be taken into consideration simultaneously in a more .... correlation is unsuitable in these cases, and a new model, in ..... nates defined in figure 7, that is to say.
Semicond. Sci. Technol. 10 (1995) 592-600. Printed in the UK
I 1
Oxide charge space correlation in inversion layers 11. Three-dimensional oxide charge distribution
F Gamiz, J A Lopez-Villanueva, J Banqueri, Y Ghailan and J E Carceller Departamento de Electronica y Tecnologia de Computadores, Universidad de Granada, Facultad de Ciencias, Avd. Fuentenueva d n , 18071 Granada, Spain Received 2 August 1994, in final form 30 December 1994, accepted for publication 26 January 1995 Abstract. A Coulomb scattering model which includes the effect of the space correlation of oxide charges on the effective mobility of electrons in an NMOS transistor channel has been proposed. A general three-dimensional oxide charge distribution has been considered in this paper. Using this new model in a one-electron Monte Carlo procedure provides the chance of theoretically studying some effects o! the electron effective mobility in silicon inversion layers (such a s bulk impurity charge, oxide charge distribution profile and charged-centre sign) that cannot be studied with our previous model (1994 J. Appl. Phys. 75 924). These effects have been found to noticeably influence electron effective mobility, which is in agreement with previous experimental results.
1. Introduction
In their 1972 theory of electron scattering by surface oxide charges in semiconductor inversion layers, Ning and Sah [ l ] presented a systematic study of the effects of (i) screening by mobile caniers, (ii) spatial distribution of electrons, (iii) distribution of charged centres in the oxide and (iv) space correlation of charged centres. They investigated all the above effects separately and stated that although some of them are more important than others, none is completely negligible. They therefore concluded that 'none of the effects studied could alone account for the discrepancy between experiment and the theoretical value of electron effective-mobility, This would indicate that a combination of these effects should be taken into consideration simultaneously in a more refined theory of electron transport in semiconductor
surface channels.' In 1986, Yokoyama and Hess [Z], following up on the work of Stem and Howard [3], provided a model for Coulomb scattering in the A1,GaL-,As/GaAs system in which effects (i)-(iii) are taken into account simultaneously. They also considered the effect of the remaining scattering mechanisms, and the effects of screening by mobile carriers populating more than one subband. This model can easily be extended to silicon inversion layers. In 1993 Fischetti and Laux [4] completed such a model, presenting a detailed study on the scattering mechanisms for electrons in silicon inversion layers that allows for the full dependence of 0268-1242/95/050592+09$19.50 0 1995 IOP Publishing Ltd
the screening parameter and accounts for intersubband Coulomb scattering. In addition, the IBM group paper [4] goes into more detail on the electron-phonon interaction and on scattering with interface modes. Joshi et al [5,6], on the other hand, investigated electronic transport in doped semiconductors using a coupled-ensemble Monte Carlo molecular dynamic scheme. In this scheme, each mobile carrier interacts simultaneously with all charges present in the system via the bare Coulomb potential. The combined effect of all such interactions, with contributions from both the ionized impurities and the free carriers, then provides an exact real space calculation of Coulomb scattering, incorporating multi-ion contributions and carrier density fluctuations. However, it requires a powerful computer and is time consuming. In recent papers [7-91 we proposed a comprehensive model for Coulomb scattering in inversion layers that includes the effects of the oxide charge and interface trap space correlation. This model thus takes into account the random nature of the charged centres in the Born approximation (and as we show in this paper, it allows one to distinguish between negatively and positively charged centres), and without the high computational requirements of Joshi et al's scheme [5,6]. In addition, using our Coulomb scattering model in a Monte Carlo simulator detailed elsewhere [7], we have been able to accurately reproduce experimental effective mobility curves in certain situations, as shown in figure 1.
Space correlation of oxide charges
T=300
I
( . . . . . . . I
1o4
1 o5
,
.
subjected to Fowler-Nordheim tunnelling injection (if the substrate doping concentration is low enough in both cases). However, there exist other important situations in which the above assumption is not applicable:
K
.....
