Surface mobility measurements in germanium

J. Phys. Chem. SOWS

Pergamon Press 1960. Vol. 14. pp. 186-192.

SURFACE MOBILITY A. MANY, Department

Printed in Great Britain.

MEASUREMENTS

N. B. GROVER,

Y. GOLDSTEIN

of Physics, The Hebrew University,

and

IN GERMANIUM E. HARNIK

Jerusalem,

Israel

Abstract-Measurements are reported of the surface mobility in accumulation layers of germanium at low temperatures, and results obtained are compared with the theoretical calculations of SCHRIIWXR. Advantage is taken here of the relatively slow thermal release of charge from many of the surface states into the majority-carrier band at sufficiently low temperatures. By the use of the pulsed field-effect technique, a rapid decrease in majority-carrier concentration within the spacecharge region is effected. The change in surface conductance measured at the onset of the pulse, before the deficit induced can be redistributed between the states and the majority-carrier band, then becomes a direct measure of the majority-carrier surface mobility. The results for n-type samples show that SCHRIEFFRR’Scorrections for electrons are overestimated. On the assumption that the theoretical values are valid for accumulation layers, it is concluded that the surface is not a completely diffuse scatterer.

INTRODUCITON

Assuming the surface to be a completely diffuse scatterer, SCHRIEFFER@) has calculated the surface mobilities as a function of barrier height. The nature of his approximations makes the results adequate only for strong inversion layers. Several attempts(4-i) have been made to improve these calculations. In particular, Fw.NKL(~*8) has been able to extend the application of the theory to small ( V8( values and to accumulation layers. It would be highly desirable to check experimentally whether the diffuse-scattering model is A0 = q(t.+@+hsAN) (1) valid. Such measurements are rendered difficult, however, by the presence of fast surface states. where pPs and pns are the effective surface mobiliWere these states absent, the surface mobility ties associated with AP and AN. would have been easily deducible from field-effect The surface mobilities are expected to be lower measurements. In this case AN and AP would be than the respective bulk values due to the scattergiven directly by the known induced charge.(l) ing present at the surface in addition to the normal bulk scattering processes.@) For small values of But, because of the surface states present, the fraction of the induced charge which is mobile is 1V, 1it is common practice to use the bulk mobility not known, and the measured change in surface values in equation (1). For strong inversion or deep conductivity yields only the product ~&P or accumulation layers, however, either the holes or p&N. Additional information is thus necessary the electrons are constrained to move along a deep to separate the two unknowns. One possibility is potential well and, as a result, the reduction in to measure the space-charge capacitance,@* 10) and mobility may become very considerable. Hence from this to evaluate AP or AN. Such measurewithin this range a knowledge of the dependence ments are difficult as the series geometric capaciof the surface mobility on Vs is essential for detance cannot be made large enough to permit ducing V, from measurements of ho. More imaccurate detection of changes in space-charge portant still, such information should shed light capacitance which, in the region of interest, is of on the nature of the scattering processes at the the order of O-1 pFjcn-12. Another possibility, surface. 186

SURFACE conductivity has been extensively employed in the study of semiconductor surfaces as it is the most convenient and accurate measure of the barrier height Vs. The value of Vs can be determinedcl-a) from a knowledge of Al’ and AN, the surface hole and electron densities in excess of the densities at flat-band conditions (V# = 0). These latter are in turn connected with the change in surface conductivity ha by the equation

