activated channel conductivity in silicon inversion layers at high temperatures

Surface Science 58 (1976) 91-97

0 North-Holland Publishing Company

ACTIVATED CHANNEL CONDUCTIVITY IN SILICON

INVERSION LAYERS AT HIGH TEMPERATURES *

Teresa COLE, M.E. SJOSTRAND and P.J. STILES

Department of Physics, Brown University, Providence, Rhode Island 02912, USA

The effects of a reverse substrate bias on the resistivity of silicon inversion layers has

been studied at low carrier densities for 4.2 < T <, 77 K. At low temperatures the activa-

tion is small and decreases when a reverse substrate bias is applied. For T > lo-20 K the

activation energy increases [assuming p a po exp(+A/kr)] and the effect of substrate

bias is more complex. The activation energy increases for p-channels and decreases for n-

channels when substrate bias is applied.

1. Introduction

This paper concerns itself with conductance measurements on silicon inversion layers in MOS structures at intermediate and high temperatures (4.2 5 T 2 77 K) at low carrier densities (n, 2 1012 cmp2). At low temperatures the system is described in the electric quantum limit in which only the lowest subband in the potential well at the Si-Sio, interface is populated at low carrier densities [I] . As the temperature is raised, inversion layer electrons (holes) will be excited into higher subbands and multi-band effects must be taken into account in the description of the conductivity. Stern [l] has calculated the energy levels, relative populations, and charge distribu- tions for n-type inversion layers on Si. He finds that for a (100) surface the majority of electrons are in higher subbands for low carrier densities at 77 K. Calculations of subband splittings for p-channels have been made by Bangert et al. [2] and by Ohkawa et al. [3].

The effect on the carriers of inhomogeneities at the Si-SiO2 interface is to en- hance carrier localization at the bottom of each subband [4]. The energy splittings might be modified if the smearing of the sharp band edge into a tail were taken into account. Because they penetrate further into the semiconductor, carriers in the higher subbands should be less affected by the fluctuating potential than those in the lowest subband.

The application of a reverse substrate bias voltage between the source and sub-

* Supported in part by the National Science Foundation under Grant No. GH40477.

91

92 T. Cole et al. /Activated channel conductivity in Si inversion layers

strate will also modify the energy level splittings since to maintain a constant carrier

density the surface field must be increased when substrate bias is applied.

2. Results and discussion

The channel resistivity for an n-type Si inversion layer is shown in fig. 1 as a func- tion of inverse temperature for carrier densities between 0.4 X 1011 and 4.3 X 10’ 1 cmP2 (as determined from the threshold in transconductance at 77 K [ 51). The resistivity increases with decreasing temperature, and for l/T> 0.10 K-l the slope is approximately constant for each carrier density. In this range the conductivity can be described as an activated process of the form u = u. exp[-A(n,)/kT] where the dependence of A(n,) on n, is the same as previously reported [6]. The activation

energies are in the range 0.2-l meV. The temperature dependence of the conduc-

tivity is consistent either with nearest neighbor hopping, or if the mobility edge varies with carrier density, with thermal promotion to states above the mobility edge. However, in the case of thermal promotion, theory predicts that the prefactor

u. should be a constant, urn,,,. We do not observe this. We therefore believe that

A(n,) is a measure of the average activation energy for nearest neighbor hopping. In the temperature range 10 2 T 5 20 K there is a characteristic knee in the ps versus

l/T curves. For T > Tknee, the resistivity drops rapidly and has a much larger slope

than in the intermediate temperature range. All the n-channels and p-channels we

.I0 .20 .30 .40 .50

+ (K-l)

Fig. 1. Channel resistivity for a n-channel device, surface (loo), plotted versus l/T. (in cm-?): (a) 0.4 X 10”; (b) 0.5 X 10”;(c) 0.9 x 10”; Cd) 1.3 x lo”;(e) 2 x (f) 2.8 x 10”; (g) 3.6 x 10”; (h) 4.3 X 1Or’.