I
1
1 o6
Effective Field (V/cm)
Figure 1. Experimental effective mobility curves (full curves) taken from [20],and simulated ones by the Monte Carlo method (symbols) versus the transversd electric field after successive Fowler-Nordheim tunnelling injection series. (Nit in cm-? (1) 1 x loTo,(2)5 x loio,(3) 2 x lo", (4) 3 x 10"; for sudace-roughness L = 15 A, A = 4.2 A.)
To obtain this model we drew upon the work of Stem and Howard [3] and Yokoyama and Hess [2]. Nevertheless, since Ning and Sah showed, the charged centre space correlation has a noticeable effect on electron effective mobility. Therefore, we incorporated the hard sphere model in our simulation. No previous models, with the exception of the papers by Ning and Sah [l], and Siggia and Kwok [lo], have taken this effect into consideration in silicon inversion layers. Nevertheless, there exist in the literature other papers studying on the influence of space correlation of ionized donors in GaA-AIGaAs heterostructures [l I. 121. In the models in which space correlation is ignored, however, the charged centres were treated as independent, thus overestimating the Coulomb scattering, since the interaction among the charges themselves causes the effect of one of them to partially overlap the effects of another and the potential fluctuations responsible for electron scattering are weaker. We systematically studied the effect of the space correlation of oxide charges on the electron effective mobility in [SI. Results were obtained by assuming different degrees of space correlation for several temperature values and various oxide charge concentrations and positions. Ln this way we qualitatively obtained the same conclusions as di2 Ning and Sah. Nevertheless, while Ning and Sah considered this effect in isolation, we considered it simultaneously with the other effects and in the presence of the other scattering mechanisms. We thus showed that the conclusions obtained by Ning and Sah, when the correlation alone was considered, are not significantly modified or masked by the rest of the effects. In our previous model of Coulomb scattering, we assumed that all the charged centres were located inside the oxide in a plane parallel to the S i S i 0 2 interface or right at it [7,81. This is true for the common and important case of thermally grown oxides, where the oxide charge is mainly located in a sheet at the SiSi02 interface [13], and it is also the case of a sample
(i) In order to avoid small-size effects, the reduction of the dimensions in scaled MOS devices requires the use of high bulk impurity concentrations. Therefore, in submicrometre devices, the Coulomb scattering due to semiconductor bulk impurities may be as important as Coulomb scattering due to oxide charges [14], and, as a consequence, the two types of charge may be taken into account simultaneously, and therefore the interaction between them. (ii) Exposure of samples to ionizing radiation gives rise to a positive oxide charge and an increase in interface-trapped charge. This oxide charge is expected to be located at distances of about 20-35 from the interface [ 151. However, the non-negligible centribution of this oxide charge to mobility degradation has already been experimentally reported [16,19]. This means that both the oxide and interface charges must be taken into account in the simulation if one intends to reproduce the experimental effective mobility data in such samples.
A
Our previous model of the oxide charge space correlation is unsuitable in these cases, and a new model, in which an arbitrary charge distribution may be taken into account, is therefore required. In this paper we have generalized our previous model to include any charge distribution, including the two situations described above. This allows us to study different effects on the electron effective mobility, such as the influence of (i) bulk-impurity charge, (ii) the charge-distribution profile, both in the oxide and in the semiconductor bulk, and (iii) the charge spin. The importance of these effects has already been studied experimentally [16-191. 2. Coulomb scattering model
The effects of Coulomb scattering can be obtained by evaluating the influence that charged centres have on the position and occupation of the subbands in which the quasi-two-dimensional electron gas in an inversion layer is contained. For simplicity, we assumed that the external charge distributions responsible for Coulomb scattering were conceptually divided into two-dimensional sublayers parallel to the insulatorsemiconductor interface 171. Thus, the actual profile of the oxide charge distribution could easily be taken into account. If Az, is the thickness of the ttb sublayer and u,(Q,z,) is the Fourier transform of the charge density per unit area. the total Coulomb scattering rate for an electron with wavevector k in the ith subband with a final state in the jth subband, is given by
593
F Gamiz et a/
x IM;?(Q,
:,, z,)lz)lS(E' - E ) ) dk'
1
x IM:yCQ,zr,z,)12 J ( E - E ' ) d k '
(1)
as obtained in [7]. i i t ( Q , z r ) is the Fourier transform of the charged centre density fluctuations in the rth sublayer, and qt the charge of each of these centres. z I ) i ; ( Q ,zI)& is the density correlation function of charged centres in the same sublayer, and (Z,(Q, z i ) f i t ( Q ,zi)Lvg, r # U , is tbe density correlation function of charged centres in different sublayers, defined in [I]. The remaining terms in equation (1) are also defined in [7]. In [SI, where the effect of the oxide charge space correlation was analysed, it was assumed that all the charged centres were located in a plane parallel to the interface and at a distance of zo from it. In this case, the second term in expression (1) vanishes, and the sum in the first one is reduced to only one term. Therefore, it is only necessary to calculate tbe density correlation function of charged centres in the same sublayer. In accordance with the hard-sphere model [l, IO], ( k ( Q ,zl)Z;(Q, z t ) h V gin [ I ] was calculated to be
(%(e,
I
.