SURFACE

MOBILITY

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employed by PEINTZ and ZEMEL(11-14), consists of measuring the surface Hall coefficient and magnetoresistance. These authors conclude that the surface is essentially a diffuse scatterer as assumed by SCHRIEFFER. A different approach to the problem would be to attempt to reduce the density of the fast states on the surface by suitable chemical treatments. In this manner, most of the induced charge in the field-effect experiment would appear in the spacecharge region and would thus contribute to the change in surface conductivity. Measurements along these lines were carried out by MILLEA and HALL(‘s) on low resistivity samples, and their results indicate that the reduction in mobility predicted by SCHRIEFFER is overestimated. In this paper the experimental procedure is chosen with the purpose of eliminating the effect of the surface states rather than reducing their density. Advantage is taken here of the relatively slow thermal release of charge from the states into the majority-carrier band at low temperatures. By the use of the pulsed field-effect technique, a rapid decrease in majority-carrier concentration within the space-charge region is effected. The change in Au is then measured at the onset of the pulse, before the deficit induced can be redistributed between the band and the states. In this manner Au can be obtained as a function of the majoritycarrier excess density, and from this the surface mobility can be evaluated as a function of V#. Results of surface mobility in accumulation layers are presented for several germanium samples. The results obtained are compared with theoretical values based on SCHRIEFFER’Scalculations.(a) It is found that for deep accumulation layers in n-type samples the experimental points usually lie well above these calculated values. On the assumption that the theoretical data are valid in this range, it is concluded that the surface is not a completely diffuse scatterer. For P-type samples, no definite conclusions can be drawn from the data obtained so far. -AL The germanium tangular filaments and O-3 mm thick. were mounted on spacers separating

METHOD samples studied were recabout 10 mm long, 3 mm wide Following a chemical etch they a jig with thin (10~) Mylar the two larger surfaces from the

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field electrodes. The jig was inserted into a cryostat in which the temperature could be controlled between 100 and 300°K. The measurements were carried out in vacuum except those at 80°K which were made while the jig was immersed directly in liquid air. The purpose of the measurement is to obtain the change in Au at the onset of the pulsed field as a function of pulse amplitude. Instead of varying the pulse amplitude, however, a constant-amplitude pulse 8 V is used which can be switched on and off the field plate in series with a variable biasing a.c. or d.c. voltage, as shown in Fig. l(a). The same pulse independently activates the Wheatstone bridge consisting of the filament, a decade resistor, and two fixed resistors. The CR0 serves as a null indicator and also to display the changing filament conductance throughout the duration of the pulse. The measuring sequence is as follows. The bridge is balanced and the filament resistance Rf read on the decade resistor. The pulse 6 V, of polarity such as to repel majority-carriers from the surface, is then switched on at the field plate. The decade resistor is readjusted to balance the bridge as close to the onset of the pulse as the circuitry permits (~10 psec.). The pulse is then switched off the plate and the plate voltage (a.c. or d.c.) is adjusted so as to restore bridge balance. The pulse is now switched on again and the process repeated. Figure l(b) illustrates this sequence for the case where the biasing voltage is a 50-cycle alternating source. The pulse is synchronizedus) with the crest of the a.c. voltage. In this manner a series of values of filament resistance (Rf)n is obtained, each corresponding to the number of times n that the pulse has been switched on. This is equivalent to a measurement of the filament resistance as a function of nSV, the voltage of a pulse whose amplitude is varied in steps of 6V (with no biasvoltage). The maximum resistance of the filament, corresponding to the minimum of surface conductance Aumrn, is derived from the usual d.c. field-effect. The various (Ao)~ values corresponding to (RI)* are then computed by reference to Aumrrr. If one assumes, as pointed out above, that no surface states are involved in this process, then the mobile charge induced is proportional to n8V. This enables the evaluation of the barrier height V, for each value of n.

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RESULTS In the measurements described minority-carrier effects are completely absent, so that the samples studied constitute in effect single-carrier systems. This is insured by using extrinsic material at low temperatures and confining the equilibrium value of the barrier height to the appropriate region.

f

I_---

L-

FIELD

HALF-CYCLE

-TIME

E.

HARNIK

In Fig. 2 are shown the results of measurements on an n-type sample at 80°K. In Fig. 2(a) ha is plotted as a function of the effective pulse amplitude n8v (6V = - 24 V) applied to the field plate as described in the previous section. As all the induced charge is assumed to appear as a change in the concentration of mobile majority-carriers

,

SINE-WAVE

and

(ID m SeC)

-

PLATE

4

-

FIG. 1. (a) Schematic outline of the experimental set-up. (b) Illustration of the measuring sequence for deriving ho as a function of n8v. The heavy solid lines correspond to the display on the CRO. Only when the minimum”of surface conductivity is being determined do minority carriers play any role. In this latter procedure time is allowed for the surface states and the two bands to attain equilibrium conditions. Actually, however, even at the minimum the surface excess of minority carriers is negligible compared to the surface excess (or rather deficit) of majority carriers. Once Aa,in has been determined, the measurements are made under conditions where the majority-carrier band is in equilibrium with the induced charge whereas the surface states and minority-carrier band are not. These conditions exist for a.c. and pulsed fields at sufficiently low temperatures. In such measurements it is possible to swing the majority-carrier band over a considerable range in the depletion region, even far beyond the position corresponding to humin.