Values of ns

10” ;

T. Cole et al. /Activated channel conductivity in Si inversion layers 93

studied had the same qualitative features, although the p-channels in general had a

smaller slope, lower mobility, and less marked knee. We interpret the break in the resistivity curves at 10 5 7’5 20 K as an indication

that other conduction mechanisms are becoming important in addition to nearest neighbor hopping. There are at least two possible mechanisms which could cause this behavior. Current can be carried by electrons (holes) thermally excited into ex- tended states above the mobility edge so that the number of carriers is activated rather than the mobility. The extent to which this takes place depends on the energy separation between the Fermi level and the mob~ity edge and, therefore, on the de- tailed shape of the band tail. A second possibility is that carriers are beginning to be excited into higher mobility states in the next subband. The splitting between subbands determines when this mechanism becomes important. Calculations by Stern [l] show that for an inversion density of (1-5) X 10” cmP2, the fraction of carriers in higher subbands is about 30-50% at 77 K for a (100) silicon surface with substrate doping of lOI5 cm -3. For a lighter substrate doping, the potential well is wider and the splitting between subbands is smaller. The sample in fig. 1 had a bulk doping of ~10”~ cme3, so a smaller subband splitting and a greater population of

the higher subbands than that calculated by Stern can be expected. Carriers excited into higher subbands will, on the average, penetrate further into the semiconductor than those in the ground state so the effect of the fluctuating Fotential should be weaker and the mobility ought to be enhanced.

In fig. 2 we plot the data of fig. 1 against (l/7’) 1/3 in order to see if the conduc-

tivity can be described in terms of Mott’s variable range hopping in two dimensions over the entire temperature range as was suggested by Hartstein and Fowler [8]. The data do not fit in p 0~ T-Ii3 at all temperatures, but only at lower temperatures and carrier densities. In any case, in Mott’s theory of variable range hopping there is a critical temperature above which variable range hopping will go over to nearest neighbor hopping (or possibly to thermal promotion to a mobility edge). This critical tem-

perature occurs when the hopping distance, R 0~ [ l/nc~V(E~)kT] lj3, becomes com- parable to the average distance between localized states, R. [CY is the tunneling factor, N(E,) the density of states at the Fermi level.]

The effect of applying a reverse substrate bias is shown in figs. 3 and 4. Fig. 3 shows the resistivity of an n-channel for 3 different carrier densities for substrate biases of 0 and -45 V. The effect is drastic; the slope as well as the resistivity decrease for all temperatures except at very high temperatures where the curves cross. With an increase in carrier density the crossover shifts to higher temperatures. The analogous results for a p-channel, shown in fig. 4, are surprising. In this case the slope of the resistivity curve increases at high temperatures and decreases at low temperatures when a 30 V substrate bias is applied. Except at very low temperatures, the resistivity is larger with substrate bias than without. The crossover temperature

again shifts to higher temperatures as the carrier density is increased. Several p-channels were studied and the features were essentially the same.

When a reverse substrate bias is applied between the source and the substrate, the

94 T. Cole et al. /Activated channel conductivity in Si inversion layers

.40 .50 .60 70

Fig. 2. Same as fig. 1 but plotted versus (1/7’)1’3 instead.

surface field must be increased to keep the carrier density constant. This leads to a larger splitting between subbands and the electron (hole) gas is forced closer to the Si-Sio, interface. A generalized model of localization by potential fluctuations [4] has a great deal of appeal since a mobility edge has been observed for both n- and p- channels regardless of the fact that the fixed charges at the interface are positive in both cases. This model predicts that the effect of substrate bias on the carriers should qualitatively be the same for both n-channels and p-channels. In this light, the results shown in fig. 3 and 4 are puzzling.

At low temperatures the trend in the activation energies is the same for both n- channels and p-channels. If the interpretation of the conductivity at nearest neighbor hopping is correct, the decrease in activation energy when substrate bias is applied implies that the range of energies of the localized states is decreased as if the potential fluctuations are weakened.

At high temperatures, the fact that substrate bias has the opposite effect on the slopes of n-type and p-type samples, is perhaps indicative that a different conduction mechanisms is dominant in each case. If the tail of localized states is shortened, as

T. Cole et al. /Activated channel conductivity in Si inversion layers

n- CHANNEL

I - I I I I I I

.04 .06 .I2 .I 6 .20 .24

95

Fig. 3. Channel resistivity versus l/T with and without reverse substrate bias (-45 V) for a (100) n-channel with substrate doping ~10’~ cme3. (0, p) n, = 0.9 X 10” cm-*; (A, Q) ns = 1.4 X 10” cm-*; (0, p) n, = 2.5 X 10” cm-*.

the low temperature results indicate, then the difference between the Fermi energy and the mobility edge will be smaller and the energy for thermal promotion above the mobility edge will decrease. This could explain the results for the n-channel. The splitting between subbands increases with substrate bias, so if condition is due to carriers excited to a higher subband, the activation energy should increase. This is the trend that is observed in the p-channels.