(4)
Assuming that Coulomb scattering mainly assists intrasubband transitions as justified in [7], we finally have
(5) where Q 2 2ksin(0/2), 0 being the angle between initial and final wavevectors, andm; the effective parallel mass of the electron in the ith subband. C,: is 0 if the distance between the rth distribution and the uth distribution is nrn-inr
6,CLLLCL
rh-r
LII'WI
D.
1,".
Expression (5) shows that Coulomb scattering depends on: where NI is the external charge density per unit area, JI the first-order Bessel function and Cl = T R ~ N a, parameter which is a measure of the degree of space correlation. C,represents the ratio of the minimum area zR: to the average area N;' occupied by a charged particle. We have also adopted the hard-sphere model [I, IO] to calculate the density correlation function of charged zJZ;(Q, centres in different sublayers, i.e. with f # K . As detailed in the appendix, for two sublayers separated by a distance 'd', we obtained
(&(e,
(i) The carrier concentration through the factor !M:'"'(Q;zJ*; (ii) The total number of charged particles in the structure, through the sum in N I ; (iii) The charge of each charged centre, due to the factor ]qrl; (iv) The sign of the charged centres, through the product qrqu; (v) The distance of the charges to the interface due to the factor IMj"'(Q,z1)1*; (vi) The space correlation between the different .-h~~.-..rl
*a,.+--C L " l I ~ C U CCLl"GiJ.
If the charge space correlation is ignored, expression (5) is reduced to
- f,l ~ ' , , ~ " ~ (2) , , . , ,*.\;*(A ~ , ~ , ,7.1\ ,"~~ -"
rl 3- R . 4 /;.(A ..U'
where Ne, = J N ~ ( z l ) N l ( z z ) ,RA =
\"I
and S is the area of the interface. C' is a constant which determines the interaction between the two distributions and is given by C' = i r R f N , ~ . J I is once again the first-order Bessel function. From (I), (2) and (3), the Coulomb scattering rate is given by
That is to say, the same expression used by the rest of the models recently reported in the literature [2,4]. In this expression there is no influence of the charge sign. Therefore, when the charge space correlation is ignored, the role of the charge sign in Coulomb scattering is incorrectly estimated. The hard-sphere correlation model was proposed - ^ - ~ .I.^..
lllUIG
LUaI1
.im v
.._^_^ JCar
e:..--
^ I ^
ngw.
0,LlC.c
A.."
LLlCll,
-^-_ U,U,G
..^.-",a.,
"w"'pc"
models, which may have more physical basis, have been proposed [ 11,121. Nevertheless, we think that the hard-sphere model is still very useful for the following reasons: 594
Space correlation of oxide charges (i) The model, in contrast with other models in the literature [ 11, 121 is conceptually very simple. It can thus easily be incorporated into models for Coulomb scattering for use in Monte Carlo simulations. (ii) In addition, this model provides very good agreement between experimental and calculated data, for all the sets of samples we have considered. (iii) Space correlation depends on the history of the sample, technological process of fabrication and method of degradation. We do not intend in this paper to perform a characterization of space correiation from this point of view, Until such characterizations are made, Ro remains an input parameter of the model, like other models in the literature [ 1I]. In addition. this model easily provides us with a chance of quantitatively studying different phenomena that strongly affect the electron effective mobility in inversion layers.
