Surface quantization effect on the macroscopic characteristics of semiconductor space-charge layers

329

Surface Science 147 (1984) 329-342 North-Holland, Amsterdam

SURFACE QUANTIZATION EFFECT ON THE MACROSCOPIC CHA~CT~RISTjCS OF SEMICONDUCTOR SPACE-CHARGE

LAYERS

Ch.M. HARDALOV Instituie of Solid State Physics. Bulgarian Academy of Sciences, 1184 Sofia. B&aria

and E.P. VALCHEVA

and K.G. GERMANOVA

Solid State Physics Department, Received

2 April 1984; accepted

Sofia Unioersity, I I26 Sofia, Bulgaria for pubhcation

23 July 19X4

A numerical study of the dependence of the space-charge density Q, on the surface potential $, in a tw~dimensionai electron inversion layer on Si(100) is presented. The dependence Q,(#,) being a macroscopic characteristic of ~miconductor space-charge layers is found to be considerably influenced by the surface quanti~tion in a wide temperature range up to room temperature. The conditions for the appearance of appreciable differences in the behaviour of the Q,(+,) curves in the quan!um-m~hanical and quasi-classical cases are determined and discussed.

1. Introduction Electrons or holes in space-charge layers in metal-insulatorsemiconductor structures are bound in discrete energy levels with respect to their motion perpendicular to the surface. The charge carriers are free to move parallel to the surface thus forming a quasi-two-dimensional electron (or hole) gas (2DEG) [l]. Recently the experimental and theoretical investigations of the electronic properties of 2DEG have greatly expanded because they contribute significantly to the understanding of various fundamentally interesting phenomena and the operation of many modern devices [1,2]. Nevertheless, there exist a number of problems that can be considered to remain still open. For example, little is known about the way in which the macroscopic characteristics of the space-charge layers f3] can be affected by the surface quantization [4--Q. The detailed study of this problem is important because of the following reasons. First, it is necessary for a further improvement of the quantitative analysis of a number of electric properties of the quatized surface layers (such as capcitance, conductance, mobility, photoelectric phenomena, etc.) in the absence of high 0039~6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

330

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et al. / Surface quanttration effect

magnetic field [4-71. Second, new possibilities for the experimental verification of the quantum nature of the 2DEG can be revealed [4]. In this paper we present a computational study of the influence of the quantized carrier motion normal to the surface on one of the main macroscopic characteristics of the space-charge layer - the dependence of the charge density Q,, on the electrostatic potential 9, at the surface. The main features of the function Q,($,) under thermal equilibrium conditions have to be calculated by a self-consistent solution of the Schriidinger and Poisson equations in the effective-mass approximation [4,8] for the case of an n-type inversion layer on Si(100). In order to estimate the importance of the quantization of the motion perpendicular to the surface the results of calculations for the two-dimensional (2D) case are compared to those obtained by a conventional calculation of the dependence Q,,(+,), the quantization being neglected (i.e. 3D case) [3]. So far as the Boltzmann statistics is not valid at the conditions when the quantization occurs [1,4], the Fermi-Dirac statistics is used in the 3D case calculations [9]. The conditions for the appearance of appreciable differences between the curves Q,,(+,) in both of the considered cases are determined and discussed.

2. Problem fo~ulation

The calculations are carried out under thermal equilibrium conditions for an inversion layer in a MOS capacitor based on p-type Si(lO0) containing only shallow acceptor impurities. The acceptor concentration is assumed to be independent of z (the distance normal to the interface and measured from it into the semiconductor). The space-charge density is given by (1)

Qsc = - 4( Ninv+ dna ). N,ny is the total number of charge carriers in the inversion NA is the net concentration of ionized acceptor impurities. d=

2.5 -+-r”+ WJ I [ 4 NA(~

layer per unit area. The quantity

r/2 (2)

is the depletion-Iayer width, where E, is the semiconductor permittivity, E, is the Fermi level and W, is the distance between the conduction band edge EC and the Fermi level in the bulk. In order to calculate the function Q,,(+,) it is necessary to find N,,, and the Following potential value 4, = G( z)]~,~ at the surface of the semiconductor. refs. [4] and [S], these quantities are determined by a self-consistent solution of