(electrons), the barrier height Vs can be computed at any point along the curve from the experimentally-known induced charge and from the theoretically-known(l* 2) value of AN corresponding to humin. The initial bias voltage is so chosen in the figure that the origin corresponds to flat-band conditions (Vs = 0). Consider first the negative quadrant (V,
SURFACE

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line obtained for smaller values of 1V,l. The slope of this straight line is in remarkable agreement with that computed from the electron bulk mobility. Besides the obvious conclusion that the surface mobility is equal to the bulk mobility for large negative barriers, these results prove conclusively that in this region there are no surface states 0

0

-moo -25

-4000

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units of kT/q. For any particular value of v8 the ordinate of the experimental curve is equal to qAiVps. The corresponding value of qANpB is the ordinate of the point in question on the extension of the straight line that represents Au for V8<0. This value is just what Aa would be had the carriers continued to move with their bulk mobility

a k t

*

-6000

-2

-800 -IO00 -1200 -400

-300

-200

-100

100

0

APPLIED VOLTAGE Y

200

V

FIG. 2. (a) Typical results [circles) of Au versus the effective pulse-voltage V, for an n-type sample (T = 80°K). The crosses represent a replot of Au versus where V8 is the barrier height. ( VG is the v-v,, voltage drop across the geometric plate-to-surface capacitance Co)

BARRIER HEIGHT

y fkr/gl

FIG. 3. (a) The same as Fig. 2(a), but with the accumulation region in greater detail. Here C6V is the induced charge corresponding to the pulse used. Values of the barrier height w, in units of kT/q are shown. (b) A plot of the experimental values of ps/pa versus va, together with the corresponding theoretical curve.

v = vo+vs qAN=

CC

VG

AN~const.

=

cG(v-vs)

XdV,

(b) V, versus effective pulse-voltage, as computed from the equations appearing above.

characterized by release times smaller than the resolution time of the experimental apparatus (~10 psec). It should be noted moreover that it was possible to elevate the conduction band up to about 60 times the forbidden gap value (Fig. 2b) without the measurements at the onset of the pulse being disturbed by minority-carrier effects. The surface mobility begins to decrease as the barrier moves toward accumulation-layer conditions (positive V and VP). This region, of principal interest in the present work, is shown in greater detail in Fig, 3(a). Here Au is plotted as before but the abscissa is marked with the corresponding values of barrier height w8 measured in

PB. Thus t~s/t~~ at any value of vs is just the ratio of the two ordinates at that point. The ratio of surface to bulk mobility, as computed from the experimental curve of Fig. 3(a), is shown in Fig. 3(b) as a function of vg. Also shown is the theoretical curve according to SCHRIEFFER~~), slightly corrected as suggested by FRANKL@).It is seen that the experimental points are well above SCHRIEFFER’Svalues. Figure 4 represents results of measurements of pn~/~nn as a function of vs made on an n-type sample at three different temperatures. Also shown are the corresponding theoretical curves according to SCHRIEFFER. It is seen that the theoretical curves in all three cases cross the experimental curves, thus showing that the former are overestimated for deep accumulation layers. The experimental curves are lower, the lower the temperature, as is to be expected from the increasing mean free path of the electrons as the

190

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GROVER,

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GOLDSTEIN,

and

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HARNIK

“777

I

I

5

3

I

7

9 y lkT/ql

BARRIER HEIGHT

FIQ. 4. Experimental results of /.~ns/~~s versus er, for an n-type sample (Nn-NA = 6.6 X 1013) at three different temperatures, together with the corresponding theoretical curves. -

Experimental,

-

-

-

Theoretical

temperature is decreased. Quite a different behavior is exhibited in Fig. 5, which represents measurements of I_L~S/~L~B versus v8 for another n-type sample. Here the results at the two temperatures 80°K and 152°K almost coincide, whereas the theoretical curves are of course different. Figure 6 shows surface mobility results for four n-type samples of three different impurity contents.