3. Results and discussion In order to illustrate the above method we have applied it to a one-electron Monte Carlo simulation described elsewhere [7] and calculated the. electron effective mobility in different conditions of temperature and oxide charge and interface trap distributions. Electron mobility is obtained from the electron drift velocity (which is calculated with an estimated uncertainty of 5%) extrapolating to zero longitudinal electric field. Figure 1 shows experimental mobility curves (full curve) taken from [ZO], and simulated ones by the Monte Carlo method (symbols) versus the transverse electric field after successive Fowler-Nordheim tunnelling injection series. An initial growth and a later decrease of the effective mobility curves can be observed as the effective field increases. This behaviour of the effective mobility curves, also experimentally observed, has been attributed to experimental uncertainties. Since this study relies on simulation, however, this phenomenon cannot be attributed to any experimentai uncertainties. Tine shape of the effective mobility curves can be nevertheless justified by the superposition of two tendencies: (i) the tendency of the phonon and surface roughness scattering to increase with an increasing electric field, and (ii) I. . . J . P n-..,--L UIC tcruency 01 LUUIUIIIV SCdLLCmg io demease as the effective electric field rises because the higher the electric field, the bigher the electron concentration in the inversion layer and the higher the screening of the oxide charge. -...&..A__
r-
^_^.. :-..^ papGL*, w c -'....J:*,l
111 p G * ' v " "
-----r
^I.
JLUUIC"
^.l*
: .o
-c,.-:,l-
^^ U,= L l l l l Y C l l c c U, U*lUG
charge space correlation on electron effective mobility in inversion layers, assuming that all of the charge was located in a plane parallel to the intekace or right at it. The effect on the electron effective mobility was found !o be notewor?hy. We have now stEdied ?he effect of charged-centre space correlation when there is more than one charge distribution. In such a case we must consider not only the interaction between charges in the same distribution, but the interaction between charges
in different distributions as well. We have therefore assumed an n-type Si(100) inversion layer with an oxide layer thickness of 200 A. It has been assumed that there exists a uniform charge distribution throughout the per oxide with a concentration of No, = 5 x IO1' unit volume. Doping concentration has been taken to be N A = 9 x 10l4 ~ 1 1 7 low ~ ~ .enough for the bulk impurity charge not to modify electron effective mobility. To take into account the charge space correlation we have assumed an Ro value of 56 A, which is to say there cannot be a charge at a distance iess than 56 from any other charge. This Ro value is in agreement with our previous results since it is the value necessary for reproducing the experimental behaviour in figure 1. This fact can be seen as being experimental evidence for the space correiation of oxide charges in MOSFETs. in addition to Coulomb scattering, we have considered phonon and surface roughness scattering [7,8]. More details of the simulation can be found in [7] and [SI. In order to apply the Coulomb scattering model given in expression (S),the oxide must be divided into sublayers. The greater the number of sublayers, the better the approximation of the oxide charge distribution to the actual profile. In order to settle on the optimum number of slices that must be considered in order to reproduce the effects oi the oxide charge disiribution, the following situations have been considered: (i) The oxide is assumed to consist of only one slice. This slice is modelled by a charged layer located in the middle of the oxide, z = 100 A? with a concentration of No, = 1 x loL2 (ii) The oxide is divided into two slices each 100 A thick. Each slice is modelled by a charged layer in its middle, with a concentration of No, = 5 x 10" ctW2. (iii) The oxide is divided into 10 slices each 20 A thick. Each slice is now modelled by a charged layer with a concentration of I x IO" cm-' in the middle of the slice. (iv) The oxide is divided into 20 slices each 10 A thick. Each slice is modelled by a charged layer located in the middle of the slice, and with a concentration per unit area of 0.5 x 10" cm?. Effective mobility curves in situations (i)-(iv) have been obtained at different temperatures using the Monte C-A-
La,,"
--A--%."p,"cG"Y'G.
117-
""G
Lo.,-
11'l"c:
-L..s-.,aA
UVJCl Y C U
rh..+
Ul'n
-$Po-+;.,CLIGbLIYC
mobility curves are very distinct when only one or two slices are considered. However, effective mobility curves coincide when 10 or more slices are taken into account. This means that dividing the oxide in 10 0.1II0
;Aao plv"A"u"
Y
E",,"
YyyY.VA"1LY"VL.
nF VI
. , U -
-hOmn
C"YL6'
concentration. At this point it is now worthwhile studying the effect of charged centre space correlation when there is a threedimensional oxide charge distribution. In the model develond r-- in this paper, this is equiv> to gEdying the effect of the interaction between charges belonging to different charged layers. To see this, in figure 2 we have calculated effective mobility curves corresponding to the above structure at room temperature, taking 10 slices 595
F Gamiz et 21
200
io5
.
