Carrier mobility and field effect in thin indium antimode films

Thin Solid Films - Elsevier Sequoia

CARRIER MOBILITY ANTIMONIDE FILMS

C. H. LING,

AND

FIELD

- Printed

EFFECT

267

in Switzerland

IN THIN

INDIUM

J. H. FTSHER AND J. C. ANDERSON

Materials Section, Department (Received

S.A., Lausanne

ofElectrical Engineering, Imperial College, London, S. W.7 (Gt. Britain)

June 28, 1972)

Thin polycrystalline InSb films in the thickness range 1000-3000 A, have technique. A linear relation is found to been prepared by the flash evaporation exist between the electron mobility and the mean crystallite size. A positive gate field, applied orthogonally to the film, increases the electron mobility by a factor of 30, at 77 “K. It is proposed that a potential barrier exists at the grain boundaries, which dominates the scattering process. This model could also explain the temperature dependence of the electron mobility. Typical potential barriers are found to be -0.04 eV.

INTRODUCTION

In thin, polycrystalline films of semiconductor materials a variety of scattering mechanisms exist and limit the carrier mobility’. This paper reports a thorough study of polycrystalline indium antimonide films, from which the dominating mobility-limiting scattering mechanism is found to be due to potential barriers at the inter-crystalline boundaries. Measurements have been made of Hall constant and of d.c. field-effect conductivity variations on both 12 and p-type films. SPECIMEN

PREPARATION

Flash evaporation was employed using zone-refined, polycrystalline, n-type InSb, having a net donor impurity, Nd-N,1 x 1014 cm-j and a mobility, p - 1 x lo5 cm2/V set at 77 “K. The finely crushed powder was loaded onto a V-shaped chute, which could be mechanically agitated from outside the vacuum chamber. The source temperature was kept at about 1500 “C, as determined by an optical pyrometer. Prior to evaporation, the source was baked for a few minutes to expel any absorbed gases. The deposition rate was difficult to control. However, previous work 2*3 showed that this was of no consequence, since the “goodness” of films depends, to a large extent, on the post-deposition thermal treatment. An evaporation time of about 1 min was needed to produce a 3000 A film. The substrate temperature was kept around 250”-300 “C. A vacuum pressure of -2 x lo6 torr was used and was found not to be a critical parameter. Thin Solid Films, I4 (1972) 267-288

C. H. LING,

268

J. H. FISHER,

J. C. ANDERSON

A post-deposition annealing is essential, and was achieved, in this case, by first letting the system up to air, while the substrate was still hot, so that a few mono-layers of protective oxide were formed, and then baking it to as high a temperature as possible. In this case, 400 “C was about the limit, if appreciable loss of Sb was to be prevented. The lower temperature could be compensated by a longer annealing time. For annealing beyond about three hours. only marginal improvement in the mobility was obtained. p-type films were prepared in essentially the same manner by flash evaporation on to a substrate at about 380 “C, from a mixture of In + 3Sb + 7 x 10e4 Cu4. The excess Sb was to compensate for the increased re-evaporation of Sb at this higher substrate temperature. Often it was found necessary to evaporate a very thin “ seeding ” layer first, at a lower temperature of - 250 “C. No annealing was found necessary. Film thickness was determined by measuring the height of the step produced by the film on the substrate, using an interferometer. To study the surface features of these films, a carbon replica was made, and was shadowed by carbon-platinum about 200-300 8, thick by evaporation of a mixture of carbon and platinum from an oblique angle. Figure 1 shows a shadowed carbon replica. The lack of surface topography is evident, the mountains and valleys being typically 50 8, in height.

Fig.

1. Carbon

replica.

( x 80 000)

FILM STRUCTURE

Samples for film structure microscopy, and it was therefore Thin Solid Films, 14 (1972) 267-288

study were usually too thick for transmission necessary to thin them down to less than about

CARRIER

MOBILITY

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269

1000 A, by prolonged etching. Figure 2 is a micrograph of an unannealed film; the average crystallite size is not more than N 100 A. The ring diffraction pattern is typical. Figure 3 is a micrograph of a well annealed film, clearly demonstrating

Fig. 2. Micrograph ( x 140 000)

of

an

unannealed

film.

Fig. 3. Micrograph ( x 70 000)

of a well-annealed

film.

the polycrystalline nature. Selected area diffraction shows that the individual crystallites are single crystals which grow with the (111) planes parallel to the substrates. The corresponding diffraction pattern over the entire area of the film, shown in the figure, is indicative of the fact that these crystallites are well oriented about the (111) axis, and not, however, of a single-crystal film. Evidence of a certain amount of twinning is suggested by the presence of satellite spots around the main (bright) spots. From the size of the spots, an upper limit of the angle of misorientation between crystallites can be estimated. This is found to be about 10”. It is fairly obvious from the micrographs that these films contain a high density of defects. Dislocation lines and stacking faults are clearly discernible within individual crystallites. Films grown on mica substrates, particularly at higher substrate temperatures and after prolonged annealing, reveal a certain amount of epitaxial growth. The crystallites tend to have triangular or hexagonal grain boundaries, having straight edges. This is absent on amorphous glass substrates, which lack the hexagonal symmetry of mica. The crystallite size is an important parameter, being a good indication of the film mobility. Because of the distribution in the size, the average must be taken. Using a method due to Williamson5, a series of random lines is drawn on the micrograph. From the intersections with the grain boundaries the crystallite size is obtained from the relation, d = EL/EN, where EL is the total length of the lines and CN, the total number of intersections. To eliminate any bias, Thin Solid Films, 14 (1972) 267-288

