THE ANDERSON TRANSITION IN SILICON INVERSION LAYERS

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THE ANDERSON TRANSITION IN SILICON

INVERSION LAYERS

C. Adkins, S. Pollitt, M. Pepper

To cite this version:

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Abstract. — Carriers in inversion layers show a well defined Anderson transition with a mini-mum metallic conductivity at the mobility edge of about 3 x 10-s S. Deep in the band tail, the density of localized states falls exponentially. The tunnelling exponent of the localized state wave-functions (deduced from observations of variable-range hopping) depends on energy as (Ea —E)s where s rises from 0.5 deep in the tail of localized states to 1.0 as (Ec —E) -> 0. When the total density of localized states is high, correlation effects become important as £j? -»- Ec with the result that Ec rises as the concentration of carriers is increased. Experiments with substrate bias support the assertion that, when long range potential fluctuations dominante, the inversion layer becomes inhomogeneous and ceases to show an ideal Anderson transition.

1. Introduction. — Investigations using insulated

gate field-effect transistors have yielded what is perhaps e(e ctron '^ depletion layer >

the most detailed information yet obtained about an energy '^inversion layer conduction band Anderson transition. In a single device, it is possible to f ^^-~

vary the Fermi level, and also the nature and magni- ; y"^[^ — — tude of the random electrostatic potential giving rise ' / ^

to localization. The main features of the system, which /

// acceptors has been described in detail elsewhere [11, are as fol- Ec -/ . • • • • * * '

lows : ^>^~ ' — In a field-effect transistor, an electric field is applied J ' ' " ~~~ ^ ^ ^ band '

normal to a semiconductor surface. This causes ben- .//

ding of the energy bands and, when this bending is >ft sufficiently large, a roughly triangular potential well is /

formed at the surface in which minority carriers can distance ^ be bound. The surface is then said to be inverted.

Figure 1 illustrates the situation when negative charge FIG. 1. — Formation of an n-type inversion layer at the surface is induced in the surface, of a p-type material. The of a p-type semiconductor. The broken curves show the effect of potential well is so narrow that the spacing of the substrate bias (see section 6).

energy levels for motion normal to the surface is large [2, 3] and, at low temperatures, only the lowest

level is populated. There is no restriction, however, on b^ a t h m msulating layer. Changing the gate voltage

motion parallel to the surface, so that the carriers form c h a nSe s the density of carriers and hence also the

a strictly two-dimensional gas which, at low tempera- F e r m i l e v e l r e l a t l v e t o t h e b a n d e dSe

-tures, is degenerate. The field is applied to the surface I n a s i mPl e field-effect transistor, the insulator is a

by varying the potential of a thin metal film, the gate X^&x o f thermally grown S i 02. Charges associated with

electrode, which is separated from the semiconductor d e f e c t s l n t h l s «l a s s a n d a t t h e interface give rise to the

random electrostatic potentials which cause Anderson (*) On leave from The Plessey Company, Allen Clark Research localization in the inversion layer.

Centre, Caswell, Towcester, Northants, U. K. In an unperturbed two-dimensional band with

23

THE ANDERSON TRANSITION IN SILICON INVERSION LAYERS

C. J. ADKINS, S. POLLITT and M. PEPPER (*)

Cavendish Laboratory, Cambridge, U. K.

JOURNAL DE PHYSIQUE Colloque C4, supplément ou n° 10, Tome 37, Octobre 1976, page C4-343