Ch. M. Hard&o

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331

the Schrodinger and Poisson equations in the effective-mass approximation, The Schrbdinger equations for the subband energies E,” and the corresponding wave functions t;(z) are

where mi_ is the effective mass for ith energy level for electrons in functions Q(z) are supposed to be points of the surface potential well dz=l,

@(.-,I2

t:(z)=0

at

the motion in the z-direction and Ei is the the jth valley. The corresponding wave normalized and to turn to zero at the end (wave function boundary conditions), z=O

IJJ+~~(z) is the potential due to the ionized determined in the Schottky approximation,

Z-M).

acceptors

(3b) in the depletion

layer

-%(zdSt”), .\

I&&)=

where d is given by eq. (2). 4 n(z) is the Hartree potential Poisson equation

with boundary

t&(4=0,

of the inversion

electrons

determined

by the

conditions

2

The electron given by 4)

and

=0

at

concentration

= CWllrcz,j2? 3.j

z=d.

n(z)

per unit volume

in the inversion

layers is

(6)

where

is the total number of electrons per unit area in the subband E/‘ and rn$ is the density-of-states effective mass for the motion parallel to the surface. Ni,, in eq. (1) is given by Ninv = ch$ ‘.I

(8)

Ch.M

332

A suitable self-consistent repeating the different N,,,

Hardalou et al. / Surface

quantiration

eJJecf

numerical procedure [lo] has been used in order to find a solution of eqs. (3)-(7). The dependence Q,,( #\) is obtained by calculations for a number of different E, values resulting in and +, and using eqs. (1) and (2).

2.2. Calculation of Q,, (I),) for a quasi-classical space-charge

luyer

The calculation of Q,,( 4,) in a space-charge layer with the conventional continuum motion of the carriers in the z-direction is carried out by integrating the one-dimensional Poisson equation only. The same MOS capacitor as in section 2.1. is considered. Thus using the Fermi statistics and taking into account the incomplete ionization of acceptors we find [9]:

1 +gA

exd&-

’ +gA

Table 1 Parameters used in the calculations layer on Si(100) Bulk

EA+qk)/kT]

exd(EF- EA)kT]

of the dependence

1

Q,( +,) for the case of an electron

parameters

Energy

gap Es

(W

Longitudinal effective mass, m, /m, Transverse effective mass, m, /m, Electron density-of-states effectwe mass, M nd /m0 Hole density-of-states effective mass, m nd/mO Permittivity of semiconductor, ~~/ce Spin degeneration factor of the acceptor level, g, Effective densities of states in the conduction band, A, (mm3 Km3”) Effective densities of states in the valence band, A, (mm3 Km3j2) Energy position of the acceptor level, EA (eV) Surface

1.1557-7.021~10~‘?~/(1108+T) 0.98 0.19 1.0839 0.5586 11.6 f/4 5.45 x 102’ 2.21 x 102’ 0.045

parameters

2 Valley 1

Valley 2

Valley degeneracy factor, n v Effective mass perpendicular to the

2 0.98

4 0.19

surface, m,/m, Subband density-of-states

0.19

0.43

Number

of valleys with different

effective masses

effective

mass, md/ma ee is the permittivity

of vacuum;

me is the free electron

mass.

inversion

Ep is the band-gap energy, EA is the energy position of the acceptor level, g, is the spin degeneration factor of the acceptor level, F,,,, are the Fermi integrals of order 3/2, Actvk = Nctvt T-j/*, where NC and NV are the effective densities of states in the conduction and valence bands respectively, and T is the absolute temperature.