270

C. H. LING,

J. H. FISHER,

J. C. ANDERSON

inevitable in this method, perpendicular lines were drawn, 1 cm apart, across the entire micrograph and a distribution graph was obtained. The total length taken on an 8 in. x 10 in. micrograph was about 1 m. The mean of the distribution curve was taken to be the average crystallite size. Figure 4 shows such a distribution for a film whose thickness is N 1000 A. The curve peaks around 700 A, whereas the mean size is about 850 A. It can be thickness

loo0

0

Fig. 4. Crystallite

size distribution.

d

1000

A

2000

ta)

seen that the distribution has a considerable tail and there are few crystallites with dimensions less than about 200 A. A second curve, obtained in a similar manner but taking into account all visible defects (as indicated by changes in intensity on the micrograph), is also shown in Fig. 4. This curve shows a greater asymmetry and peaks at a lower size, about 300 A. This parameter is a rough measure of the total density of defects present in these films. Figure 5 is a plot of the mean crystallite size as a function of film thickness. There is considerable scatter, but, generally, the mean crystallite size of a well-annealed film is approximately equal to its thickness. There is a better correlation between mobility and mean crystallite size, as shown in Fig. 6. The scatter, in this case, is greatly reduced. CONTACTS

Six pairs of contacts of Ag, Al, Au, In, Sn, and Sb on both n- and p-type InSb films were studied on the Tektronix curve tracer (type 575). The metal contacts, of approximate dimensions 0.1 cm x 0.1 cm, were vacuum-deposited side by side on a single InSb film, through wire masks. The semiconductor gaps were wide. Initial measurements showed wide variation in the sample resistance, Thin Solid Films, 14 (1972) 267-288

CARRIER

MOBILITY

AND

FIELD EFFECT

40001

InSb

IN

271

FILMS

1

8, 0

d

01 0

d

I

I

I

1

2

3cx30A

I

I

I

0

1

2

t

Fig. 5. Crystallite

size as a function

I

I J

30008

of film thickness.

4000 cm2/Vsa

0

3-

0

0

p2-

a

0

0 l-

0 /I!,:

0 0

1

2

3000A

t

Fig. 6. Mobility

as a function

of crystallite

size.

as shown in Table I. This can only be attributed to the different contact metals, since the resistivity does not vary over the film by more than about 10%. Figure 7 shows a set of I-V characteristics for an n-type film, at 77 “K. Only In, Sn and Ag exhibit ohmic behaviour, Sb being the most non-ohmic in this case. The behaviour at higher temperatures is essentially similar. For P-type films, none of the metals studied prove to be satisfactory. The Z-V characteristics are shown in Fig. 8. Sb which forms the most non-ohmic contact in n-type films now appears to make the best contact. HALL

EFFECT

MEASUREMENTS

Figure 9 shows the physical shape of the Hall sample used, having a width, w = 0.1 cm and a length between contacts 2 and 3, I= 0.3 cm. The length to width ratio, I/w, must be greater than about 2.5 if the shorting effect (of the Hall voltage) is to be negligible. In or In/Bi alloy (the latter having a melting point of 75 “C for a 33 % Bi content) was soldered on to the five contacts. A d.c. current, typically 100 PA, Thin Solid Films, 14 (1972) 267-288

212

C. H. LING,

TABLE

I

CONTACT

RESISTANCE

AT 77

Contact metal

“K

(MEASURING

Resistance lkQl

Ag

13 150 210 10 11 250

Al Au In Sn Sb

Fig. 7. I-V characteristics contacts--n-type sample; Sn, Al. Ag, Au, Sb.

configuration

Thin Solid Films, 14 (1972) 267-288

J. C. ANDERSON

1.5 V)

Resistance

(p-type)

(kQl 48 46 24 38 25 24

of metal-InSb-metal Top (left to right): In,

u

4 Fig. 9. Sample

(n-type)

VOLTAGE

3. H. FISHER,

Fig. 8. I-V characteristics contacts-p-type sample; Au, Sn, In. Al, Ag.