01

3

I

I

5

7 v,CkT/gl

ESARRER HEIGHT

FIG. 6. Experimental results of pns/pne versus vI for n-type samples of various impurity content (T = 80°K). All the samples were etched in CP-4A, and sample (b) underwent an additional etch in warm HaOa.

q n l

0 -

No-N4 ND-NA No-N4 ND-NA

Experimental,

= = = =

2.9 4.4 4.4 6.6

-

-

~101~ ~10~3 (a)

X lOi (b) x 1Or3 -

Theoretical

One expects that due to the decreased barrier thickness, the higher the impurity content the greater the reduction in mobility. This trend is not fully substantiated, as the curve for the purest sample lies between the other two. The curves for the two different samples of equal resistivities however, coincide at deep accumulation layers. Sample (a) was etched as usual in CP4A whereas sample (b) underwent an additional etch in warm H202.

I 0

I 2

I 4

I 8

I 6

BAWER HEW

I

q fiW

FIG. 5. Experimental results of pns/wzr versus ws for an n-type sample (ND -NA = 4.4 x 101s) at two different temperatures, together with the theoretical curves.

-

0 = 152°K Experimental,

l = gO”K, -

-

-

Theoretical

In Fig. 7 are shown results obtained on a p-type sample at two temperatures. The theoretical curves here lie above the experimental curves, but in both cases their separation decreases at deep accumulation layers. CONCLUSIONS

Results have been presented of surface mobility versus barrier height in accumulation layers of germanium at low temperatures. In the analysis of the data it has been tacitly assumed that the experimental procedure employed eliminated

SURFACE

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effects arising from surface states. Actually, the occurrence of any states with release times shorter than the experimental resolution time (~10 psec) would result in the mobile charge being less than the total charge induced by the pulse. The effect of such states, if present, would then be to raise the surface mobility above its measured values on the one hand and, on the other hand, to lower the

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materially changed. Even in the extreme case where it is assumed that no reduction in mobility takes place over the entire accumulation region, the reduction in the maximum value of v6 is only from 9.5 to 7.2 kT/q. It is thus seen that the experimental curves shown in Figs. 3-7 represent the lowest possible limit to the actual values. For deep accumulation layers in n-type samples, these

0.6

0

-2

-4

-6

BARRIERHEIGHT

q=

136”K,

0 -

= SOoK, -

-

2

4

6

BARRIERHflGHT

g&J

FIG. 7. Experimental results of pps//+e versus wr for a p-type sample (Na-ND = 13.2 x lOIs) at two different temperatures. -Experimental,

0

-6

Theoretical

values of et, below those calculated on the basis of the absence of surface states. As er, for deep accumulation layers depends only logarithmically on the induced mobile charge CJAN, however, the deduced values of vg would not be appreciably affected in this range. This argument is illustrated in Fig. 8. The experimental curve represents mobility results on an n-type sample obtained, as in the previous cases, on the assumption that no states are involved in the measurement. The “corrected” curve was constructed on the rather arbitrary assumption that states whose release time is too short to be detected were present with a density of 2 x 1011/ems at 6 kT below the conduction band. (The Fermi level at flat bands lies 12 kT below the conduction-band edge.) The parameters of the states were chosen so as to make PnS as near as possible to pnn for small vg values. It is seen that the “corrected” values lie above the experimental curve but that the v, values are not

6 ys/kr/q)

FIG. 8. Experimental results of Jb$/~nB versus v8 for an n-typesample(Nn-NA = 6.6 x 1013, T = 250°K). The dash-dot curve is computed from the experimental data on the assumption that surface states are present with time-constants shorter than the resolution time of the experimental apparatus.