I 5

I

I

I

I

I

T

I

2

-

1 2

-

l&2 -

7 .G=

-

f

s

c

2 e2

-

1c3 -

t 5

a

/ E

2 k

/(, a9

1.0

, , 1.1

1.2

1.:3

Fig. 1. Space-charge density Q,, as a function of the surface potential +, in the inversion layer of a MOS capacitor on p-type Si(100). Curves 1 represent the two-dimensional case and curves 2 the quasi-rlassical one. Marks a indicate the onset of the strong inversion region; marks b indicate the fulfillment of the condition E, > E, - q#s - kT, marks c indicate the fuIfillment of the condition E, > EC - q$% f Ei - kT; marks c’ indicate the fulfilIment of the condition E, > EC - qlts + EA. N,=1.65~10~’ mW3 and T= 300 K.

The comparison between the results for the quantized space-charge layers and for the quasi-classical ones for various physical and technological reasonable parameters are given below.

3. Results and discussion The parameters used for the calculations are listed in table 1. Moreover acceptor concentration NA and temperature T are also needed. Cases for N, = 1.65 X 102’, 5 X 10” and 1 X 1O23 mP3 at each of the temperatures T = 300, 77 and 4.2 K are considered. The dependences Q,,( 4,) obtained in the 2DEG case (curves 1) and by the conventional calculations (curves 2) are shown in figs. 1 to 9, respectively. All energies are counted from the top of the valence band with the exception of the levels E,J which are counted from the bottom of the potential well. In all the

I

I

I

I

I

I

I

I

5

5

2

EC

I--1G4

7!trk+d

Fig. 2. As fig. 1, but for IV,=~X~O~~

3

mw3 and T=300

K.

Ch.M.

Hordalov

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ef’fect

335

cases appreciable differences in the shape of the curves 1 and 2 appear for I/J, values for which the condition E, > EC - qqs - kT is satisfied. The differences enhance with the +, increasing and become best pronounced for 4, values where E, a EC - qGs + Ei - kT. In order to understand the origin of these differences we consider the following influences. On the one hand the subband energy splittings AE; increase with the GcI,increasing [1,4], i.e. the quantum-mechanical nature of the 2DEG becomes better expressed. This suggests an increase of the differences between the curves 1 and 2. On the other hand, as 4, increases the inversion electron gas degeneration in the 2D and 3D cases changes in a different way. The electron gas in the 3D case goes into degeneration if the condition E, > EC - q$, - kT is realized. With the realization of the condition E, a EC - qGs + E; - kT,

I

I

I

0.8

0.9

I

5

2

I

a

IL?

0.6

0.7

I

I

I

I

336

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Surface quankwtion

effect

the electron gas in the 2D case begins to degenerate while in the 3D case the electron gas is already completely degenerated (now E, B EC - q$, - kT, since E; > E,). In the range of $,, where E, z+ EC - q+, + Ed - kT, it is possible to obtain approximate analytical expressions for the dependence Q,( J/,), As in this range the inversion is strong enough we can assume that the total charge in the space-charge layer is determined only by the free electron charge. At the same time the gas formed by these electrons is strongly degenerated. Thus eqs. (I) and (9) can be approximated by:

(11)

I

I

I

I

I

I

r

2

1

~ c

b /y

,~~~ 0.6

0.7

0.8

0.9

1.0

Fig. 4. As fig. 1, but for NA = 1.65 x 10”

1.2

1.

mv3 and T = 77 K.

Ch.M.

Hard&o

et al. / SurJace quantiralion

effect

331

The dependence Q,,(+,) given by eq. (11) follows the power law with an exponent of 5/4. The first term in eq. (10) depends linearly on #, while the second one in a more complex manner. The second term, however, is one order of magnitude lower than the first one and is subtracted from it. So, evidently, in the 2D case the space-charge density increases relatively more slowly with 4, than in the 3D case. In the range of I,!J,values where EC - q+, - kT < E, < EC - q#s + EA - kT the behaviour of the 2D and 3D curves Q,, (4,) is difficult to compare analytically; moreover it changes in a complex manner with the temperature (figs. l-9). At T = 300 K some differences in the shape of the curves 1 and 2 are observed also for E, < EC - q#, - kT (figs. l-3). They are probably due to the fact that the electron gas transition from non-degenerate to degenerate state at high temperatures is not abrupt. With temperature decreasing at NA = const. or acceptor concentration in-

I

I

I

I

I

r

I1

I

/ / 2 1

~ C

b 5

-

2

-

lt?

