Degeneracy effects on semiconductor surfaces

Degeneracy effects on semiconductor surfaces

SURFACE SCIENCE 13 (1969) 17-29 0 North-~o~iand Publishing Co., Amsterdam DEGENERACY EFFECTS ON SEMICONDUCTOR SURFACES * J. N. ZEMEL and M. KAPL...

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SURFACE

SCIENCE

13 (1969) 17-29 0 North-~o~iand

Publishing Co., Amsterdam

DEGENERACY EFFECTS ON SEMICONDUCTOR

SURFACES *

J. N. ZEMEL and M. KAPLlT** The Moore

School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104. U.S.A.

Received 6 June 1968; revised manuscript

received 21 June 1968

Two types of degeneracy effects are discussed in this paper. In the first case, degenerate surfaces on degenerate bulk semiconducting materials are reviewed. It is pointed out that epitaxic thin films are a prerequisite for any reasonable study of this type of substance. Recent work on degenerate lead chalcogenide films have illustrated the need for epitaxic films. The results show that the scattering at the surface is quite specular. Extremely high surface charge densities were generated by oxygen adsorption. It was found that coulombic interactions caused spreading of the surface state levels comparable to the band gap of the semiconductors. A model based on an oxide with high defect densities of ionized oxygen appears more reasonable. In the second case, a degenerate layer is formed on a nondegenerate bulk by means of an external electric field in a metal-oxide-semiconductor field effect transistor (MOSFET) structure. At low temperatures, an n-type inversion layer formed on a (100) p-type silicon MOSFET behaves like a two-dimensional electron gas as shown by Fowler, Fang, Howard and Stiles using Schubnikov-de Hass oscillations. Recently, an oscillatory behavior was observed in the gate voltage dependence of the capacitance of a MOSFET in a high magnetic field (> 50 kOe). Potential utility of these measurements is discussed.

1. Introduction In recent years there has been a growing interest in examining the properties of very high carrier concentration materials, primarily, metals, semimetals, and degenerate semiconductorsl). The surface properties of metals have long been of interest and the recent work with bismuth represents the first attempt to examine the behavior of semimetal surfacesss a). With but few exceptions degeneracy effects on semiconductor surfaces have been ignored4+5). This paper describes some aspects of degeneracy effects in semiconductor surfaces. The first section presents some details of the theory of these effects. The next two sections describe several aspects of degenerate * Supported by the U.S. Army Research Office-Durham under Contract No. DAHC0468-C-0023. Work supported in part by an Advanced Projects Research Agency grant to the Laboratory for Research on the Structure of Matter at the University of Pennsylvania. ** Part of this work was performed while this author was a Guest Scientist at the National Magnet Laboratory which is supported at M.I.T. by the Air Force Office of Scientific Research. 17

18

J. N. ZEMEL

AND

M. KAPLIT

surfaces in materials where the bulk is either degenerate or nondegenerate. Finally new trends in the study of degeneracy effects in semiconductor surfaces are described.

2. Degenerate semiconductor

surfaces

For a semiconductor to be degenerate the Fermi level must be within 1 to 2 kT of the pertinent band edge, i.e., the conductance or valence band edge. The same criterion holds for surface phenomena, i.e., in order for the surface to be degenerate the Fermi level must be within 1 to 2 kT of the pertinent band edge or actually be in the band itself. Since most semiconductors have density of states effective masses on the order of half of the free electron mass or less, it is easy to show that the minimum carrier concentration required for degeneracy is approximately lo’* carriers/cm’. If the bulk of the semiconductor happens to be degenerate, it is clear that thicknesses below 1 micron are required in order to have observable effects on the conductivity of the specimen. In the case of a nondegenerate bulk material, no special requirements are placed on the thickness, and for the case of an inversion layer the problems of observation are considerably reduced. The first question to be asked is “What type of phenomena would be associated with high concentrations of charge in a particular substance?“. For a metal, there does not seem to be any problems in understanding the shielding of an external field by the free charge. Simple calculations based on the Poisson equation, yield a shielding length which is of the order of a fraction of an angstroms). In effect, only a small redistribution of the charge in the conduction band is required to shield the external electric field. In the case of a semiconductor, however, the bulk charge densities are considerably lower and the wavelengths of the carriers in either the conduction or valence bands are considerably larger. Under these extreme conditions one simply cannot use Poisson’s equation to derive the width of the shielding layer. Instead, a simultaneous solution of the Schroedinger equation and Poisson equation are required. A simple model that can be assumed is a linear potential well for which the Schroedinger equation can be solved directly. While not exact, this approach can give an adequate insight into nature of the surface quantization problem. For the linear potential the wave functions of the Schroedinger equation in one dimension are Bessel functions of order 3, the Airy functions. and the eigen-energies E,, for the different quantum sub-bands are