1o6
Effective Field (V/cm) Figure 2. Effective mobility curves for a three-dimensional oxide charge distribution, modelled by 10 slices 20 A thick 2nd assuming: (i) space correlation exists both among charges in the same layer and among charges in different layers, (ii) there is no space correlation between charges in differentlayers and (iii) there is no space correlation at all. ( N =~5 x 10" cm-3, do, = 200 A, NA= 9 x 1014 and assuming that (i) there exists space correlation both among charges in the same layer and among charges in different layers, (ii) there is no space correlation between charges in different layers, but there is among charges in the same layer (curve Z), and (iii) there exists no space correlation at all (curve 3). The results show a significant effect of space correlation that could lead to an error of about 15% in the effective mobility dataif not adequately taken into account, which clearly points to the necessity of properly considering space correlation. With our previous Coulomb scattering model, we analysed the effect of different physical magnitudes on the electron effective mobility. With the improvement of our model in this paper, it is now possible to study a whole new set of effects as well.
3.1. Influence of bulk impurity charge on the effective mobility One of the main advantages of the model presented here is that it makes it possible to study the effect on the effective mobility of simultaneous charge distributions. This is the case of samples with high bulk impurity concentrations, in which the effect of bulk impurity charges cannot be ignored [ 141. Therefore, on one hand there is the charge trapped in the oxide and on the other charge trapped in the semiconductor, which for high doping levels can be even higher than the former. Figure 3 shows the different contributions to the total effective mobility of the different charge distributions for two bulk-impurity concentrations (see figure caption for symbols used): phonon and surface-roughness scattering only; the effect of Coulomb scattering due only to silicon bulk impurities; the effect of the oxide and interface charges, neglecting bulk impurities; and total effective mobility. In both 3(a) and 3(b), an oxide charge concentration of No, = 5 x 10" cm-2 located right at 596
the interface has been considered. The figure shows a very light contribution of the bulk impurity charge for low doping levels. However, this contribution is even greater than the oxide charge one for high bulk impurity concentrations. In addition, another important effect can be observed. Figure 3 shows that the way in which oxide charges affect the electron effective mobility strongly depends on the bulk impurity concentration. For a fixed effective transverse electric field, when the concentration of bulk impurities increases, the depletion charge also increases, thus decreasing the amount of charge in the inversion layer. This involves the loss of screening of the charge centres by mobile carriers. As a consequence the increase in bulk impurity concentration causes a reduction of effective mobility at low electric fields in two ways: (i) an increase of Coulomb interaction produced by an increase of charges in the bulk and (ii) the reduction of screening caused by the loss of charge in the inversion layer.
3.2. Influence of the charge distribution profile on the effective mobility Let us consider the following three oxide charge distributions: (i) A two-dimensionale charge distribution located inside the oxide at 100 A from the interface with a concentration of N , = 1 x 10'' cm-* (figure 4, full curve). (ii) A c h q e distribution formed by two charged layers located inside the oxide. One layer located at 50 13 from the interface with a concentration er unit and the second at 150 with a area of 5 x IO" concentration of 5 x 10" cm-' (figure 4, broken curve). (ii) A uniform distribution throughout the oxide with a concentration per unit volume of No, = 5 x IO" (figure 4, symbols). In all the above cases the total amount of charge trapped in the oxide is Qo = eA(1 ~ 1 0cm-'), ' ~ e being the unit charge and A the transverse area of the structure. The mean position of the distribution is located inside the oxide at 100 A from the interface. As both the total charge and its mean position are the same for all the above distributions, tbe threshold voltage shift? produced by them is also the same in all cases. Therefore, all these charge distributions would appear the same to C-V measurements, thus making it impossible to use a C-V method to distinguish between them. However, from the effective mobility viewpoint, they show very different behaviour, as can be seen in figure 4. This figure is useful for noting the important effect that the oxide charge distribution profile has on the electron effective mobility in semiconductor inversion layers: the effect of oxide charge on electron effective mobility cannot be characterized by the total charge and its mean position. It is absolutely necessary to know the actual profile of the charge distribution.
