The Anderson transition in silicon inversion layers

Surface Science 0 North-Holland

58 (1976) 79-88 Publishing Company

THE ANDERSON TRANSITION

IN SILICON INVERSION

LAYERS

S. POLLITT, M. PEPPER and C.J. ADKINS Cavendish Laboratory,

Cambridge CB3 OHE, England

The Anderson transition has been studied in MOS and MNOS structures. The conductivity, magnetoresistance and Hall effect of this low temperature phenomenon are discussed. When the devices are metallic, that is, when the Fermi level lies in the extended states, a peaked structure is observed in the field effect mobility. The dependence of the structure on the reversible charge trapped within the gate dielectric of n and p-channel MNOS transistors has been investigated in an attempt to explain the mechanism giving rise to the structure.

1. Introduction The inversion layer produced at the silicon surface in MOS and MNOS transistors has proven an ideal system for investigating the Anderson transition [l-6] . In 1958 Anderson [7] published a paper entitled “The Absence of Diffusion in Certain Random Lattices”. In this paper he considered a tightbinding band of bandwidth B and introduced a random potential V on each well such that -i Vu < V< i VO. He showed that if Vu/B is greater than a critical value then electrons would not freely move through the lattice in extended (Bloch) states, but would be localised in space, electrons moving from one site to another by exchanging energy with phonons, a process called thermally activated hopping. When the states are localised the electron wave functions will decay exponentially as e-“’ where (Yis the decay constant of the wave function. Mott [8] pointed out that if the Anderson criterion is not satisfied then states will be localised in the tail of the band and an energy EC (the mobility edge) exists separating localised and extended states. At absolute zero o=O

for E,
of0

for E,>E,.

The smallest value that the conductance can take before an activation energy occurs is the minimum metallic conductance, umm. In two dimensions this is [9] umm = 0.12 e2/h = 3 X 1O-5 fin-l 0. Using the Mott [lo]

ratio g defined by

g = ‘V(EF)IN(EF)free

, 79

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S. Pollitt et al. /Anderson

transition in Si inversion layers

whereWEI: 1free is the density of states at the Fermi level in the absence of disorder, we can write the conductance of a two-dimensional length, S,:, and mean free path, I, as S,e21g2 (J=------47&l

electron gas with a circular Fermi

(1)

.

AS the metallic conductance cannot be reduced below u,~, we set I = a, k, = n/a and equate u to urn,,, , and we find g - 0.8 for localisation and g - 0.4 for the practically important case of the (lOO)Si surface. As the free electron density of states is constant in a two-dimensional system (depending on the surface orientation and carrier type for the case of an inversion layer) the density of states near E, will always have the same value, no matter how far up the band localisation sets in. Mott [ 1I] initially suggested that tails of localised states might exist below the 2D subbands in the inversion layer of silicon, charges within the silicon dioxide producing a random electrostatic potential within the inversion layer, giving rise to the localisation. By sweeping the voltage on the gate of a MOSFET we can push the Fermi level through the localised states into the extended states and observe an Anderson transition from activated to unactivated conduction. When E,
exp [-(EC - EF)lkT]

.

(2)

At low temperatures hopping conduction between the localised states dominates and this always proves to be variable range hopping giving u = a0 exp[-3,2/3/(dV(E,:)kT)1/3]

2. Experimental

.

(2)

results

The devices used for our experiments have been both n and p-channel MOS and MNOS field effect transistors fabricated on the (100) and (111) surfaces of silicon. The devices used for conductivity measurements were of an enclosed geometry, diameter 200 pm, source drain separation 20 q, source drain voltage less than 50 V m-l being used to avoid carrier heating. The conductivity of a n-channel MOS transistor fabricated on a (100) surface is shown in fig. 1 plotted as log u versus l/T. As the gate voltage becomes more positive the Fermi level moves through the localised states into the extended states and a transition from activated to unactivated conduction is observed. At higher temperatures activated conduction is by excitation to E, deviating from this relation at the lowest temperatures due to variable range hopping. The value of the conductance at the transition of 1.5 X 10e5 R-l 0 is in good agreement with the predicted value of3 x 10-S a-1 0.