of metal-InSb-metal Top (left to right): Sb,

CARRIER

MOBILITY

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IN

InSb FILMS

273

was passed between contacts 1 and 4 of the sample; its value was accurately determined by measuring the voltage drop across a standard 100 Q resistance, in series with the sample. The conductivity voltage, Vz3, between contacts 2 and 3, was measured on a Solartron digital voltmeter which had an input impedance varying between 10 MQ to greater than 5000 MQ, depending on the voltage range. Such high input impedance is necessary, since the resistance measured at 2 and 3 was usually a few hundred kQ. The resistance could easily increase to a few MS2 at 77 “K, particularly for thinner samples, when correction to the voltmeter reading became necessary for measurements made on the lower input impedance range. The sample was mounted inside a brass tube provided with external leads and placed inside a Thermos flask, between the pole-pieces (6 cm gap) of a permanent magnet. The magnetic field was 2 kG. To measure the Hall voltage, Vn, between contacts 3 and 5, the sample was rotated in the magnetic field and the maximum and minimum meter readings were noted. This was done to eliminate the standing voltage (due to electrodes 3 and 5 not aligned directly opposite one another), which could be 10 times as high as the actual Hall voltage. The difference between the two readings is twice the Hall voltage. No correction was necessary as the measurements were made on the highest impedance range. The Thermos flask could be filled with liquid nitrogen, thus enabling the temperature variation to be made continuously down to 77 “K. A brass heating tube was also available for measurements up to 150 “C (just below the melting point of In solder). Magnetic field-dependence experiments were carried out on a variable gap electro-magnet, capable of a maximum field of 20 kG. A large number of Hall samples were prepared, under various evaporating conditions, the results of some of them being shown in Table II. It can be seen that ,L+,and n do not depend critically on the pressure (for pressure < 2 x lo- 5 torr) and the substrate temperature, T,. Rather, these parameters are critically dependent on the post-evaporation annealing temperature, T,. Samples annealed below 360 “C are generally poor in mobility and have high carrier concentration - 2 x 101’ cm- 3 at 296 “K. Annealing at about 420 “C and using a somewhat elevated K (in the nF series) produce much better films, with correspondingly larger crystallites. In this case, the carrier concentrations are generally lower -5 x lOi cme3 at 296°K. The highest mobility obtained was 4600 cm2/V set in a sample of thickness 1880 A (nF18). This sample was annealed for 14 hours at 390 “C and followed by slow cooling. In polycrystalline films, Hall effect measurements give the carrier concentration in the crystallites. For films of the thickness range investigated (lOOO3000 A), the space charge region at either surface represents a significant portion of the film’s total thickness. This space charge is always in depletion, the surface band bending being roughly two-thirds up the band-gap. Hence, the Hall effect does not give a true measure of the carrier density within the crystallites. A correction, based on the simple Schottky solution of the Poisson equation, is used. We assume a depletion region completely devoid of carriers and having a uniform distribution of ionised impurities, ND and a depletion depth, z,. Assuming the film to be depleted to the same extent at both the mica/InSb and the free surface, Thin Solid Films, 14 (1972) 267-288

C. H. LING,

274 TABLE

J. H. FISHER,

J. C. ANDERSON

II

Sample

~~(296, 77 “K) fern’/ V set)

n(296, 77 “K) (1O’6 cme3)

T,J’CI

TJ’Ci

Press. (tori- I 1 lO-h 2 3 2 2 2 2 3

nA5 nA12 nA19 nA20 nA26 nA33 nAl4 nA78

2000 3000 500 1000 700 3000 3000 2500

300 960 440 600 600 1170(200) 1400(1100) 1600( 1400)

31 4.5 15 18 20 7 (3.0)

200 220 200 230 230 230 230 230

300 360 360 360 350 370 360 360

nB4 nB5 nB34

1800 2250 1150

1520 3200(80) 1200

5.2 3.3(2.3) 5.6

260 250 260

380 400 400

2

nC24”

3000

2860(2100)

21(18)

360

400

10

nF8 nF9 nFl1 nF15 nF16 nF18b nF20 nF27C

1100 1600 1380 1500 2000 1880 2100 1500

1200(120) 1700 1160 1580(80) 1300(160) 4600(90) 2560(360) 3010(830)

6.8(2.0) 5.2 4.9 4.9(1.1) 8.9(5.3) 4.0(3.5) 5.2(3.X) 9.7(7.0)

300 310 320 290 290 310 300 290

420 410 420 430 430 390 430 500

x 15 6 6 12 2 4 6

Jd

3-5000

6p 15000 ( l-4000) (105)

(10’6-10”) (2 x 10’4)

bulk’ ’ b ’ d e

Evaporated from a 1:3 In/Sb mixture, at a higher Annealed for 14 hours, followed by slow cooling. Coated with SiO. From ref. (2). Polycrystalline InSb, from R.R.E.