It is not clear whether one of these two mechanisms should be expected to dominate over the other, or that there is not another possible mode of conduction. The energy splittings for n-channels are of the same order of magnitude as the estimated tail length so the two effects should become important at about the same tempera-

96 ‘T. Cole et al. /Activated channel conductivity in Si inversion layers

I f f f f I

p - CHANNEL

ClOA ‘d,=oV

f+‘Q+ V,= +3OV

I I I I I I

.04 .06 .I2 .I6 .20 .24

f (K-l)

,

c

i

Fig. 4. Channel resistivity versus l/T with and without reverse suostrate bias (+30 Vf for a (100) p-channel with substrate doping =10’s cme3. (0, Q) ns = 1.7 X 10” cmM2; (o,q) ns= 2.7 X 10” cm-‘; (A, 9) ns = 5.8 X 10” cm-‘.

tures. Calculations by Ohkawa et al. [3] indicate that for a (100) surface at 0 K the energy splittings for p-channels are smaller than those for n-channels. If this remains

true at higher temperatures excitation to a higher subband should be more important for p-channels than for n-channels. Since the subband splitting depends on surface orientation, and it is probable that the length of the band tail depends on the oxide charge density, it is not clear that there should be a consistent difference between n-channels and p-channels. All of our samples had approx~ately the same oxide charge density.

Fowler [ 1 I] has suggested that the increase in conductivity can be explained by assuming that there is an overlap between the tail states of the first excited subband and the ground state subband. The increase in splitting when a substrate bias is applied causes electrons to move from these tail states to the higher mobility states in the lowest subband, thus increasing the conductivity. For this effect to be noticeable, it must dominate over any increase in localization. This explanation could account for the p-channel results if the increase in the subband splitting with increasing tem-

T. Cole et al. /Activated channel conductivity in Si inversion layers 91

perature is large enough that the tail states of the higher subband are already above

the Fermi energy for temperatures above the crossover. An alternative explanation for the n-channel results is that the capture cross sec-

tion of the fixed oxide charges in the interface increases when the electron gas is squeezed closer to the interface and subsequent trapping of electrons will neutralize some of the positive ions and thus weaken the fluctuations and increase the mobility. Since the interface charges have the same sign as the carriers in a p-channel freeze- out effects are not expected. This explanation does not account for the crossover in the p-channels unless some other mechanism is invoked as well.

A coherent picture which accounts in detail for the behavior of the conductivity of both n-channels and p-channels over the entire temperature range is lacking at present.

Acknowledgements

One of us (M.E.S.) would like to thank S. Nagel for many valuable discussiosn.

References

[l] F. Stern, Phys. Rev. B5 (1972) 4891. [2] E. Bangert, K. von Klitzing and G. Landwehr, in: Proc. 12th Intern. Conf. on Semiconductor

Physics, Stuttgart, 1974 (Teubner, 1974) p. 714. [3] F.J. Ohkawa and Y. Uemura, Prog. Theoret. Phys. Suppl. (Kyoto), to be published;

F.J. Ohkawa, T. Ando and Y. Uemura, to be published. [4] N.F. Mott, Phil. Mag. 22 (1970) 7. [5] F.F. Fang and A.B. Fowler, Phys. Rev. 169 (1968) 619. [6] M.E. Sjostrand and P.J. Stiles, Solid State Commun. 16 (1975) 903. [7] D.C. Tsui and S.J. Allen, Jr., Phys. Rev. Letters 32 (1974) 1200. [8] A. Hartstein and A.B. Fowler, J. Phys. C8 (1975) L249. [9] N.F. Mott and E.A. Davis, Electronic Processes in Non-crystalline Materials (Oxford Univ.

Press, London, 1971). [lo] N.F. Mott, Electron. Power 19 (1973) 321. [ll] A.B. Fowler, Phys. Rev. Letters 34 (1975) 15.

activated channel conductivity in silicon inversion layers at high temperatures

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