1'
t The threshold voltage shift is the shift produced in the threshold voltage of a ~zosnrwhen the charge distribution considered is trapped in the oxide with respect to the lhreshold voltage when there is no charge trapped inside the oxide,
Space correlation of oxide charges
T=300 K
Effective Field (V/cm) Figure 3. Different contributions to the electron effectivemobility, showing the effects of charges in the oxide and silicon bulk, for two bulk-impurity concentrations: *,phonon and surface-roughness scattering only; A ? effect of Coulomb scattering due only to silicon bulk impurities; * effect of the oxide and interface, neglecting bulk impurity charges; W, :he total effective mobility. (Nox = 5 x 1O'O cm-2 right at the interface, NA = 9 x 1014
charged centres next to an inversion layer are not all of the same sign. Various authors have previously investigated the effect of Coulomb scattering of electrons in inversion layers in the presence of charges of both signs. Kassabov et al 1171 experimentally concluded that the presence of negative charge trapped at the interface screens part of the positive fixed oxide charge, which leads to an increase in electron effective mobility. This conclusion is the opposite to that obtained, also experimentally, by Zupac et al [i6,18,19], in which electron effective mobility decreases as oxide-trapped and interface-trapped charge concentrations increase, in both n-type and p-type semiconductor inversion layers exposed to ionizing radiation. To investigate tbis effect, we assumed the following two charge distributions:
1000 N
0
s 0 0
s
100
1os
10'
Effective field (V/cm)
(i) a two-dimensional negative charge distribution located right at the interface with a concentration of N I= 1 x 10" cm-'; (ii) a second two-dimensional charge distribution in the oxide at 30 A from the interface, with positive charges in one case and negative ones in the other.
1 os
1o5
We obtained effective mobility curves for the above two cases using the Coulomb scattering model given in C:"..rr n Cln^+ilm "'"Y"'Ly m^l.ilit.. t.-nr..nrr" ' y Y ' r _.. r"I"ra expression (5j, having Ro = 56 A. Resuits are shown electric field for different profiles of the oxide charge in figure 5. The curve in full squares corresponds to distribution: full curve, a two-dimensional charge the case in which the charge inside the oxide is negative distribution at 100 A from the interface Nm = 1 x 10l2 c m 2 ; (with the same sign as the charge trapped right at the broken curve, two charged layers with a concentration interface) while the curves in full circles corresponds of 5 x 10" c m 2 located at 50 hi and 150 ri from the ..^-:*:..^ -L"--- ,*Le --..":*- ":-.. ifi;e&ce inside :he oxide; +U;; &&j, a ufiifoix&;iiⅈio~ Lu a puJLLLyL vyyu~~LL as the charge throughout the oxide with a concentration per unit volume trapped at the interface) inside the oxide. Figure 5 of N, = 5 x IO" ~ m - ~(4, . = 200 A. NA= 9 x l O I 4 ~ m - ~ . ) reveals that when charges of a different sign exist in the structure, electron effective mobility is lower than 3.3. Effect of the charge sign when ail the charges have the same sign. This is in Another _____. effect that ..._. m n he chidied -.__.__ with ...... this _... " new .._.. mnArl ...___ a p e g e n t Kith ex~fip-egta!resu!cs of &pac e! g! is the influence that the presence of charged centres [16,18,19], and is due to the greater local fluctuations of different sign (i.e. positive and negative charge) of the band bending when charges of both signs are has on the electron effective mobility. In fact, the present, which means greater scattering and thus lower Effective field
I
L l l r r u V r
(V/cm)
Yrl=U=
L I Y I I I a I r I ~ r
I^
__
^
597
F Gdmiz et a/
400
300
I
. N,=9xln"
"3
Effective Field (V/cm)
$.;u:e 5, EHective mobi[i!y cupiss vers.5 !he effetftjyg field assuming two oxide charge distributions, one (Nl) located right at the interface and formed by negative charges, and the other (N2)located inside the oxide at 30 A from the interface. In one mse (full circles), N2 is formed by positive charges, and in the other (fuli squares) by ii2ga:ive ones. (Nl = Pi2 = ? x !GI' cm-2). 15
L
e
T=300
io
K
space correlations, which allows us to physically examine the effects of charged-centre space correlation, bulk impurity charge and charge distribution profile, both in the oxide and in the semiconductor bulk. The effect of simultaneous charge distributions of both signs can also be studied when including correlation, though not when charged centres are treated as independent sources of scattering. It has thus been possible to distinguish between negatively and positively charged scattering potential in the Born approximation, without the need for the high computational requirements of other schemes. Finally, we have found these effects to have a noticeable influence on electron effective mobility. These effects had already been shown experimentally to play an important role in carrier transport properties in semiconductor inversion layers.