Note-Figures

in brackets

the corrected n cm

=

carrier

are for liquid nitrogen

density

is calculated

substrate

temperature

of about

3

380 “C

temperature.

as

%wJl(t - 2z,)

and t is the film thickness. For where nmea is the measured Hall concentration N,-2x 1017 cmP3, 22,~ 800 A, hence this correction becomes of doubtful validity when the thickness is less than about 1000 A. For films less than about 800 A, the space charge regions from either surface overlap, thus depressing the Fermi level further into the band-gap. In the absence of overlapping, the Fermi level in the bulk crystallite is determined solely by the impurity concentration. This no longer holds for thinner films, which are therefore expected to exhibit lower carrier density. Moreover. it is not possible to extract any bulk properties from measurements on such a film. The temperature dependences of mobility of some typical n-type films are shown in Fig. 10, the general behaviour of these curves being in agreement with is also included in the figure for the previous reports ‘3 ‘3 7. The bulk behaviour Thin Solid Films, 14 (1972) 267-288

CARRIER

MOBILITY

AND FIELD EFFECT

IN

InSb FILMS

275

‘nF15

-I

.5 2

4

Fig. 10. Temperature

6

variation

8

10

of Hall mobility

12

14

in n-type films

purpose of comparison. This measurement was made sions 0.5 x 0.5 x 5 mm3 and with 5 grain boundaries traversing the entire cross-section of the rectangular using a measuring field of 0.1 V/cm. The apparent observed mobility of 3 x lo4 cm’/V set at 77 “K - lo5 cm’/V set, could be explained by the fact that used was sufficiently high to produce a reduction in of 3 (ref. 8). IMPURITY

on a sample having dimen(as revealed by CP4 etch) slab, and was carried out discrepancy between the and the quoted figure of the magnetic field of 2 kG mobility of about a factor

CONCENTRATION

The variation of electron concentration (l/Rue) after correcting for the space charge layers at the surfaces, with temperature is shown in Fig. 11. After correction, most of the films have carrier concentration in the 10” cm- 3 range. It is seen that below about 200”K, the electron concentration is relatively insensitive to temperature, being equal to the excess donor impurities (i.e. Nd - NJ, Thin Solid Films, 14 (1972) 267-288

C. H. LING,

2

I

I

I

I

I

I

4

6

8

10

12

14

Fig. 11. Temperature

variation

of electron

concentration

.I. H. FISHER,

J. C. ANDERSON

in n-type films.

which do not have any detectable ionization energy in this temperature Above 300 “K, all samples show intrinsic carrier concentrations. HALL

range’.

MOBILITY

Mobilities observed in the films are all in the range of 1000-4000 cm2/V set at room temperature. Compared to the bulk figure of 3 x lo4 cm’/V set at 300 “K and for the same impurity concentration, this represents a reduction of some lo-30 times. At 77 “K, the reduction is even higher, by as much as a factor of 103, since mobility increases with decreasing T for bulk and decreases with T for thin films. A temperature dependence of the form, ,um T’” is applicable, m being approximately - 1.6 for bulk (for T> 200 “K) and O.W.7 for thin films (for T- loo”-300 “K). A mobility peak has been observed in the vicinity of 350 “K for the higher carrier density films and around 450°K for lower carrier density films. The decrease,at temperatures higher than these, marks the onset of bulk dominated behaviour and might suggest that, at these temperatures, films behave in much the same way as bulk. This perhaps is the case, since optical phonon scattering and to a lesser extent, inter-carrier scattering dictate carrier transport in bulk InSb at high temperatures, and are also expected to be the most significant Thin Solid Films, 14 (1972) 267-288

CARRKR

MOBILITY

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FIELD EFFECT

IN

InSb FILMS

277

mobility-limiting mechanisms in films. At lower temperatures, the mobility reduction must be interpreted in the light of thin film defects. We now consider those scattering processes peculiar to thin films, bearing in mind that the background scattering (due to bulk) is always present. Surface

scattering

Generally, mobility decreases with decreasing thickness, i.e. with decreasing crystallite size. This is compatible with the surface scattering theory of Fuchs and Sondheimer”, ” and of Schrieffer12 : the largest surface effect (geometry and surface potential) is observed in crystals with dimensions comparable to the carrier mean free path. Many et aLI3 show that the reduced average mobility, ji, is approximately given by

(1) where pt, is bulk mobility. For a film with thickness, t = 1000 A, the typical observed mobility is 1000 cm2/V set at room temperature. Taking an appropriate value of p,, - 3 x IO4 cm2/V set, the mean free path, A, is found to be approximately 2200 Al’. This gives a reduced mobility of 6000 cm2/V set for the film, which is 6 times too high. Clearly surface scattering cannot be the dominant mobility-limiting mechanism. Indeed, Schrieffer’s theory predicts significant reduction only for surfaces that are in strong accumulation, as, for example, in the case of CdS films on SiO I4 . The depleted surfaces in InSb films tend to repel electrons and prevent them from reaching the surfaces. Dislocation

scattering

The sphalerite structure of InSb can be visualised as consisting of a series of (111) planes of In and Sb atoms, with alternate large and small spacings between them. Slip dislocations occur most readily between the widely spaced pairs of (111) planes, since fewer bonds are broken in this case. When such a dislocation occurs, an extra half-plane of atoms (In or Sb) is introduced into the lattice. The atoms at the edge of this half-plane would consequently have free or dangling bondsr5,r6. Screw dislocations, however, do not lead to dangling bonds and are unimportant here7. The dangling bonds can be considered to be acceptors capable of capturing electrons from the conduction band. Thus the dislocation acts as a negatively charged line, with a cylindrical space charge of positively charged ions around it. If the dislocation lines are perpendicular to the current flow, then since the conduction electrons have to follow curved paths that wind around them, a reduction in electron mobility is to be expected. This, however, is not the case if the charged cylinders are parallel to the current flow. The mobility of minority carriers can actually increase in the presence of dislocationsr7. If the space charge cylinders are parallel to the current flow, then the minority carriers are constrained to move in the vicinity of the dislocation line with an enhanced mobility. Thin Solid Films, I4 (1972) 267-288