Résumé. — Les porteurs de charge dans une couche d'inversion présentent une transition d'Ânderson bien définie avec une conductivité métallique minimum d'environ 3 x 10-3 S au seuil

de mobilité. Pour des niveaux situés profondément dans la queue de bande, la densité d'états loca-lisés décroît exponentiellement en énergie. La décroissance exponentielle dans l'espace des fonctions d'onde des états localisés se fait avec un exposant qui dépend de l'énergie comme (Ec — E)s où ^ croît de 0,5 loin dans la queue de bande des états localisés à 1,0 lorsque E ->• Ec. Quand la densité totale des états localisés est élevée, les effets de corrélation deviennent importants lorsque Ev -> Ec, avec pour résultat le fait que Ec croît quand on augmente la concentration en porteurs. Les expé-riences faites en appliquant une différence de potentiel entre le substrat et la couche d'inversion sont en accord avec le point de vue que nous avons déjà présenté : quand les fluctuations de potentiel à longue distance dominent, la couche d'inversion devient non-homogène et ne présente plus de transition d'Anderson idéale.

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C4-344 C. J. ADKINS, S. POLLITT AND M. PEPPER constant effective mass, the density of states is cons-

tant ; but the random potentials cause smearing of the band edge so that the density of states in an actual device will be as shown in figure 2. We expect a mobility edge at some energy E, separating localized states in the tail from extented states above. According to theory, E, should lie where the density of states has fallen to about one fifth of its value in the unperturbed band [l].

extended

I

density of states

FIG. 2.

-

The density of states near the bottom of the conduction band in the presence of a random electrostatic potential.

2. The mobility edge.

-

If the Fermi level E, is above E,, we expect the conductivity to be metallic, that is, it would tend to a constant finite value as T 4 0 . If E,

<

E, and the temperature is not too low,

we expect that the conductivity will be dominated by carriers excited from the Fermi level to extended states just above the mobility edge, giving

o = o,, exp(- W J k T ) (1)

with

W = E ,

-

E F .

Such behaviour is illustrated in figure 3 which shows log o plotted against 1/T for various gate volta- ges. A clear transition from activated to non-activated conduction occurs as the Fermi level is raised. We note that the results in the activated region can be fitted to straight lines which intersect at cm, = 2 X lO-' S , in

close agreement with theoretical estimates which give

3 X 1 0 4 S 11, 41.

3. The density of Iocalized states.

-

The change in carrier concentration An for a change in gate vol- tage AV, is given by

An = C,, AV,/e

where C,, is the capacitance per unit area of the gate- oxide-inversion layer structure. If the corresponding

FIG. 3. - Plots of lg a against 1/T for various gate voltages (positions of the Fermi level) in the temperature range where conduction is dominated by excitation of carriers to the mobility edge. The activation energies are also shown. o m m is about 2 X 10-5 S. This device had a total density of localized states of about 1.8 X 1015 m-2 and a net oxide charge density (as reflected at the interface) less than 1 X 1015 positive charges m-2. The

oxide thickness was 60 nm.

change in activation energy is A W, then the density of localized states N(EF) is given by

Figure 4 shows the density of states derived in this way for the same device as was used for the measure-

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THE ANDERSON TRANSITION I N SILICON INVERSION LAYERS C4-345 ments shown in figure 3. A logarithmic plot shows

that, deep in the tail, N(E) falls off exponentially with W in agreement with the calculations of Zittartz and Langer for a two-dimensional system [5].

4. The tunnelling exponent. - At low temperatures such that k T 4 (E, - E,), there is negligible excitation to the mobility edge and conduction is by variable- range hopping at the Fermi level. This gives a variation

of conductivity of the from

where a is the tunnelling exponent of the localized state wavefunctions [6, 81. Hence, knowing N(E), the gra- dients of the 0. K. against l/T1/' plots in the variable- range hopping rCgime yield values of a. Typical results for two specimens are shown in the double logarithmic plot of figure 5. It has been proposed that, near the mobility edge, &(E) should vary as

a(E) cc (E,

-

E)" . (3) Figure 5 shows that s -+ 1 as (E, - E) becomes small. This result is in disagreement with Abram [9] who predicts s = 0.75 for a two dimensional system ; but it is in agreement with a recent conjecture of Mott [l01 who has argued that s 1 is necessary if the metal- insulator transition is to be discontinuous at T = 0.

