Hartree calculations for n-inversion layers on stressed silicon surfaces

Solid State Communications, Vol. 26, pp. 349-352. 0 Pergamon Press Ltd. 1978. Printed in Great Britain.

00381098/78/0508-0349

$02.00/O

HARTREE CALCULATIONS FOR n-INVERSION LAYERS ON STRESSED SILICON SURFACES H. Tews* 1. Institut fur theoretische Physik, Universitat Hamburg, 2000 Hamburg 32, Federal Republic of Germany (Received 18 August 1977; in revised form 8 December 1977 by ikiollwo)

The energy splittings and occupation number densities in n-type inversion layers are estimated in a Hartree calculation with a parametrized exponential potential. Good agreement with experiment is obtained for low temperatures and for electron concentrations 1 x 1012 cmm2
[

I

-2%g2+YIAz) + vwrt(z) 3

externally applied stress. In the usual approximation the density of ionized impurities p&p1 (z) is COnStaM in a layer between z = 0 and z = z&pr, and zero outside [ 11. Equation (1) is a one-dimensional equation with an effective mass m3 normal to the surface, parallel to which translational invariance is assumed. For Vr, (z) an exponential ansatz with two parameters Ve and c is considered as was done previously by Duke [7] for accumulation layers: V,,(z)

= [

V, [ 1 - exp (- cz)] m

for z > 0; forz
The boundary conditions for the J/i (z) are Jli (0) = 0 and J/i (-) = 0. The surface field F,, fixed by the gate voltage and by the oxide thickness, yields a correlation between the three UnknOWII quantities Fe, c and z&&,1: (3) Thus only two of them are independent. E is the static dielectric constant of silicon. Using the definition of an electrical field, equation (3) can be replaced by an equivalent expression: Voc =
(4)

e

$i tz) = E;:J/i (Z)

(1)

contains a potential Vi,” (z) produced by the inversion layer electrons, and a part Vpert (z) caused by the ionized acceptors in the depletion layer and by an

where N, = 7 Nr, and Ni is the occupation number density of the ith energy level in a two-dimensional system. If Vpart (z) in equation (1) is small enough, it can be treated as a perturbation. Equation (1) with initial values 6 and co then can be solved by Bessel functions: tip(z) = AYIY[&$exp

* Present address: MPI fti Festkorperforschung, 7000 Stuttgart 80, Federal Republic of Germany. 349

(2)

(-$)I.

(5)

n-INVERSION

350

,,.,, ,_..,.......

LAYERS ON STRESSED SILICON SURFACES

Vol. 26, No. 6

Vpert (2)

....‘..‘.

0

50

I

100

150

Ando

200

Fig. 1. The total potential V(z) is shown for a (lOO)surface with N, = 5 x 1012 cm-?. The first excited level of the lowest subband system and the ground level of the second subband system coincide.

y2

Kneschaurek

I

z(A)

The A, are normalization

EPM

l

-

..““.

,(,,,,,,,.......

0

factors. For Y’ we get

- 8m30’: -6’)

_

L

0

I 1

I 5

2 N S t:o12 crnf2

6

1

Fig. 2. The calculated splitting El - E. is compared with experimental data of Kneschaurek [8,9] and with a theoretical curve of Ando [6] . (EPM = .exponential potential model). Ndepl = P&pi z&pi is the two-dimensional density of ionized acceptors in the depletion layer.

h2 c2

The eigenvalues .f$’ are improved in a first-order perturbation calculation. One finds new values VA and c1 from the Hartree potential of the JIy (z) [ 1,2]. By use of a numerical two-parameter interaction one can show that there exists a unique set I’,, , c such that (i) I’, and c come out self-consistently, and (ii) equation (4) is fulfilled. Because equation (1) has to be solved for each of the conduction band minima, it leads to up to three subband systems in the case of silicon. The procedure described above is only useful if Vpert (z) really is a small quantity. Figure 1 shows the total potential V(z) = I&(z) + I’Mrt (z) at the interface for N, = 5 x 1012 cmW2. The dashed and the dotted lines indicate the potential I&.,“(z) and I&&z), respectively. One finds that for small z the total potential deviates only little from V,,(z). Thus, the low-lying eigenvalue E,, will be fairly correct. For higher levels of Ei the error caused by Vpert (z) becomes important. An analysis of the perturbation method used in equation (1) shows that our calculations with a simple parametrized exponential potential should yield good results for the energy splittings and the occupation number densities at low temperatures T < 100 K and at inversion electron concentrations 1 x 1012 cmm2
temperature dependence of Ei - E,, for small A$ which is in agreement with experimental results [9]. The splitting EL -E. between the ground states of the two subband systems on a (100)surface is confirmed by Shubnikov- de Haas data [ 10 1: (E6 --Eo)

theor. = 47 meV; (PO -E,)

exp. = 46 meV

at N, = 7.4 x 1012 cms2. The influence of stress is considered within the deformation-potential model of Herring and Vogt [ 1 l] . To first order a deformation F caused by a uniaxial compression P has the effect of shifting the conduction band minima energetically without changing its shapes or positions in k-space. For the ith valley we get AEi = C ~ !i,’ E,, p,s