-2-m 1(n+$)% n=0,

E = 1 (37&Q2 ” [ 2

+

1, 2, ..*.

(1)

DEGENERACY

Fig. 1 shows these energy

19

EFFECTS

levels in the linear

well. It is evident

that those

states with energies less than E,, the first quantized state in the linear well, will be eliminated due to quantization. The net effect will be to decrease the amount of charge (i.e., the screening) at the surface, allowing the field to penetrate further into the bulk than is predicted by classical theory. Another result of the quantization of the surface space charge region (SSCR) is to

i

CONDUCTION BAND

LEVEL

Fig. 1. Linear potential well model for the semiconductor space charge region showing the first electric sub-bands due to surface quantization. Ed and Eo2 are the levels associated with different valleys in a multi-valley semiconductor.

increase the average energy of the carriers. Because of this increase in the average energy of the carriers, one expects that the shape of the surface potential well will be modified. This modification of the shape of the potential well produces an increase in both the value of the surface potential and in the width of the space charge region. A more realistic model such as that of Stern and Howard7) would give the details of this variation. Fig. 2 shows the shape of the surface well calculated by Howards) but presented by Stern and Howard7). Included also are the positions of the energy levels associated with the (100) face of silicon and the shape of the wave function associated with the lowest electric sub-band. This wave function has a single maxima and was obtained assuming that the wave function disappeared at z=O. As a result of the nonpropagating character of the wave function in the z-direction, the carriers in the inversion layer of p-type silicon will behave like a 2-dimensional electron gas. The significance of this will be discussed later on at greater length.

20

3. N.

2 :

ZEMEL AND M. KAPLIT

- - - - - FERMI LEVEL 6

w” I

w

-40

Fig. 2.



Potential well and probability density of charge in the first electric sub-band, y(z), for an n inversion layer on p-type Si. E, is the conduction band edge. After Stern and Howard s).

3. Epitaxic Blms of the lead salts

Epitaxic films of the lead-salt compounds (lead sulfide, lead selenide and lead telluride) show both degenerate bulk and surface behavior. To date, there has been no explicit evidence that these materials show surface quantum effects, although there are recent measurements that might indicate the existence of such effects. Lead sulfide has been studied by BrodskyQ), lead selenide, by Brodsky and Zemell’J) and lead telluride by Egerton and Juhaszlr). In every case, large surface charge densities on the order of 5 x 10r3/cm2 were observed. The film thicknesses needed for these measurements were I-4000 A. Also, Hall has been studying the properties of the magnetoresistance on lead sulfides). The PbSe case will be discussed in greatest length. The Hall mobility of 1000 A thick p-type PbSe epitaxic films was substantially independent of the incremental charge induced by a surface oxide as shown in fig. 3. The resistivity was directly proportional to the surface charge. The variation of the Halt coe&ient with resistivity is shown in fig. 4 and was found to be almost linear. A normal temperature dependence was observed for the mobility in the film studies. Variation of the surface charge was produced by exposing the films to oxygen and was as large as 5 x 101s holes/cm2. The oxygen was removed by pumping on the films and turning on the ionization gauge on the vacuum system. It is now assumed that this generated suf’hcicnt carbon monoxide (CO) to completely reverse the effect