FIG. 5. - Double logarithmic plot of the tunnelling exponent against (E, - E ) for two different devices. Putting a co(E,

-

E), we see S + 0.5 deep in the tail and S -+ l as E -+ Ec.

As (E,

-

E) increases, figure 5 shows s decreasing to about 0.6. For large (E,

-

E), s must, of course, go to 0.5 in agreement with E,

-

E = Fz2 a2/2 m*.

5. Correlation , effects.

-

Results described above

were obtained on high quality field-effect transistors fabricated using a technology which minimizes inter- facial and SiO, disorder. The total density of localized

states was less than 3 X 1015 m-'. With higher localized state densities, electron-electron correlation effects become important before (E, - E,) -, 0. As the Fermi, level is raised and more of the localized states are populated, the added charges themselves significantly increase .the random potential. When this happens, raising E, also causes E, to rise : the mobility edge floats above E,. Plots of In a against 1/T still intersect at G,,, for the minimum metallic conductivity imme- diately above E, is unchanged even though the position of the edge is rising .The effect of correlation is apparent, however, when the conductivity results are processed to obtain the density of localized states. The calculated values of A W will be smaller than the true change in E,, so that the calculated values of N(E) will be too large. Such " results " are shown in figure 6, where the apparent density of localized states rises to nearly 20 times the value in the unperturbed band (1.7 X 10'' meV-l m-2), which is obvious nonsense.

band d e n s ~ t y

2

0 1 2 3

FIG. 6. -The apparent density of localized states calculated according to equation (2) for a sample in which the total density of localized states is so high (about 10 X 10'5 m-2) that correlation effects are important. Within the range shown here, the apparent density of states rises to nearly 20 times the value in

the unperturbed band.

Correlation effects may also account for the low values of s obtained in earlier experiments

1111

which were carried out on samples with relatively high total localized state densities.

6. Long range potential fluctuations. - In their inversion layer experiments, the- Bell group [12, 131, and the IBM group [l41 have obtained results which differ from ours in that

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C4-346 C . J. ADKINS, S. POLLITT AND M. PEPPER

(b) the minimum non-activated conductivity they

observe is from one (Bell) to three (IBM) orders of magnitude greater than 3 X 10-5 S as seen by us in agreement with the theoretical estimates.

We have suggested that the most likely explanation of these authors' measurements is that their transistors may contain long range inhomogeneities such that parts of the inversion layer would go metallic before others. This would result in observation of an increased conductivity while activation is still present in parts of the current path. Our suggestion appeared to be supported by the fact that the Bell group observed weak de Haas-Shubnikov oscillations in their devices while the conductivity was still activated : observation of de Haas-Shubnikov oscillations implies the presence

nf metallic regions.

Recently, however, we have carried out experiments in which we have varied the range of the potential fluctuations seen by the carriers and so produced at will either type of behaviour in a single device. To understand these experiments we must look in a little more detail at the origin and nature of the potential fluctuations.

In insulated gate field effect transistors (fabricated under clean conditions), the net charge in the SiO, is always positive with a value of about

+ 1015 e m-,.

The total density of localized states, however, may reach values as high as 2 X 1016 m-' for either electrons or holes. This can only be possible if both positive and negative charges are present, in which caze, some of the potential fluctuations will be relatively short range (dipolar). Then the characteristic spatial scale of the potential fluctuations seen by the carriers would increase if they could be moved back, away from the Si-SiO, interface.

The mean distance of the carriers from the interface can be changed by applying substrate bias. If a potential difference is applied between the inversion layer and the underlying semiconductor, the shape of the potential well is changed : the carriers can be pushed up against the interface or drawn away from it. The effect is illustrated in figure 1 for a positive bias applied to the substrate, which results in a widening of the potential well for electrons in the inversion layer. By this means, we might hope to be able to change continuously from a situation in which the potential fluctuations seen by the carriers are short range, giving rise to an ideal Anderson transition, to a situation in which long range fluctuations predo- minate so that the channel becomes inhomogeneous and displays results like those of the Bell and IBM groups.