(7)

where g(i) IS * the deformation potential tensor. The AEi are treated as perturbations on the eigenvalues of equation (1). For an arbitrary orientation of the stress, the shifts AEi are different for all pairs of valleys i, thus partly removing the degeneracy of the energy levels. It is assumed that internal stresses at the Si-SiOZ interface are much smaller than the applied ones. 3. RESULTS The effect of a 001~stress along a (100)surface is discussed for temperatures T = 4.2 K and T = 77 K. In the stress-free case there are two subband systems at the (100)~surface: E. , El , . . . and El,, E; , . . . , resp. A compression in the [OOl]-direction removes the 4-fold degeneracy of the primed levels to two two-fold

n-INVERSION LAERS

Vol. 26, No. 6

ON STRESSED SILICON SURFACES

351

P (Ntm1-~1

E;-

E, 0 150 300

T -4.2 K OOl-Stress si (1001

L /

I

I

~_

5

0

‘”

NS

(10'2cm-21

Fig. 3. The energy splitting EA -E. between the ground states of the two subband systems is calculated as a function of N, for some values of the compression P. At about Ns = 1 x 1012 cmm2and P = 300 N mme2 the two energy levels coincide.

Q

100

260

Pressure

300

I N rnmm2 1

Fig. 5. The occupatjon number density A$is plotted vs pressure for the E. and the E&level. At 4.2 K a total transfer is found for P = 350 N mm-*. The difference N, -No - NA is due to the occupation of excited levels.

60P (Nmms2 300 150 0

5

l&5

r

60-

Si

(100)

100

100 Pressure

8 2QQ

1 300

(N n~tn-~)

Fig. 4. ‘l’heshift of the ground state energies as a function of the compression P is shown for two tem~ratures T = 4.2 and 77 K. At 77 K the energy shifts are linear with the stress, at 4.2 K one finds a break where the occupation of the .&-level starts.

0

5

lo

NS

(10'2cm-2~

Fig. 6. The sp~t~g Er -E. within the lowest subband system is plotted vs N, for some values of P. The stress dependent change in the splitting increases with N, and with P.

n-INVERSION

352

LAYERS ON STRESSED SILICON SURFACES

degenerate levels. The energy levels of the 00 1-valleys C&E:,. . .) are shifted down, those of the OlO-valleys (Eb’, E’,‘, . . . ) are pushed up. Thus the splitting EA - EO is reduced as a function of the applied stress. In Fig. 3 this is demonstrated for two compressions P = 150 and 300 N mm-‘. At N, = 1 x 10” cm-*, the change in the energy splitting EA - EO between zero stress and P = 150 N mm-* is equal to the value predicted by the deformation potential model. With increasing electron density the influence of the stress is reduced as a consequence of the self-consistency to about half that valueatA!,=l x 1013cm-2.TheP=300Nmm-2 curve shows at N, = 1 x lOI* cm-* a shift of the Ehlevel beyond EO; because of the larger density-ofstates effective mass of Eh, more than 75% of the inversion electrons then populate the EA-level. In Fig. 4 the shifts of the ground states are shown as a function of pressure for two temperatures. In both graphs the electron density is N, = 1 x lOI* cm-*. The corresponding occupation number densities Ni are plotted in Fig. 5. At g c;mpression of about 350 N

Vol. 26, No. 6

mm-’ , a complete transfer is achieved from EO to Ei . This transfer is well confirmed by Shubnikov-de Haas data [ 121. In Fig. 6 the splitting E, - EO is plotted vs A’, for various pressures. One finds a stress-dependent shift of El - EO, which is small for small concentrations but which increases with A$. It should be possible to observe this effect directly with optical intersubband resonance experiments. Summarizing, we state that the effect of a mechanical stress along the surface of an Si-MOS-transistor on the splitting EA - EO is strongly reduced for high inversion electron concentrations. For small N,, a total transfer from one subband to another can be achieved by experimentally reasonable pressures. The energy splitting within a subband is found to depend on stress.

Acknowledgements - We would like to thank Professor J. Appel and Drs M. Pfuff and P. Hertel for many valuable discussions.

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