Interface vs. bulk transport

1. The potential required to form the 'channel' at Si-Si$O$₂interface confines the electrons in a region so narrow as to be comparable to the wavelength of the electrons. This fact has at least two major consequences:
   • The component of the electron momentum normal to the interface can take only discrete values. In the parabolic-band approximation, this translates into a 'quantization' of the energy associated with the motion along the direction normal to the interface. This results in energy 'subbands', their discrete number being caused by the discrete allowed values of the normal component of the momentum, and the continuum of states within each subband being associated with the motion on the plane of the interface. See the review by Ando, Stern, and Fowler[1] for an exhaustive discussion of the physical properties of the 2-dimensional electron gas.
   • The usual approximation invoked to treat electrons as semiclassical particles breaks down. Yet, since transport along the 'normal' direction is precluded, one can still make use of the Boltzmann transport equation, but now in two dimensions, since electrons are free to move only on the plane of the Si-Si$O$₂ interface.
   The band-structure, the velocity, the scattering rates, and the screening properties of the electron gas are altered, with respect to what happens in the bulk, because of the reduced dimensionality of the system. Only electrons which gain sufficient kinetic energy and 'escape' the grip of the confining potential behave like 'normal' bulk, three-dimensional particles, free to move in all directions.
   As a consequence, Monte Carlo simulations of nMOSFETs must account for these effects. DAMOCLES, in this case, solves the three-dimensional (3D) Boltzmann equation for high-energy electrons, a 2D Boltzmann equation for the low-energy carriers confined by the interface potential, a 1D Schrödinger equation in order to determine the energies of the discrete levels (subbands) in the channel: the full 2D solution of the Schrödinger equation including scattering is still beyond our computational means. Yet, as long as the potential along the channel does not vary too fast, compared to the electron wavelengths, the 1D-solution is sufficiently accurate. In the simple case of pure parabolic bands (but DAMOCLES can also handle the quite important non-parabolic corrections), for every 2D-location R in the plane of the interface, the Schrödinger equation to solve is:
   

   (Symbols are defined at the end of the document. Note that, by our convention, lower-case symbols are vector components along this direction, upper case bold letters denote the vectors lying on the plane of the interface). Note that for the 'usual' <001> surface in Si, there are two sets (or 'ladders') of subbands, since the quantization mass, $m$_z, can take the value of 0.91 $m$₀ ($m$₀ is the free electron mass) for the 2 X-valleys with longitudinal axis perpendicular to the interface (the so-called 'unprimed' subbands), or the value $m$_z=0.19$m$₀ for the other 4 X-valleys ('primed' subbands).Thus, there are two Schrödinger equations to solve, one for each ladder (or mass). The wavefunctions are needed in order to obtain the charge density of the quantized particles, by summing over all subbands µ in both ladders the electron occupation (known from the Monte Carlo simulation) multiplied by the spatial distribution of the electrons in each subband (i.e., the squared amplitude of the envelope wavefunctions):

   

   This charge density is obviously needed to solve the Poisson equation. The 2D electrons are subject to ballistic motion in free flights determined by the structure of the subbands and to scattering processes whose rates are computed from matrix elements evaluated as in the bulk, but replacing the 'plane waves' $eikr$ along the confining direction with the envelope wavefunctions obtained from the solution of the Schrödinger equation.

   

   where $V$_s is the scattering potential. All this in a full self-consistent way: The potential is fixed by the charge density, which is fixed by the wavefunctions obtained from the Schrödinger equation, which depends on the electrostatic potential...

2. The presence of the interface so close to the moving electrons causes additional scattering processes: Electrons can now scatter with phonon-modes characteristics of the interface ('surfons', interface modes arising from the coupling of the 2D plasmons with the optical phonons of Si$O$₂), with charges at or near the interface (charged interface traps, fixed oxide charge), with the roughness of the interface, etc.

Figure 1. (Left): The confining electrostatic potential (black solid line, left scale) and the envelope wavefunctions associated to the confined electrons, as computed in DAMOCLES. The blue solid lines correspond to the first two eigenstates in the 2-fold degenerate ellipsoidal valleys whose longitudinal axis is normal to the plane of the interface (the so-called 'unprimed' subbands), the red dashed lines are the envelopes of the two lowest-lying eigenstates in the 4-fold degenerate valleys with a light mass along the direction normal to the interface ('primed', higher energy subbands). (Right):Snapshot of the Si channel simulated by DAMOCLES. Particles are displaced vertically above the bottom of the conduction band (grey surface) according to their total kinetic energy and are color-coded according to the subband they belong to: The more numerous cyan electrons are in the first primed subband, blue particles in the lowest-energy unprimed subband, green and red electrons are in the second unprimed and primed subbands, respectively. Yellow electrons are in higher subbands, while most of the higher-energy particles are not shown. Notice that the distance of the particles from the interface (located at the front surface) has no physical meaning, nor it plays any role in DAMOCLES. For graphical purposes only, electrons are placed away form the interface with a probability distribution given by the squared-amplitude of the wavefunctions. Thus, higher-energy particles are -- on average -- deeper in the substrate.
Click on the pictures to open a new window displaying 607×645 and 792×612 jpeg versions.