S. Pollitt et al. /Anderson

transition in Si inversion layers

0

o 0.48 0

81

0.29 0 0

0 0

W=[Ec-E,]

1

2

meV

3

K/T

Fig. 1. Log conductance of an MOS device, versus l/T. Vg refers to the applied gate voltage, the energies are indicated.

The rate of change of activation energy, AIV = EC - EF, with change in carrier concentration, An, is roughly equal to the density of localised states, An/Aw N(E,). If the results of fig. 1 are used to determine N(E), then we find that the density of localised states within - 1.5 meV of EC is greater than the free electron density of states for the twofold degenerate ground state subband on the (100) surface. This cannot be the case, and indicates that EC increases as EF increases probably due to increased fluctuations in potential in the inversion layer as the localised states are populated. Fig. 2 shows the excitation to the mobility edge regime on a (100) MOS device with -1.5 X 1015 mm2 localised states. Because there are few localised states the effect of tilling the states is likely to have little influence on EC. We find no anomaly in the density of localised states as EF approaches E, in agreement with the predic-

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transition in Si inversion layers

vg=2.0 Q

"ne~ncc

0

oc=,;--3 0

;

‘J 1.4 0 1.0

m_..

"

0

.I

0

0.2

0.4

0.6

0.0

K/T Fig. 2. Log device current

of an MOS device, versus l/T. Source-drain

voltage

is 0.25

mV.

tion. The density of localised states near the edge is 6 X 10-17/m2/eV in good agreement with the value calculated previously. The increase in E, as EF is raised in samples with 1016 m -2 localised states fits into a general pattern of electron correlations enhancing localisation. The second conduction mechanism which is dominant at low temperatures is variable range hopping. Both n and p-channel MOS and MNOS transistors have been used to investigate the validity of the T-1/3variable range hopping law. Both the pre exponential term, uo, and the exponent contain the density of states; hence the temperature corresponding to the onset of variable range hopping for similar values of (E, - EF) is greater for devices fabricated on (111) surfaces than those on (100) surfaces. A statistical fit to data taken on an n-channel (111) MNOS device in the variable range hopping regime shows the power of the temperature dependence to be 0.32 + 0.02. At the lowest temperature, 0.17 K, we have been able to obtain we have seen no deviation from the T-II3law.

S. Pollitt

et al. /Anderson

transition

in Si inversion

layers

83

The fact that the T-‘13 behaviour extends up to 3 K indicates that the density of localised states varies only slowly within -0.5 meV of E,. This is a finding confirmed by the extraction of the density of localised states from l/T plots using devices with little localisation. The density of localised states appears fairly constant within - 1.5 meV of the mobility edge, at energies greater than this the density of states decreases rapidly with energy. A slowly varying density of states within kT of E, is of course a pre requisite for observation of l/T and Tp113 laws, both of which contain the implicit assumption that the Fermi energy is independent of temperature. When E, lies in a region of rapidly varying N(E), l/T plots are not obtained, a curvature is present which we attribute not to variable range hopping but rather to E, varying with temperature. We believe this to be the cause of results published recently by Hartstein and Fowler [ 121 showing a “T-1/3” relation up to ~77 K. Clearly this temperature is too large for the mechanisms to be variable range hopping. At temperatures higher than those corresponding to excitation to E, we believe that we have observed excitation to a higher subband. The activation energies observed are in reasonable agreement with calculation, the higher mobility resulting from a more extended wave function normal to the surface causing the electron transport to be Bloch-like rather than the diffusive motion which occurs at and just above E, in the ground state subband. When the Fermi level lies just above E, a small negative magnetoresistance is observed (fig. 3) and is possibly due to spin dependent scattering. This effect is small as umm is overwhelmingly determined by disorder scattering. On lowering the Fermi level below E, the magnetoresistance changes sign and becomes positive. This we believe is due to the shrinking of the localised wave functions (a-’ decreases) and hence the exponent in eq. (3). The change of sign of magnetoresistance, as the Fermi level is moved into the localised states, is also observed in doped semiconductors [13]. The Hall effect has been studied at the transition on an n-channel (100) device * and preliminary results are presented. At 77 K the number of electrons, l/RHe, in the inversion layer varies linearly with gate voltage. On cooling to 4.2 K the same dependence is observed in the metallic regime although shifted slightly to higher gate voltages; which can be accounted for by carriers going into localised states and not contributing significantly to the Hall effect. The change in carrier concentration with gate voltage is shown in fig. 4a, and in the linear region is in agreement with the value calculated from the known capacitance of the device. As E, approaches E, from the metallic side the carrier concentration deviates from linearity with gate voltage and becomes slightly larger than the appropriate free electron value, as the Fermi energy is reduced. The carrier concentration drops sharply to a value too small to measure (<1014 mv2). In this reg’ion the carrier concentration for a particular gate voltage decreases strongly with decreasing temperature. The Hall mobility, p”, * We wish to thank