DEGENERACY

2

EFFECTS

700

20~ 600 < hl 3 ; I5 5 9 2 Id

500 400 300 200 100

RESISTIVITY

(lO-3 OHM CM)

Fig. 3. Variation of the Hall mobility as a function of film resistivity in three 1000 8, thick p-type PbSe epitaxic films on NaCl. The films denoted by 0 and 0 were two halves of the same film while the one denoted by @ was another film grown at a different time. Resistivity variations were due to ambient gas changes.

6.0 -

RESISTIVITY

(lO-3 OHM CM)

Variation of the Hall coefficient as a function of film resistivity in three 1000 8, thick p-type PbSe epitaxic films on NaCI. The films denoted by @ and 0 were two halves of the same film while the one denoted by @ was another film grown at a different time. Resistivity variations were due to ambient gas changes.

22

J.N.ZEMELANDM.KAPLlT

of the oxygen, This assumption is based on the observations of Berezhnayal2) that carbon monoxide would reversibly change the resrstivity of lead sulfide po~ycrystalline films exposed to oxygen. Using the data shown in table 1, Brodsky and Zemel observed that the band bend~llg was of the order of 6-7 kT at the surface. The films were initially degenerate p-type and the effect of the oxygen was to drive the films even more degenerate at the surface. TABLE

Physical

1

properties of

Pb salts

PbS

PbSe

PbTe

190 0.485 0.40

380 0.340 0.28

450 0.217 0.31

One of the more obvious problems associated with such a large charge density as 5 x 10’3/cm2 is the question of the state responsible for the surface chargers). A rough calculation based on the nearest neighbor interactions yields an energy term Einc of the form E

int

-

4g2 %vEO

es ,i’

(2)

4

where 4 is the electronic charge, E,, is the static dielectric constant averaged over the oxide and the bulk %V=

bC%xide

+

%I

>

(3)

where E,,ide and csc is the static dielectric constant of the oxide and semiconductor respectively, e0 is the permitt~vity of free space and Q, is the surface charge density. The factor 4 takes into account the nearest neighbor was assumed to be 20. The values for the interaction interactions. Here ~~~~~~ energy are listed in table 1 for the three lead salt compounds. As can be seen these interaction energies are comparable to the listed energy gaps of the material. As a result, it becomes questionable whether a surface state model will provide an adequate explanation for the charge generation at the surface. Another model based on oxide growth ~~ouId appear to be somewhat more satisfactory. The oxide, however, would have to include high density of charge defects in order to account for the observed surface charge in the semiconductor. Using a model suggested by Fromholdl5), a calculation has been carried out which suggests that oxides in the range from 5-20 A could account for the observed surface charge with oxide defect densities near 1020-102’~cm3 (ref. 13).

DEGENERACY

The simple discussion research.

presented

First, is gas-surface

above points

interactions

23

EFFECTS

out two important

of lead sulfide compounds

areas of and the

resulting surface charge generation mechanism. Second, is the effect of large surface charge densities on the surface potential well. As indicated earlier, it would be expected that there should be a marked distortion of the surface well (leading to a larger surface potential resulting from the high surface charge densities). A contact potential experiment might be employed to observe this distortion. If surface quantization makes an important contribution to the surface potential, it would be expected that the classical value of the barrier height would be lower than the observed value. The difference depends on the choice of effective mass component perpendicular to the surface. It has been shown that the simple linear model of eq. (1) fails when multi-valley effects are includedrs). It then becomes necessary to obtain values for the quantum correction from a solution of the coupled Poisson and Schroedinger equations. The fact that the mobility is independent of surface charge in PbSe may indicate that surface quantization is present at these surfaces. If surface quantization is present, the wave functions of the carriers in the electric sub-band vanish at the surface. Therefore the overlap integral between the carriers in the electric sub-band and the scatterers at the surface should be reduced and the mobility of these carriers should be largely independent of surface potentialrs). The high dielectric constant of the lead salts should also favor restricting the charge carriers to the interior with little overlap into either the oxide or vacuum.