0

EC

I

I

I

I

0.6

0.7

0.8

0.9

I 1.0

1 1.

/rl 1.2

1.:

qY,ieV) Fig. 5. As fig. 1, but for NA = 5 x lo** mm3 and T = 77 K.

338

Ch.M.

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et al. /

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quantiration

effect

creasing at T = const., the strong inversion region shifts towards higher values of $J,. This region in all cases considered begins at $, values for which the steep increase of the Q,,($,) curves starts (figs. 1-9, marks a). At T = 300 K and T = 77 K the onset of strong inversion coincides with the conventional criterion I+,1 > 2)$,,j [3,11] d erived under the assumption that the Boltzmann statistics is valid (4 ,, is the potential difference between the Fermi level in the bulk of the semiconductor and the intrinsic Fermi level). At T = 4.2 K for all NA the onset of strong inversion in the 3D case occurs at \c/, values for which EC - 44, -C E, < EC - q+, + Ed, while in the 2D case it is markedly shifted to higher I/J,, where E, a EC - qx,bs + EJ (figs. 7-9, marks a). Therefore, at 4.2 K, right at the beginning of strong inversion, the inversion electron gas is completely degenerated in both 3D and 2D cases. On the contrary, at 300 K, in the beginning of strong inversion, EC - q$s - kT < E, < EC - qqs + EA - kT, the curves 1 contain a region corresponding to the Boltzmann approximation and the curves 2 contain a region related to an incom-

I

I

I

I

I

I

I

I

/ / 2 1

~ C

-b

2

ll+

-

EC

I

0.6

I

I

I

0.7

0.8

0.9

I

1

1.0

1.’

/r[ 1.2

1.:

qY,(eV) Fig. 6. As fig. 1, but for NA = 1 X 1O23 md3 and T = 77 K.

Ch.M. Hardalooet al. / Surfacequantirationeffect

339

plete degeneration of the electron gas (figs. l-3). At 77 K a transitive situation occurs: the regions mentioned above decrease (figs. 4-6) and disappear completely at 4.2 K (figs. 7-9). For all NA the transition from the weak inversion or depletion region to the strong inversion region becomes more abrupt as T goes down. The effect of T and NA on the Q,,($,,) behaviour in the range of strong inversion can be understood from the following considerations. At 4.2 K the stepwise shape of the Fermi-Dirac functionf( E) at E = E, is quite steep. The density of states LI;‘( E) in the 2D case has also a steep step shape at E = Ei (only the lowest level is occupied [1,4]). So in the 3D case with +, increasing the inversion electrons in the conduction band will appear when the Fermi level enters the band (E, > EC - q#,) and in the 2D case when the Fermi level crosses the bottom of the zero subband (E, > EC - qqs + E,j). It is obvious that the inversion electron gas will be degenerated in both cases. Since EA > EC the onset of the strong inversion region in the 2D case is shifted to the higher

I

1

I

I

I

I

I

r

5

2

IO3

5

a(301

2D):

164 -

0.6

0.7

0.8

0.9

C' .i

1.0

1.1

3

qYs(eV)

Fig. 7. As fig. 1, but for MA =1.65

X

102’ rnw3 and T= 4.2 K.

#,_

The steep run of the functions f(E) and L.)/(E) explains the sharp transition from the depletion to the strong inversion region. As the temperature increases the energy spacings between the levels E,’ become comparable or smaller than the thermal energy kT, which leads to the occupation of the higher energy levels as well [1,4]. So the quantum-mechanical case approaches the quasi-classical one. Then the onset of the strong inversion region at high and moderate temperatures becomes common for the 2D and 3D cases, it shifts to the lower 4, and coincides with the classical criterion /+/,/ = 21$hl. Its energy position changes with T and NA according to the requirements of this criterion [3,11] (figs. l-6). Besides with temperature increasing the Fermi-Dirac function changes smoothly with the energy. The width of the energy interval where the change occurs, increases. This effect and the approaching of the quantum-mechanical density of states to the quasiclassical one explains the smooth transition from the weak to the strong inversion at moderate and high temperatures.