Dr. A.B. Fowler

for these devices.

S. Pollitt et al. /Anderson transition in Si inversion layers

84

./

8

vg= -1 7 . 6

/

p-p,% PO

5

/ .

3 4

/ .

1 2

/

kG

Fig. 3. Magnetoresistance of an n-channel MNOS device, fabricated on a (111) silicon surface, versus magnetic field. Arrow indicates increasing carrier concentration.

obtained from the product of the Hall constant and the conductivity is shown in fig. 4b. /.L” falls rapidly as E, approaches EC from the metallic side but becomes con stant for gate voltages corresponding to an activated conductivity; that is, when the Fermi level lies below EC and conduction is due to carriers at the mobility edge. Thus we conclude that the mobility of carriers at the mobility edge is constant (-600 cm2/V s for this sample) and it is an activated carrier concentration which gives rise to an activated conductivity. This is, of course, expected when conduction is by excitation to the mobility edge. According to Friedman [ 141 , the value of the Hall mobility in the 2D system when carriers are excited to the mobility edge is given by /.&=0.5

(g) $-) ,

where a is the distance between localised states, @/Vu) - 4/3, the critical value for localisation, 77- l/3, z the coordination number -4. As we know the number of localised states (which is equal to the number of carriers to drive the device metallic) a reasonable value for a is 200 A. Putting this value in Friedman’s formula we obtain a Hall mobility of 4000 cm2/V s at the mobility

S. Pollitt et al. /Anderson

85

transition in Si inversion layers

T=4.2K 0

0

0

0

~

0

0

0 0

0 0

0

0 0”

0

0

0”

020

0

Cl

I

1

t

1

2

3 “Q

(a)

I

,

I

I

1

2

3

4

L

4

0

“g (b)

Fig, 4. (a) Carrier concentration, l/RHe, of an MOS device versus gate voltage, Vs. (b) Hall mobility of an MOS device versus gate voltage, Vg.

edge, which is a factor of -6 greater than the experimental value. The Hall mobility in the expanded fluid mercury system investigated by Even and Jortner [ 1S] , which shows an Anderson transition in a similar manner as the inversion layer, is a similar factor smaller than the value predicted by Friedman’s theory. Although a large hopping Hall effect is predicted by Holstein [ 161, we observe no contribution to the Hall effect due to carriers in localised states. Amitay and Pollak’s [ 171 investigation of the Hall effect in doped silicon and germanium shows that the Hall effect was unmeasurable in the hopping regime contrary to Holstein’s prediction. Various authors have reported a peaked structure in the field effect mobility of MOS devices [ 18,191. We have used both n and p-channel MNOS devices to investigate this phenomenon. Using a MNOS device it is possible to tunnel charge in and out of deep traps which exist at the nitride oxide interface and in the nitride and so change the random potential with&r the inversion layer. The field effect mobility of an n-channel (111) MNOS device with positive (attractive) charge in the dielectric is shown in fig. Sa, the peaked structure having the same characteristics as those previously reported. The field effect mobility of the same device but with negative (repulsive) charge in the dielectric is shown in fig. Sb, the peaked structure is absent. In p-channel MNOS devices we have examined, we find that although the peaked structure is reduced when repulsive charge is introduced into the dielectric, it does