4. Silicon surfaces In 1966, Fowler et al.17,1s), observed an oscillatory magnetoconductance in n-channel enhancement mode MOSFET’s. These devices had 100 Q-cm p-type bulk conductivity but an n-channel or inversion layer was generated by an electric potential between the gate and the contact electrodes. The insulator was 5330 A thick oxide film. Interpretable data were obtained only for (100) surfaces. Howard and Fanglg) have shown theoretically that the strong electric field lifts the 6-fold valley degeneracy of the (100) surface producing 2 sub-bands having 2-fold and 4-fold degenerate valleys respectively. The 2-fold degenerate sub-band arises from those valleys with longitudinal effective masses perpendicular to the surface and the 4-fold degenerate sub-band arises from those valleys with transverse effective masses perpendicular to the surface. The calculations also indicate that the energy difference between these split off electric sub-bands is sufficiently large so that at low temperatures (= 1 to 4 “K), the conduction electrons should be in the lowest band for surface field strengths normally achievable (< lo7 V/cm).

24

3. N. ZEMEL

AND

M. KAPLIT

Pippardsl) has discussed in some detail the dynamics of electron conduction in a normal three-dim~nsiona1 electron gas when a magnetic field is present. In such a system, the oscillatory magnetoresistance is a direct result of the periodici~ in the density of states produced by magnetic quantization. The density of states increases with energy as s* in a threedimensional gas, The density of states in each magnetic sub-band (or Landau level) therefore increases because the average density of states must be perserved in a magnetic field. On the other hand, in a two~dime~sional electron gas, the density of states is independent of energy so that the number of states in each Landau level will be constant. In a MOSFET structure, the amount of charge in the channel is directly proportional to the gate voftage and, as a result, the gate voltage is a function of the Fermi level in the channe1 region. The dependence of the Fermi level on the gate voltage is a function of whether the gas behaves Iike 2- or 3dimensional gas, For a 3-dirne~si~~al gas, 8,&V+, whife for the 2-dimensional gas

where N is the bulk density of charge and ;iv,is the surface density of charge. As the Fermi level crosses the Landau levels, charge is stored in the level leading to an initial rise in conduction. As the Fermi level increases, the Landau Level becomes tilled and the conduction decreases. Any additionat charge must be stored in the depletion layer until the next Landau level is reached by the Fermi level. fn the three-dimensional case the spacing between the conduction maxima would increase since the gate voltage would be proportional to $, whereas in the two-dimensional case the spacing will be uniform since &cc+. In fig. 5 we show the density of states in a magnetic field illustrating the 2~dimensional character of the carriers in the inversion region of a (100) oriented silicon n-channel MOSFET. The separation energy between these levels is given by ds=liCD,, (4) where fi is PIanck”s constant ad w, = gBfmr is the cyclotron frequency with 4 the electronic charge, B the magnetic induction and rrz* the effective mass in the piane perpendicular to the direction B. Since there are two valleys with the longitudinal effective mass normal to the surface, the valley degeneracy for the lowest quantized level is expected to be 2. Using this value of the valley degeneracy and the observed spacing of the oscihations, FowIer et aI. were abfe to show that the effective mass obtained from the osciliations

DEGENERACY

25

EFFECTS

ENERGY

Fig. 5.