I

I

I

I

I

I T

1

I

-

2

1 a(301

~

c1(2IY:c’

I 0.6

1 0.7

I 0.8

I

I

0.9

1.0

1 2,

I.’ I

1.2

1.3

qU:IeVI

Fig. 8. As fig. 1, but for NA = 5X1O22

me3 and T=

4.2 K.

Ch.M.

I

I

I

Hardalov

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et al. /

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Surfnce

I

quantization

effect

341

I

/ /

YE 0 v a

5 :: 2

12 a(2Dl:c’

l$

2

0.6

0.7

0.8

0.9

Fig. 9. As fig. 1, but for N, =1x10*’

1.0

1.’

mm3 and T=4.2

1

K.

4. Conclusion

We have calculated numerically the dependence of the space-charge density Q,, on the surface potential #, for an n-type inversion layer on Si(100) with quantized motion of the electrons normal to the surface. The results indicate that the dependence Q,( #,) being a macroscopic characteristic of semiconductor space-charge layers is considerably affected by the quantization up to room temperature. This characteristic is commonly used in the analysis of different electrical properties of the semiconductor surface [3-7,111. Thus, it should be concluded that for the better understanding of this properties the effect of quantization must be taken into account even at room temperature. Furthermore, the study conducted here enables us to demonstrate in a new manner the quantum nature of the 2DEG in inversion layers in a wide temperature range. This can be done comparing the experimental Q-U curves measured by the

342

Ch.M. Hardalou et al. / Surface quantiration effect

precise static technique [12,13] and the corresponding curves computed as it was discussed above.

theoretical

Q,,( +,)

Acknowledgements It is a pleasure to acknowledge many valuable discussions with Dr. S.P. Alexandrova on the experimental aspects of this problem. The work is partially supported by the Alexander von Humboldt Foundation.

References [l] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. [2] For recent reviews, see: Proc. 4th Intern. Conf. on Electronic Properties of Two-Dimensional Systems, New London, NH, 1981 [Surface Sci. 113 (1982)]; Proc. 5th Intern. Conf. on Electronic Properties of Two-Dimensional Systems, Oxford, 1983 [Surface Sci. 142 (1984)]. [3] A. Many, Y. Goldstein and N.B. Grover, Semiconductor Surfaces (North-Holland, Amsterdam, 1971). [4] J.A. Pals, Philips Res. Rept. Suppl. 7 (1972) 1. [5] D.R. Choudhury, A.K. Choudhury and A.N. Chakravarti, Phys. Status Solidi (a) 59 (1980) K69. [6] V.A. Zuev, A.V. Sachenko and K.B. Tolpigo, Nonequilibrium Processes at the Surface of Semiconductors and Semiconductor Devices (Soviet Radio, Moscow, 1977) (in Russian). [7] CD. Kohl, Appl. Phys. A30 (1983) 127. [8] F. Stern and W.E. Howard, Phys. Rev. 163 (1967) 816. [9] Ch. Hardalov, K. Germanova and K. Marinova, Bulg. J. Phys. 10 (1983) 415. [lo] J.H. Wilkinson and C. Reinsch, Linear Algebra (Springer, Berlin, 1971) pp. 227-240. [ll] E.H. Nicollian and J.R. Brews, MOS Physics and Technology (Wiley, New York, 1982). [12] K. Ziegler and E. Klausmann, Appl. Phys. Letters 26 (1975) 400. [13] K.I. Kirov and S.P. Alexandrova, Phys. Status Solidi (a) 49 (1978) 781; K. Kirov, G. Minchev and S. Alexandrova, Compt. Rend. Acad. Bulg. Sci. 29 (1976) 1749.