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S. Pollitt et al. /Anderson

transition in Si inversion layers

-8

-7 vg volts

-6

-5

(a)

not completely disappear possibly due to a background of negative charge in the dielectric of the MNOS structure. The peaked structure is observed in the metallic regime, so it is not associated with the localised states. From the results we conclude that the peaked structure is due to attractive charges in the oxide, but the exact mechanism giving rise to fluctuations in either the density of states or mobility so causing the structure is not yet fully understood, but is thought to be related to resonant scattering. It is not an inhomogeneity effect because peaked structure is observed in devices which, from magneto-conductance oscillations, are known to be very homogeneous. If, as we believe, the peaked structure is due to attractive charges within the dielectric, then these charges must be distributed at discrete distances from the interface, since a continuous distribution of charge would not give rise to any structure.

S. Pollitt et at. /Anderson

transition in Si inversion layers

87

Fig. 5. (a) Field effect mobility of an n-channel MNOS device versus gate voltage, with positive charge in the dielectric. (b) Field effect mobility of the same device shown in fii. 4a versus gate voltage, but with negative charge in the dielectric.

3. Conclusion The conductivity and Hall effect results give overwhelming evidence to support the concept of a mobility edge separating localised from extended states. Although there is remarkably good agreement between theoretical and experimental values of min~um metallic conductivity, we find the Hall mobility at the mobility edge smaller than the theoretical one of Freidman, as do workers investigating other systems. Finally, we wish to point out that studies of the Anderson localisation in inversion layers offer a means of further understanding the Si-SiOz interface. The localised state concentration in an inversion layer can be a factor of 20 greater than the net charge in the dielectric, a fact which indicates the presence of charge compensation. Under these circumstances the low temperature conductivity is the most sensitive indication of the interfacial charge density. Early results suggest that the extent of the localisation can be varied by appropriate technological adjustments although this

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et al. /Anderson

transition in Si inversion layers

cannot be appreciated from measurements of net charge. The situation may be analogous to that in many of the chalcogenide glasses where an inherent mechanism ensures that almost equal numbers of positive and negative charges are present.

Acknowledgements We wish to thank Professor Sir Nevill Mott for his interest and discussions on this work, the Plessey Company for supplying the samples, and the Science Research Council for a maintenance grant which has supported one of us (S.P.).

References [ 1 J M. Pepper, S. Poilitt, C.J. Adkins and R.E. Oakley, P’l;ys. Letters 47A (1974) 71. (21 M. Pepper, S. Pollitt and C.J. Adkins, Phys. Letters48A (1974) 113. [3] M. Pepper, S. Pollitt and C.J. Adkins, J. Phys. C7 (19’74) L273. [4] E. Arnold, Appl. Phys. Letters 25 (1974) 705. [5] D.C. Tsui and S.J. Allen, Jr., Phys. Rev. Letters 32 (1974) 1200. [6] M.E. Sjtistrand and P.J. Stiles, Solid State Commun. 16 (1975) 1200. [7] P.W. Anderson, Phys. Rev. 109 (1958) 1492. ]8] N.F. Mott, Phil. Mag. 13 (1966) 689. [9] D.C. Licciardello and D.J. Thouless, J. Phys. C8 (1975) 4157. [lo] N.F. Mott and EA. Davis, Electronic Processes in Non-Crystalline Solids (Oxford, 1961). [11] N.F. Mott, Electronics and Power 19 (1973) 321. [ 121 A. Hartstain and A.B. Fowler, J. Phys. C8 (1975) L249. [ 131 See N.F. Mott and E.A. Davis, ref. [lo], p. 181. [ 141 L. Friedman, J. Non~rysta~ne Solids 6 (1971) 329. [ 151 U. Even and J. Jortner, Phys. Rev. B8 (1973) 2536. (161 T. Holstein, Phys. Rev. 124 (1961) 1329. [ 171 M. Amitay and M. Pollak, in: Proc. Intern. Conf. on the Physics of Semiconductors, Kyoto, 1966, J. Phys. Sot. Japan (Suppl.) 21 (1966) 549. [ 181 J.A. Pals and W.J.J.A. van Heck, Appl. Phys. Letters 23 (1973) 550. [ 191 R.J. Tidey, R.A. Stradling and M. Pepper, J. Phys. C7 (1974) L353.