concentration in this sample (solid curve) is perceptibly lower than the calculated values of nj (cross­es) below about 230 OK, owing to the excess of holes. The squares show a theoretical estimate of the electron concentration obtained from the relation nB '" nj - ~ A'

Helicon phase velocity is not affected by the presence of holes except insofar as they affect electron concentra­tion via nenh=n~. Above 250 0 K the behavior of this sam-

pIe is indistinguishable from very pure InSb, as expect­ed for the intrinsic region.

The magnetic fields, and the temperature ranges scanned by the curves in Figs. 6 and 7, are indicated in Fig. 1 by the vertical lines marked A and B, respec­tively. At the low-temperature limit of the data, the experimental range extends beyond the shaded area, i.e. the error in determining A(nl / 2) from A<I> via Eq. (9) theoretically exceeds 5% at temperatures below 220"K because of neglect of the lattice dielectric con­stant K, in Eqs. (6) and (7) (see Appendix). ThiS, how­ever, does not appear to affect the data seriously. Results of similar experiments on other InSb samples show equally good agreement with the calculated elec­tron concentration.

APPENDIX

The simple form of Eq. (9) is obtained by neglecting terms in (IlnB)-2, (llhB)2, and K/WfoB/(ne) in Ct.. The er­ror inherent in this approximation is readily obtained by expanding the complete expression for (}' in terms of the above corrections. As an example, we have indicated by the shaded area of Fig. 1 the region of temperatures and fields where Eq. (9) is expected to hold to within 5%, estimated in this manner for intrinsic InSb.

It is clear from the lower bound of the shaded area that the neglect of K, (i. e. , the requirement w!e» wWce )

seriously restricts the present analysis at lower tem­peratures. This restriction can be readily removed. When K/ is retained, the transmitted phase is given by

<I>=(W/C)d(K, +ne/weoB)l/2. (A1)

If we now consider the change in <I>2 rather than <I> , we obtain

A<I>2 = <I>2(T) - <I>2(To) = (/loewtf /B)[n(T) - n(To)], (A2)

which holds for arbitrary values of K,WEoB/(ne), i.e., above as well as below the boundary line marked w! =wwc in Fig. 1, provided the other conditions in Eqs. (4) and (5) remain fulfilled. Thus the shaded area of Fig. 1 can be extended at lower fields to the dotted line.

Unlike Eq. (9), the above analysis requires the know­ledge of absolute phase <I>(To) rather than A<I> only. This is calculated simply from n(To), which must be deter­mined in any event. Subsequently, A<I> is measured interferometrically as before, yielding the absolute phase <I>(T) for use in Eq. (A2).

*Work supported in part by the U.S. Army Research Office, Durham.

1See , e.g., R. Bowers, Plasma Effects in Solids (Dunod, Paris, 1965), p. 19.

2A. Libchaber and R. Veilex, Phys. Rev. 127, 774 (1962). 3J.K. Furdyna, Rev. Sci. Instr. 37, 462 (1966). 4J.K. Furdyna, Phys. Rev. Letters 14, 635 (1965). 5See , e.g., E.D. PalikandJ.K. Furdyna, Repts. Progr. Phys. 33, 1193 (1970).

6T.S. Moss, Progress in Semiconductors, edited by A. F. Gibson (Wiley, New York, 1960), Vol. 5, pp. 210-211.

J. Appl. Phys., Vol. 43, No.4, April 1972

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