278

C. H. LING,

J. H. FISHER,

J. C. ANDERSON

The observed mobility is found to be higher than the calculated dislocation mobility by some 2-3 orders of magnitude, assuming a dislocation density of lo6 cm-‘. It therefore appears that the contribution from dislocations to mobility reduction is small. However, prolonged annealing of films followed by slow cooling usually improves the mobility. This would give some reduction in dislocation density (and also an increase in crystallite size), and is qualitative evidence for some dislocation scattering. Grain boundary scattering Grain (crystallite) boundaries are regions of lattice mismatch and affect carrier transport in two ways: (1) they scatter carriers, limiting the mean free paths to approximately half the mean crystallite size, and (2) the presence of dangling bonds traps carriers, giving rise to a potential barrier. Grain boundary scattering leads to a mobility, which decreases with increasing temperature as in bulk, in contradiction with the experimental results. POTENTIAL

BARRIER

SCATTERING

For a potential drop across each barrier due to the applied small compared with the barrier height, simple diode theory conductivity:

field, which is gives for the

where

;0

= clarity of carriers in crystallite, = b arrier height at grain boundary,

4v”Ll PO = N,kT

(4)

17= mean thermal velocity of the carriers, L, = length in which there are Nr crystallites, p. is, in fact, approximately equal to the value of mobility calculated for purely grain boundary scattering. The above equations are obtained assuming parabolic energy bands and Boltzmann statistics and for InSb this is not valid. However, similar expressions can be used so long as the potential barriers are high. Equation (3) can be written as: g=

nbqpO

(5)

where ~2~is the density of carriers with energies above the barrier height. Thus the carriers move with their bulk plus boundary-scattering-limited mobility but only ones whose energy is higher than the barrier height contribute towards the measured conductivity. In InSb it can be shown, by using the correct statistics and taking into account the non-parabolic bands, that when n, is high Thin Solid Films, 14 (1972) 267-288

MOBILITY ANO FIELD EFFECT m

CARRIER

Grain

T---l

e!ifo,

279

boundary

ly2states EC

‘i ---__-

InSb FILMS

------L-v-___-----

5

Fig. 12. Grain boundary potential barrier: (1) No bias, (2) With bias, V.

enough to remain approximately constant as the temperature is varied from 300 to 77 “K and n,s2 x lO”j, the temperature dependence can still be expressed as:

where 4 differs from the barrier height 4,, given by eqn. (2) but is proportional to it. Substituting eqn. (6) in eqn. (5) we have

( e#>

c = wwo exp -

k~

from which the measured mobility, using Hall effect and conductivity data, will be given by

( 4>

p=poexp -p

(7)

and 4 is the measured barrier height. This predicts the functional dependence experimentally observed between p and T. Calculation of ,uo for 2,000 a diameter islands and including bulk and boundary scattering for Nd = 1 x 10’ 7 crne3 gives: p. (300 “K) = 1.Ol x lo4 cm’/V set p. (77 “K) + 2.25 x lo4 cm’/V set taking eb to be equal to 0.04 eV, eqn. (7) gives p,, (3OO”K)==2144 cm2/V set Thin Solid Films, 14 (1972) 267-288

280

C. H. LING.

2

3

4

5

6

7

8

J. H. FISHER,

J. C. ANDERSON

9

z+‘) Fig.

13. Temperature

pLb (77

variation

of R, for p-type

films.

“K) == 55 cm2/V set

These are typical mobilities observed at the two temperatures. P-TYPE

FILMS-HALL

COEFFICIENT

Over 10 p-type samples were prepared from a 1:3 In/Sb mixture doped with Cu. Figure 13 shows the variation of Hall coefficient with temperature. At high temperatures, the conduction process is n-type because of the higher mobility of electrons, though the hole population exceeds the electron population. But at lower temperatures, the conduction is p-type, i.e. hole-dominated since now p>>n. R, goes through zero when the contributions due to the two species of carriers cancel out, and goes through a minimum at RHcminjwhen p = nb, where Thin Solid Films, 14 (1972) 267-288

MOBILITY

CARRIER

AND FIELD EFFECT

InSb FILMS

IN

281

b is the mobility ratio ,LLJ~~. In the low temperature region below 150 “K the Hall coefficient does not vary with T and has the value R,, ex. It can be shown that

tb- l)’ IRH =--_ mini

R H,

(8)

4b

ex

from which b is obtained. R, min varies with magnetic field B and the value used is the experimental value obtained by extrapolating to B = 0. The results of Hall measurements are shown in Table III for some p-type films. The mobility ratio of films in the thickness range 1000-2000 w is therefore around 10. For thicker films in the pm region, Potter and Wieder” obtain a higher mobility ratio of approximately 32. The bulk mobility ratio for similar impurity concentration is about 60 at room temperature. TABLE