Appendix
Let n l ( r l )and nz(rz)be parallel two-dimensional charge distributions placed at a distance d from each other. r l and rz are the respective coordinates for each distribution, and eNl(zl) and eNl(zz)the average charge densities of each distribution. If S is the area of the interface, then N I = SN,(zl) and N2 = SN1(z2)are the total number of charged centres in each distribution. Let
L
W
R
W I . r d d r t &Z
.. ..
5
rk drk G
0
I o5
1o6
Effective Field (V/cm) Figure 6. Difference between the effective mobility obtained assuming all oxide charges in figure 5 have the same sign (fuli circles) and the effective mobility obtained when there exist oxide charges with a different sign (full squares).
mobility. It is interesting to note that in the second case the mean value of the electrostatic potential is lower, and therefore the threshold voltage also. To see the effect of charge sign clearly, figure 6 shows the difference between the effective mobiiity obtained assuming ali the charges have the same sign b1). the effective mobility , assuming different signs for the charges ( p 2 ) normalized by p i . It can be seen that at low electric fields, the sign of the charges has a noticeable effect, which must be
:.....--..,...
"*
Ro
+ Irli - rz, I > ,/R;
- dZ (A1 1)
and thus the correlation function between a particle in the 6rst distribution and a particle in the second one is given by d < RO
(A 12)
* W(rI,,rZ,)
bi
and Rb = - d2. C' is where N,E = .JN1(zl)N,(z2) a constant which determines the interaction between the two distributions and is given by C' = nR:N,e. 31 is once again the first-order Bessel function.
References [I] Ning T H and Sah C T 1972 Theory of scattering of electrons in a nondegenerate-semiconductor-surface inversion layer,by surface-oxide charges Phys. Rev. B 6 4605-13 [Z] Yokoyama K and Hess K 1986 Monte Carlo study of electronic transport in AI,-,Ga,As/GaAs single-well hetemstructures Phys. Rev. B 33 5595606 [3] Stem F and Howard W E 1967 Properties of semiconductor surface inversion layers in the electric quantum limit Phys. Rev. 163 816-35 [4] Fischetti M V and Laux S E 1993 Monte Carlo study of electron transport in silicon inversion layers Phys. Rev. B 48 2244-74 151 JOSER P and Ferry D K 1991 Effect of multi-ion screening on the electronic transport in doped semiconductors: A molecular-dvnamics analvsis Phvs. Rev. B 43 9734-9 161 Joshi R P 1994 Temoerature-deoendent electron mobilitv in GaN: Effects oi space ch&ge and interface roughnkss scattering Appl. Phys. Len. 62 223-5 [7] G M z F, Mpez-ViLlanueva J A, Jim6nez-Tejada J A, Melchor I and Palma A 1994 A comprehensive model for Coulomb scattering in inversion iayers J. Appl. Phys. 75 924-34 [SI Gimiz F, Melchor I, Palma A, Cartujo P and MpezVillanueva J A 1994 Effects of oxide charge space correlation on electron mobility in inversion layers Semicond Sci. Technol. 9 110S7 [9] Gamiz F. Mpez-ViIlanueva J A, Banqueri J, Jim6nezTejada J A and Cartujo P 1993 Simulation of Semiconductor Devices and Processes ed S Selberherr et al (Vienna: Springer) pp 481-4 L
~
1
~
~~~~
599
F G h k et a/ [IO] Siggia E D and Kwok P C 1970 Properties of electrons
[ I I]
[12] [ 131
[I41
[IS]
600
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