Density of states of a two-dimensional electron gas in the presence of a magnetic fleld (Landau levels).

was in good agreement with the transverse effective mass obtained by other authorsso). The periodic density of states due to magnetic quantization of the first electric sub-band suggested another experimental approachzl). Surface capacitance, C,, is a measure of the density of states in the surface space charge region. This can be readily seen by considering the capacitance of the first electric sub-band 21) 4mn*e2

” When

= ar<

1

h2kT Ed

1

1 + exp [(.s_/kT]

cexp

at very low temperatures,

a%

[(sr - E~)/~T] atj,’

C, is substantially

(5)

zero but when

sr
c, =

hem*

a&,

h2kTG,

(6)

Thus the “turn-on” of a MOSFET capacitor at low temperatures is directly connected to the energy of the first electric sub-band in the SSCR. The surface capacity is in series with the capacitance of the oxide region, C,,. Therefore the total capacitance of the MOSFET is 1 1 --=_+__ C C,

1 C,,

Where C, is small, C is dominated by the capacitance of the SSCR. But when C, is large (E~>Q), C will be very soon close to C,. When a large magnetic field is applied to the system, there is a further quantization of the levels because of the generation of Landau levels. The

26

J. N. ZEMEL

AND

M. KAPLIT

shape of the Landau levels at these temperatures is approximately Lorentzian in character. This is shown in fig. 5. As the gate voltage increases, the Fermi level moves into the first Landau level. Using a similar argument as the one used above, we find 4nm* e2

a&,

kT at,b,1 + exp ([Q + (a + n)Aa,, - +]/kT} 1

’ 1+ exp

{[or - (t + n)Izo,

- e,]/kT)

(8)



where H is the order of the Landau level without respect to either spin at valley degeneracy. When sr Z!E~+(Ji-?l)ko,, one finds that C, is large, but C, is small when El + (3 + n)lio, < Ef < Er + ($ + Il)fio,, (9) i.e., when sr is between the Landau levels, then C, tends to zero and therefore C tends to zero. As a result, surface quantization would give rise to a series of notchlike oscillations as shown in fig. 6. These notches occur at the minimum between the Landau levels rather than at the maximum. The induced charge in the MOSFET will be stored either at the edge of the depletion region or at the electrodes contacting the inversion region. The

Device*8075C 150 kOc 1.3OK

<

( -10

0

I

IO

I

1

I

I

I

20

30

40

50

60

GATE

Fig. 6.

VOLTAGE

Variations of capacitance of a MOSFET transistor measured between the gate and the shorted source-drain-substrate in a magnetic field 150 kOe at a temperature of 1.3 “K.

21

DEGENERACY EFFECTS

surface capacitance therefore, provides as sensitive a measure of the location of the Landau levels as the Schubnikov-de Hass measurement. Examination of eq. (8) indicates that another measurement can be carried out with the surface capacitance. As noted, the capacitance of the spacecharge region is substantially zero until the Fermi level comes within a few kT of the first Landau level. The zero-order term in the energy of the Landau levels should produce a shift in the turn-on voltage of the surface capacitance, an example of which is shown in fig. 7 where the capacitance of a MOSFET structure is plotted as a function of gate voltage for several values of magnetic field. In the low field region, co,z values are quite small and as a result the line shape is dominated in large measure by the scattering processes in the SSCR. Stern and Howard7) have discussed bound states due to charges in the oxide layer and have calculated that they should have an effect on the scattering of the 2-dimensional gas. Detailed studies of the turn-on effect should provide useful information on the bound state. Finally, it should be noted that the depth of the notches decreases with increasing gate voltage. In a recent paper, Fang and Fowler2s) show that the Hall mobility of carriers in the SSCR goes through a pronounced maximum as the gate voltage increases above turn-on. At these temperatures this maximum is less than 15 V for 3000 A thick oxide MOSFETs at 1.3”K (the approximate oxide thickness and temperature used in these measure-

-0.2

0

0.2

0.4

0.6 GATE

Fig. 7.

0.6

1.0

1.2

1.4

1.6

1.6

2.0

VOLTAGE

Variation of turn-on voltage of a MOSFET capacitor similar to the one used in fig. 6 with magnetic field.