III

Sample

PC5 pc9.3 PC10 PC11 PC15 pc39.1 The mobility

(A)

thickness

pn (cm’/V

1000 1500 1200 1500 1700 1500

570 2500 2640 4500 1450 3000

SK)

values were measured

FIELD EFFECT

p”p

b

N,-N, (lOI cm-3)

40 250 180 510 170 300

13 10 14 8.8 8 15

10 8 8 6 15 9

at temperatures

corresponding

to the Hall minima

MEASUREMENTS

A critical test of the potential barrier model is a field-effect experiment. If an electric field is applied perpendicular to the plane of the film from a gate electrode, separated from it by an insulating layer, the density of carriers in the film can be enhanced or depleted. For an n-type film a positive gate voltage will induce a number, An of additional carriers per unit area of film, assumed to be equal in the crystallites and the grain boundary regions. The barrier potential can now be written as (9) where L, is the mean space-charge depth, as defined by Many et a1.13. This leads to a gate-voltage dependent mobility given by18

HV,) =

PO exp

(-40 j G(S) -

kT

where WV,) = l+(~z-B1>~,-Pl(P,-~l>V,Z+.... Thin Solid Films, I4 (1972) 267-288

(11)

282

C. H. LING,

J. H. FTSHER,

J. C. ANDERSON

whereP1,2 = b, 2CgIeLc,,rc. C, is the capacitance of the gate insulator per unit area and 8 is the fraction of total charge induced that is not trapped in the insulator and in the surface states. 2

EXPERIMENTAL

DETAILS

The field-effect samples were prepared by first cleaving the mica substrate down to about 20 pm thick, followed by a deposition of a layer of Al to form the gate, on the opposite side of the substrate to that on which the InSb Hall sample was previously deposited. The mica substrate, therefore, also served as the gate insulator. The gate voltage supply was capable of giving a maximum voltage of f 3 kV. The maximum gate voltage that could be applied is limited by the breakdown field strength of mica, which is around lo6 V/cm. For a mica thickness of - 20 pm, this implies a maximum gate voltage of - 2 kV. If a surface trap density of - IO” cmm2 can be assumed, then the minimum gate voltage needed to produce any observable change in the Hall measurement would be -0.25 kV. The Hall measurements were carried out at room and liquid nitrogen temperatures. in the presence of a constant magnetic field of 2 kG and a variable gate voltage. All the Hall samples were n-type; measurements of p-type samples were not possible in the present study. because of the very high impedance of these samples at 77 “K. Because of trapping of induced carriers at slow surface states, all measurements were made after the gate voltage was applied for about 30 mins, when the slow states were almost completely filled. THE

BARRIER

POTENTIAL

Figure 14 is a plot of Hall mobility against l/T for different gate voltages (only 3 values of V, are shown) for sample nF20.3. The sample is 2100 A thick, having ,M(Vg = 0) = 2530 cm’/V set and n( V, = 0) = 5 x 1Ol6 cm- 3, at 296 “K. The exponential temperature variation for T> 200 “K, typical of polycrystalline films, is strong evidence supporting the barrier model. From the linear portions of p(l/T) curves, the barrier potential, 4, is deduced, using eqn. (7). The variation of 4 with sg (gate field across mica = gate voltage/mica thickness), is also shown in the figure. For a positive gate field of 7.5 x lo5 V/cm, the barrier potential decreases by 12 % from its zero gate field value of - 0.04 eV. However. for negative gate fields, 4 shows a slight increase. If eqn. (2) is assumed to apply to holes as well, then a negative gate field should also bring about a lowering of c$, which has not been observed. For InSb, it has been shown by Lile3, that, even in the absence of an applied gate field, the surface is naturally depleted by the action of surface states to the point where the surface is quasi-intrinsic, with the Fermi level approximately twothirds down the band gap. A positive gate voltage reduces the surface band bending, thereby extending the space charge deeper into the bulk, i.e. resulting in an increase in the space charge penetration depth, L,. A negative gate voltage depletes the surface even further, drastically reducing L,, so that any modulation Thin Solid Films, 14 (1972) 267-288

CARRIER

AND FIELD EFFECT

MOBILI~

IN

InSb mm

283

effect due to holes would be confined to a very thin layer near the surface, the bulk conduction being still due to electrons. The negligible hole contribution to conduction could therefore account for the relative independence of C$ on negative V,. The deviation from linearity for T< 200 “K, might indicate that the barriers are now no longer the dominating mechanism in determining the effective mobility. In bulk InSb, polar optical scattering dominates from high temperatures down to 200 ‘K9, and this would cause mobility to increase with l/T. But the main scattering mechanism at temperatures in the vicinity of 100 “K is by ionised impurities 21, for which mobility varies as T312 and would actually decrease with the increase in I/T. It seems unlikely, therefore, that the deviations observed can be accounted for in these terms. An alternative explanation is possible. In InSb, the donor ionisation energy is not detectable over the temperature range investigated. It cannot, however, be assumed that the electron density, n,, is equal to the impurity concentration, Nd, since in a low band-gap material such as InSb, the Fermi energy could rise above the conduction band edge for relatively low N,. Due to the markedly non-parabolic nature of the conduction band in InSb, the density of states in nF

20.