This we have achieved. Figure 7 shows two sets of log a against 1/T plots for a device without and with substrate bias. In (a) there is no substrate bias and we ,observe ideal behaviour ; in (b) a positive substrate bias is applied, drawing the electrons away from the inter-

FIG. 7. - The effect of substrate bias. In ( a ) there is no substrate bias and we observe an ideal Anderson transition with

a m m m 4 X 10-5 S. In (b), with a substrate bias of

+

0.8 V, the mean distance of the carriers from the interface is increased and the transition becomes non-ideal with rising intercepts and a much increased minimum non-activated conductivity. This device has a total density of localized states of about 5.5 X 1015 m-2 in (a) and 8 X 1015 m-2 in (b). The net oxide charge density (as reflected at- the interface) was about 1.7 X 1015 positive charges m-2, and the oxide thickness was

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THE ANDERSON TRANSITION IN SILICON INVERSION LAYERS 04-347 face, and we observe the type of behaviour that we wavefunctions depends on energy as (E,

-

E)" where s

attribute to the development of an inhomogeneous rises from 0.5 deep in the tail to 1 as E + E,.

channel. We count these results as strong evidence in ₄₎_{Correlation effects are important when the den-} support of our explanation of the non-ideal behaviour sity of localized states is high.

observed by the Bell and IBM groups. ₅₎_{The likely cause of non-ideal transitions is the}

A more this Part our work is presence of long-range inhomogeneities in the inver- to be published shortly [ l S ] . _{sion layer.}

7. Conclusions' - 1) When short range disorder _{Acknowledgements.}

_-

_{We should like to thank}

dominates, carriers in inversion layers show an ideal _{Professor Sir Nevill}_Matt_{for his continual inspiration} Anderson transition with a minimum metallic conduc- _and_guidance._{We also express our thanks to the science} tivity of about 3 X 10-5 S.

Research Council for grants which have supported two

2) The density of localized states deep in the tail _ofUS (S. P. and M. P.) during the course of this work,

falls exponentially. _{and we are indebted to The Plessey Coinpany and IBM}

3)

he

tunnelling exponent of the localized state for manufacturing and supplying devices.

References

[l] MOTT, N. F., PEPPER, M., POLLITT, S., WALLIS, R. H., [9] A ~ R A M , R. A., J. Phys. C 6 (1973) L-379. ADKINS, C. J., PYOC. R. SOC. A 345 (1975) 169. [l01 MOTT, N. F. (1976) Comm. Phys. 1 (1976) no 7.

[2] STERN, F., Phys. Rev. B 5 (1972) 4891. _{[l11 PEPPER,}_M.,_POLLITT,_{S., ADKINS,}_C.J., J. Phys. C 7 (1974) [3] STERN, F., Phys. Rev. B 9 (1974) 2762. _L-273.

141 LICCIARDELLO, D. C., THOULESS, D. J.9 J. Phys. C 8 (1975)

1121 T ~D. C., ~ ~A, ~S. J . , ~ phys. ~ ~ ~ , ~32 (19743 ~ t . 4157.

[5] ZITTARTZ, J., LANGER, J. S., Phys. Rev., 148 (1966) 741. 1200.

161 POLLAK, M., J. Non-Cryst. Solids 11 (1972) 1 . [l31 ALLEN, S. J., TSUI, D. C., DE ROSA, F., Phys. Rev. Lett. 35 [7] HAMILTON, E. M., Phil. Mug. 26 (1972) 1043. (1975) 1359.

[S] BRENIG, W., DOHLER, G. H., HEYSZENAU, H., Phil. Mug. 27 [l41 FOWLER, A. B., Phys. Rev. Lett. 34 (1975) 15.