28

J. N. ZEMEL

AND

M. KAPLIT

ments). Beyond this maximum, there is a precipitous drop in the mobility that as o,z decreases the depth of the notch gradually rises.

so

5. Conclusions At present, there is a rapid growth of interest in surface quantization as new measurements techniques become available. For one of the more promising cases, that dealing with a degenerate surface and a degenerate bulk, specific methods for studying surface quantization are still not available. However, there are interesting problems involving surface interactions and the validity of an oxide or surface state model when large surface charges are present that require additional study. In the case of silicon, the Schubnikovde Hass and related Landau level phenomena have established that twodimensional electron gases can be generated on technologically available silicon surfaces. The behavior of such gases is not understood at all and will be the subject of increasing study. Scattering processes involving ordered and random spins can be examined by properly doping the oxide. The effect of impurities on the “bound” state of Stern and Howard can be examined in detail with the “turn-on” shift of the surface capacitance with increasing magnetic field. In both the Schubnikov-de Hass and the surface capacitance measurements, meaningful surface quantum effect was only seen on (100) surfaces. This raises serious experimental and theoretical problems since one cannot conclude that the surface quantization well established until corresponding phenomena have been observed on both (110) and (111) surfaces. These measurements should also be extended to other materials such as germanium or the III-VI compounds. In the case of germanium with its four-fold valley degeneracies, the (111) surface would give rise to an electric sub-and with a single valley degeneracy, as in the case of the silicon (100) surface. The longitudinal effective mass produces the lowest level. It should be interesting to see if either the surface capacitance or surface conductance measurements can give any information on the existence of the second quantum level. The combined conductance and capacitance measurements will give a wealth of detail concerning the detailed shape of the Landau levels. Acknowledgement The authors would Dr. A. B. Fowler, F. surface quantization. with Dr. M. Brodsky

like to acknowledge several valuable Stern, and W. Howard concerning We would also like to acknowledge concerning his unpublished work.

discussions with their work on the discussions

DEGENERACY

EFFECTS

29

References 1) A. B. Pippard, Dynamics of Conduction Elecfrons (Gordon and Breach, New York, 1965). 2) A. N. Friedman and S. H. Koenig, IBM J. 4 158 (1960) 158. 3) J. E. Aubrey, C. James and J. E. Parrott, in: Proc. Intern. Conf. on Physics of Semicond., Paris (Dunod, Paris, 1964). 4) R. Siewatz and M. Green, J. Appl. Phys. 29 (1958) 1034. 5) D. R. Frankl, J. Appl. Phys. 31 (1960) 1752. 6) J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). 7) F. Stern and W. Howard, Phys. Rev. 163 (1967) 816. 8) W. Howard, unpublished. 9) M. H. Brodsky, unpublished. 10) M. H. Brodsky and J. N. Zemel, Phys. Rev. 155 (1967) 780. 11) R. Egerton and C. Juhasz, Brit. J. Appl. Phys. 18 (1967) 1009. 12) I. A. Berezhnaya, Russian J. Phys. Chem. 36 (1962) 1500. 13) J. N. Zemel, to be published in: Proc. Battelle Kronberg Colloquium on Molecular 14) 15) 16) 17) 18) 19) 20) 21) 22)

Processes on Solid Surfaces. 0. Kellogg, Foundations of Potential Theory (Dover, New York, 1953).

A. H. Fromhold, Jr., J. Chem. Phys. 40 (1964) 3335. R. F. Greene, Surface Sci. 2 (1964) 101. A. B. Fowler, F. F. Fang, W. E. Howard and P. J. Stiles, in: Proc. Intern. Conf. on the Phys. of Semicond., Kyoto, J. Phys. Sot. Japan Suppl. 21 (1966) 331. A. B. Fowler, F. F. Fang and W. E. Howard, Phys. Rev. Letters 16 (1966) 901. W. E. Howard and F. F. Fang, Phys. Rev. Letters 16 (1966) 797. G. Dresselhaus, A. F. Kip and C. Kittel, Phys. Rev. 98 (1955) 368. M. Kaplit and J. N. Zemel, Phys. Rev. Letters 21 (1968) 212. F. F. Fang and A. B. Fowler, private communication.