3

36

0.645

eb (eV) 1

--i‘:i

34

~

i zG

-5 0.035

0

Eg(x105

32

A Eg = -7.5

\’

\,J\ \ \ \\

‘,

30

0 \p

4, \\ \

‘a,

3

0 \

\A ‘\

2.8

5

v/cm)

0

=O

0

..7.5

x 10~V/cm

1 \

1

I

I

I

I

I

I

4

5

6

7

8

9

10

Fig. 14. Temperature of gate field. Thin Solid Films, 14

variation

of Hall mobility

(1972)267-288

for different

gate fields: potential

barrier

as a function

284

C. H. LING,

J. H. F’ISHER, J. C. ANDERSON

the vicinity of the band edge is low. Thus, as the temperature is lowered, the Fermi level moves into the conduction band and some de-ionisation of donors occurs, leading to a reduction in n,. 4 is therefore expected to vary with T through n,. Using Fermi-Dirac statistics for the carriers, a non-parabolic band and partial ionisation of impurities, appropriate to InSb, Ef and n, have been numericallyevaluatedi8. ForN, = 5 x 10’6cm-3, itwasfound that Ef = -0.41 kT (with respect to the conduction band edge) and n, = 2.26 x 1016 cm- 3 at 300 “K, and Ef = +0.66 kT and n, = 1 x 1016 cmm3 at 100°K. The decrease of n, with diminishing Ttherefore decreases qf~provided that n2 varies little with temperature. HALL

COEFFICIENT

AND MOBILITY

T = 77°K The variation of Hall mobility and coefficient at 77 “K is shown in Fig. 15 for three typical n-type listed in Table IV. TABLE

IV

SW?lple

thickness

nF5.3 nF16.3 nF18.3 * Figures

as a function of gate field samples, having parameters

in brackets

iRi

carrier cow. of 77 “K (1CP6 cmm3)

mobility (77°K) icm’/ V see)

2250 2200 1880

2.33 (6)” 6.0 (14)* 3.36 (lo)*

78 200 90

are the (approximate)

corrected

carrier

concentration

The application of a negative gate field depresses the Fermi level at the surface towards the valence band edge and minority carriers (holes) are generated. These increase in number as the negative gate field increases. When the film is relatively thick and the carrier density, n,, in the volume of the film is fairly large, the minority hole generation does not significantly affect n 1. This accounts for the relative independence of R, on negative l?, for sample nF16.3. Likewise. the mobility varies little with negative Vg. For sample nF5.3, which has a low carrier density, R, falls by a factor of 2.6 over a gate field range of - 7.0 x lo5 V/cm. The induced holes contribute significantly to the conduction process in this case and R, should decrease due to the increased importance of bipolar conduction. The Hall coefficient for both the samples decreases with positive gate field, indicating an increased electron concentration. Sample nF18.3 shows a very interesting behaviour. The film actually becomes p-type for negative gate fields, and this occurs at a rather low value of V,. This is often observed in thinner films in the 1000 8, region. Because they are heavily depleted on either surface, only a low density of holes is needed to cause the entire film to become p-type. The conduction is then predominantly due to holes. As the Fermi level shifts towards the valence band edge, detrapping of charges at dangling bonds within the barrier occurs, leading to a reduction in 4. Thin Solid Films, 14 (1972) 267-288

CARRIER

MOBILITY

AND

FIELD EFFECT IN

285

InSb FILMS

1, :mYCouI 1 00

.P\

30 9 _.+._.-.a-‘-

./.p

( \i 11

Rti positive-

!

>

2 1) T-77 20-

'K

o nF5.3 A nF16.3 q nF18.3

10 -

OL -8

-6

-4

-2

0

4

6

0

&s ( lo5 Fig. 15. Variation

of Hall constant

and mobility

with gate field at 77 “K.

Hole mobility therefore increases and saturates when C$becomes negligible (for increasing negative gate field). For positive gate fields, the linear relation as predicted by eqn. (11) for small K’s, is clearly demonstrated for samples nF5.3 and nF18.3. An increase by a factor of 30 is seen. Such a large increase is usually associated with low carrier density films, in view of the fact that a higher change in carrier density can be induced. By the same reasoning, the mobility of sample nF16.3, which has a higher carrier density, does not change much from its zero gate field value. Breakdown in the mica insulator limits the gate field to about 8 x lo5 V/cm, when deviation from linearity begins to appear, due to the effects of higher order terms in eqn. (11). T = 296°K In Fig. 16, Hall mobility and coefficient for the following samples, in Table V. Thin Solid Films, 14 (1972) 267-288

variation

for T = 296 “K is shown

286 TABLE

C. H. LING,

J. H. FISHER,

J. C. ANDERSON

V

SUWlpk

Thickness (Al

Carrier cow. at 296 ‘K (IO’h c?ic3)

Mobility at 296 “K (cn?/ v xc)

nF14A.3 nF20.3 nF22.3 nF21.3

1500 2100 800 1500

4.45 5.1 5.6 9.9

1880 2530 460 3200

*Figures in brackets concentration.