O-

0 Experimental, - - - Theoretical, c]-._ . /-J Corrected

lowest limits are seen to be well above the theoretical curves. The theoretical values used here are based on SCHRIEFFER'S calculations which, as pointed out previously, are adequate for strong inversion layers. For lack of other data, the theoretical curves shown were constructed under the presumption(s) that SCHRIEFFER'S results are valid for deep accumulation layers as well. It should also be pointed out that, for the samples studied and in the temperature range covered, the mean free path of the carriers is comparable to the thickness of the accumulation layers. This seems to indicate(l7) that the calculated reduction in mobility in this case would probably be even greater than that derived by SCHRIEFFER. Thus if the theoretical curves shown in the figures are taken to represent

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GROVER,

Y.

the correct values based on a completely-diffuse scattering model, it follows from the experimental results presented here that the surface has to be at least partially specular. A definite conclusion on this issue needs await the accurate computations which are reportedlyov) being carried out by GREENE, FRANKLand ZEMEL. The question arises as to how far the actual values of surface mobility lie above the experimentally-determined lower limit. This could possibly be estimated from an extensive and systematic study over a large temperature range of samples of widely varying impurity content and surface treatment. At this stage only a few general features will be discussed. Provided that the scattering is not entirely specular, a key parameter determining the amount of mobility reduction at the surface is the ratio of the bulk mean free path h to the space-charge-layer thickness L. For accumulation layers, X/L is proportional to PB IiVo - N4 IV 2. Thus, the lower the temperature and the higher the impurity content, the greater the reduction in mobility is expected to be. This is substantiated by the experimental curves of Fig. 4, where the mobility is seen to decrease with decreasing temperature in accordance with the increasing h/L values. The experimental results at the three temperatures, for any given v8 in the region of deep accumulation layers, are roughly in the same proportion as the corresponding theoretical values. This may indicate that the measured values are not too far below the actual surface mobilities. The results of Fig. 5, however, do not appear to support this argument, as the measurements at the two temperatures coincide. Two possible explanations for this behavior present themselves. The first requires the presence of undetected surface states whose effect on the measured results counterbalances that due to the two different values of h/L. Alternatively, the surface of the sample being considered can become less of a diffuse scatterer as the temperature decreases. These two possibilities can also be applied to account for the data in Fig. 6. In p-type samples (Fig. 7) all the experimental

GOLDSTEIN

and

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points lie below the relevant theoretical curves. This is not surprising, however, in view of the characteristics of the trapped-charge release process observed in such samples. In n-type samples, the release times associated with all the states observed were far above the resolution time of the apparatus. Contrary to this behavior, the release process inp-type samples exhibited time-constants right down to the limit of the experimental resolution. It appears therefore that the states present in p-type samples are responsible for the low experimental values measured. Acknowledgements-The authors wish to thank Mr. S. ALBERT for his help in carrying out the measurements and for his contribution to the processing of the data. Thanks are also due to Mr. M. D. EICHENBAUMfor his technical assistance. REFERENCES 1. KINGSTON R. H. and NEUSTADTERS. F., J. Appl. Phys. 26,718 (1955). 2. GARRETTC. G. B. and BRATTAINW. H., Phys. Rew. 99, 376 (1955). 3. SCHRIEFFERJ. R., Phys. Rev. 97,641(1955). 4. HAM F. S. and MATTIS D., reported at Amer. Phys. Sot. Meeting, Toronto, June 22-24, 1955. 5. ZEMEL J. N., Bull. Amer. Phys. Sot. ser. 2, 3, 255 (1958). H., Ann. Phys., Lpz. ser. 7,3, 396 (1959). 6. FL;~R 7. FRANKL D. R.. Bull. Amer. Phvs. Sot. ser. 2.4. I I 179 (1959). 8. FRANKL D. R., private communication. 9. BROWN W. L., BRATTAINW. H., GARRBTTC. G. B. and MONTGOMERYH. C., Semiconductor Surface Physics (edited by KINC~TON R. H.). University of Pennsylvania Press, Philadelphia (1957). 10. BATH H. M. and CUTLER M., Bull. Amer. Phys. Sot. ser. 2,3,138 (1958). 11. PBTRITZR. L., Phys. Rm. 110,1254(1958). 12. ZEMEL I. N. and PETRITZ R. L.. , Phvs. _ Reo:llO. 1263 (1958). j. N., Phys. Rm. 112, 762 (1959). 13. Z&L 14. ZEMEL J. N. and PETRI~ R. L.. ,_ T. Phvs. _ Chem. Solid;& 102 (1959). 15. MILLEA M. F. and HALL T. C., Phys. Rev. Letters 1, 276 (1958). 16. MANY A. and GERLICH D., Phys. Rev. 107, 404 (1957). 17. GREENE R. F., this Conference, p. 291.