(15)” (13)” (24)* (20)X

are the (approximate)

corrected

carrier

180

140 R, (cm3/Ccul) 100

60

T= 296’K

1.04

B nF 14A.3

102

x nF20.3 q nF22.3 o nF 27.3

Jl(V,) P(O)

100

0 98

096

0.94

0.92 1

I

-15

-10

I

-5

0 &s ( d

Fig. 16. Variation

of Hall constant

V/cm

and mobility

I

I

I

5

IO

15

I

) with gate field at 296 “K

the Generally, there is little change in R,. since at this higher temperature, electron concentration is much higher. The number of induced carriers (after trapping at surface states and dislocation dangling sites) is significant only in the barrier region, being small compared to the number of carriers ~1~already Thin Solid Films, 14 (1972) 267-288

CARRIER MOBILITY AND FIELD EFFECT IN InSb

287

FILMS

present in the crystallite. ~zi is therefore relatively constant, while n2 increases to reduce 4. But because of the exponential dependence of p on 4, a given change in 4 produces a much smaller change in p at 296 “K than if the change occurs at 77°K. For this reason, mobility varies very little with Vs, to within 5 5 % of its zero gate field value. The decrease in mobility with positive gate field would mean a new scattering mechanism now being operative. A positive s results in (1) an increase in the electron concentration, and (2) a decrease in surface band bending. It has been seen that effect (1) produces only a mobility increase. It therefore remains to examine the effects due to a reduction in surface band bending. For a depleted surface, surface scattering is not important. However, as the surface is enhanced by a gate field, it attracts electrons, thereby increasing scattering. Based on such a model, Schrieffer’s theory predicts a mobility reduction, which has since been substantiated by the experimental data of Waxman et a1.14 and van Heek2’. It would appear that the theory could also provide an explanation for the observed mobility behaviour at 296 “K. CONCLUSIONS

It is now possible to explain the different slopes of the mobility-temperature curves (Fig. 10). The ratio n&z2 varies to a much larger extent from room temperature to liquid nitrogen temperature for films with low impurity concentration, N,w 1-2x 1O1’ cmm3. The results of Hall and field-effect measurements have all been found to be in favour of the potential barrier model proposed. The barrier potential associated with grain boundaries is attributed to dislocation dangling bonds, rather than to different impurity concentration due to stoichiometric variations, as suggested by previous workers. The exponential variation of mobility and the slight temperature dependence of 4 can all be accommodated by the dislocation dangling bond model. Further information on the potential barrier model is provided by conductivity measurements at high electric (measuring) fields. This will be the subject of a subsequent paper. ACKNOWLEDGEMENTS

One of us (CHL)

is grateful

for a London

work. REFERENCES

1 J. C. Anderson, Advan. Phys. 19 79 (1970) 311. 2 C. Juhasz, Ph. D. Thesis, London University, 1968. 3 D. L. Lile, Ph. D. Thesis, London University, 1968. 4 J. Fisher, personal communication, 1970. 5 W. J. Williamson, Solid State Electron.. 9 (1966) 213. 6 H. von Pagnia, Z. ftir angewandte Physik, 16 (1963) 209. Thin Solid Films, 14 (1972) 267-288

University

bursary

during

this

288 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

C. H. LING,

J. H. FISHER,

J. C. ANDERSON

H. H. Wieder, Intermetallic Semiconducting Films, Pergamon Press, Oxford, 1970. M. Glicksman and W. A. Hicinbothem, Phys. Rec., 129 4 (1963) 1572. 0. Madelung, Physics of III-V Compounds, Wiley, London, 1964. K. Fuchs, Proc. Camb. Phil. Sot., 34 (1938) 100. E. H. Sondheimer, Phil. Mug., 1 (1952) 1. J. R. Schrieffer, Phys. Reu., 97 3 (1955) 641. A. Many, Y. Goldstein and N. B. Grover, Semiconductor Surfaces, North-Holland Publ. Co., Amsterdam, 1965. A. Waxman, V. E. Henrich, F. V. Shallcross, H. Borkan and P. K. Weimer, J. Appl. Phys., 36 1 (1965) 168. D. B. Holt, J. Phys. Chem. Solids, 23 (1962) 1353. D. B. Holt, J. Phys. Chem. Solids, 25 (1964) 1385. H. F. Matare, Defect Electronics in Semiconductors, Wiley, London, 1971. C. H. Ling, Ph. D. Thesis, London University, 1972. H. K. Henisch, Rectifying Semiconductor Contacts, Clarendon, London, 1957. Roy F. Potter and H. H. Wieder, Solid State Electron,, 7 (1963) 253. A. S. Filipchenko and D. N. Nasledov, Phys. Status Solidi, 19 (1967) 435. H. F. van Heck, Solid State Electron., II (1968) 459.

Thin Solid Films, 14 (1972) 267-288