DAMOCLES, like all other 'bona fide' simulation codes stressing the basis physics, must account for all of these effects and, hopefully, be able to reproduce quantitatively the experimental data available on electron transport in Si inversion layers.

Past failures

Experimental information about electron transport in Si inversion layers comes in many forms. But of utmost technological interest is information related to the speed of the device, i.e., how fast carriers move. At low 'longitudinal' fields (that is, small electric fields in the plane of the interface), the velocity of the electrons is proportional to the magnitude of the field itself. The proportionality constant is called the 'effective mobility', µ$$_eff. Figure 2 illustrates how this quantity is found to vary with the electron sheet density, n_s (or, equivalently, with the strength of the field along the direction perpendicular to the interface). In pure samples (low doping in the Si substrate, negligible density of interface and oxide charge), in the limit of zero density one should recover essentially the mobility of bulk Si. Indeed this is what is found experimentally, as it can be seen in Fig. 2. Data are from unpublished early work by Fowler, reproduced in [1], by Sun and Plummer[2], Cooper and Nelson[3], and by Manzini[4]. Only in samples in which Coulomb scattering is significant one observes that the mobility decreases also at small densities, since screening becomes less effective and Coulomb scattering dominates.

As the 'normal' field is increased, the effect of scattering with phonons is enhanced, because of details of the 'overlap' integrals between the envelope wavefunctions in the subbands, and the mobility roughly decreases as $n$_s^-1/3, where $n$_s is the electron sheet density (electrons per unit area). At even larger densities ($n$_s>$1012cm-2$) scattering with the roughness at the interface dominates and the mobility decreases roughly as $n$_s^-2.

Coulomb scattering with interface and oxide charges as well as scattering with the interface roughness are technologically very important, since scaled sub-micron devices have heavily doped substrates and operate at large electron densities. However, there is little one can do from a purely theoretical point of view: The spatial distribution of the scattering centers can only be approximately determined, and the roughness of the interface can only be occasionally and roughly measured with high-resolution transmission electron microscopy (TEM). In either case, too many experimental unknowns prevent a careful quantitative evaluation of the theoretical model. It is in the range of intermediate sheet densities ($n$_s around the mid-$1011cm-2$ range) that things get interesting: This is the regime in which electron-phonon scattering controls the mobility: A stringest test for the theory. And here, historically, theory has failed. Early sophisticated theories by Ezawa and coworkers, using initially only one subband[5] but later employing many subbands and no intervalley scattering[6] ended up predicting mobilities exceeding 3000 $cm2/V\; sec$ and\; 2000$ cm2/V\; sec$ ,\; respectively,\; at$ $$$$$n_s = 10¹², about a factor 3-to-4 too large. The inclusion of intervalley scattering[7] [8] [9] could lead to good agreement with the experiments only using ad hoc values for some deformation potentials. Using the 'best' values for the shear and dilatation deformation potentials available at the time, DAMOCLES could not do much better than what shown by the open blue symbols in Fig. 2[10]. This is a 20% overestimation of the mobility: Pretty good, considering there are no adjustable parameters. But device designers would not be favorably impressed by a complicated computer program which would overestimate by 20% the current flowing in a transistor! One could have always 'adjusted' the deformation potentials and fix the situation. Unfortunately, the values needed to 'fix' this problem would 'break' the situation in the bulk. And the 'fitting' strategy, even when motivated (and, a few would say, justified) by pressing practical needs, is contrary to the philosophy DAMOCLES has always embraced.

The correct mobility, at last!

For example, under stress along the [001] crystallographic direction, the 6-fold degenerate minima close to the X symmetry points split into a 2-fold degenerate set along the stress direction, and a 4-fold degenerate set in the plane normal to that direction. The splitting of these two sets is linearly proportional to the trace of the stress tensor (for small values of the strain). The proportionality constant provides the shear deformation potential for the X-valleys. Similarly, the splitting of the valence bands provides the values of the shear deformation potentials b for stress along the [001] direction, d for stress along the [111] direction. Unfortunately, no information is obtained about the 'dilatation' deformation potentials: These are the 'absolute' shifts of the bands, unlike the 'relative' splittings we just talked about. However, one obtains information about the sum of the valence and conduction dilatation deformation potential, from the variation of the (indirect) energy-gap with strain. The quantities one can extract from these calculations are shown in red in Fig. 3. The values of the band-gap deformation potential unfortunately depends on the empirical rules used to interpolate the ionic (pseudo)potentials when dealing with dilated/distorted crystals. But having fixed these rules by calibration with experimental data on the gap of relaxed $Si$_1-xGe_x alloys, calculations on strained Si result in the following values for the deformation potentials[11]:

where a is the valence band dilatation deformation potential, b and d are the shear deformation potentials, and other symbols being defined below. Note how the second equation relates the valence- and conduction-band dilatation deformation potentials. Since carrier mobilities depend not only on the shear, but also on the dilatation deformation potentials, there is actually a way to extract the troublesome dilatation deformation potentials from the experimental mobility. Herring and Vogt[12] had already followed this strategy in 1956: They had computed the electron mobility as a function of the ratio of the dilatation and shear deformation potentials in the conduction band. Figure 4 shows essentially the result of the same exercise: The electron mobility -- normalized to the experimental value -- at 300K is plotted as a function of the dilatation deformation potential, since the shear deformation potential is already fixed. As found by Herring and Vogt, the problem one must face is that actually the 'correct' mobility can be obtained for two different values of the dilatation deformation potential. In the past, following values reported in the literature, and picking-and-chosing a few among several -- often inconsistent -- experimental data, we settled on a particular choice, corresponding to the left-most value in Fig. 4.

These values are remarkably different from those used in the past and they bring us good news and bad news. The good news is that the value a = 2.1 eV is amazingly consistent with what is reported in the literature (although not unanimously!), in particular with the recent theoretical results by Van de Walle[13] and Yoder et al.e[14]. Also, it promises at least 'changes' in our results for the phonon-limited electron effective mobility. The bad news is that things do not quite work as well as we had hoped: using these new values for the deformation potentials, µ$$_eff still turns out to be about 5 to 10% too large. An improvement, but not quite what we wanted.

Again, calculations in strained Si provide a clue: In Fig. 5 we show the electron mobility at 300 K in Si under [001]-strain. The 'in plane' mobility (in the <100>-plane) and the mobility along the [001] axis are shown by the red lines. At an in-plane tensile strain of about 1% the mobility is boosted up, but only to a value of about 2450 $cm2/V\; sec$, not quite as large as values observed experimentally, up to (or even exceeding) 3000 $cm2/V\; sec$[15]. In an attempt to resolve this new problem, we may ask whether there are additional scattering parameters, perhaps poorly known, are affecting our results. Intervalley scattering is indeed another 'dark' area. Historically, the Monte Carlo calculations performed by the group at the University of Modena[16] resulted in a set of intervalley deformation potentials corresponding to a rather weak scattering among <100> valleys (the so-called g-scattering), and a stronger f-scattering among <100>-<010> valleys. A different set was proposed later by the same group[17] in order to explain observations related to the field dependence of the longitudinal and transverse diffusion coefficients. All results reported so far have been obtained using this latter set.

The blue lines in Fig. 5 shows what happens when the mobility in strained Si is computed using the 'older' set of intervalley deformation potentials[16]: Now the peak mobility under tensile strain comes closer to the experimental value. Better yet: Using these intervalley deformation potentials, the effective mobility in Si inversion layer is (finally!) in agreement with the data, as the red dots in Fig. 2 illustrate. So, it appears that both intra-valley and inter-valley deformation potentials have to be modified in order to explain the data. With the 'new' set of intravalley acoustic deformation potentials determined theoretically and with the 'old' set of intervalley scattering parameters of Ref. [16] we are now able to explain the phonon-limited electron mobility in strained Si and in Si inversion layers.

The velocity-saturation problem

Despite the success just described, something is still amiss in the theory of electron transport in Si inversion layers: If we now look at the high-field properties, situation in which Monte Carlo techniques should excel, we discover another annoying situation: The electron velocity, as said above, grows linearly with increasing longitudinal field, as long as the electrons remain approximately in thermal equilibrium. This is the regime in which the concept of mobility has a meaning, since the distribution function is minimally and linearly perturbed away from its equilibrium condition. As the field increases beyond this 'Ohmic' regime, the velocity begins to saturate. This is also what happens in bulk semiconductors. In particular, in bulk Si at room temperature, electrons accelerated by a homogeneous field larger than about $104V/cm$ reach a steady-state saturated velocity, $v$_sat, of about $107cm/s$. In principle, one may suspect that the presence of the interface will result in a different value of $v$_sat. Experimentally, the community is divided into two opposite camps: On the one side, Fang and Fowler[18], Cohen and Müller[19], and Modelli and Manzini[20], among others, have measured a significantly lower saturated velocity, not exceeding 6 to $7\times 106cm/s$. On the other side, Cooper and Nelson[3] have measured a saturated velocity very close to the bulk value. Experiments are indeed complicated, since the longitudinal field in the channel is not uniform, unless precautions are taken. Indeed Fang and Fowler have employed very thick oxides, Modelli and Manzini resistive gates in order to minimize field nonuniformity. The very sophisticated time-of-flight experiments by Cooper and Nelson are, on the contrary, of difficult interpretation and deal with depleted, rather then inverted, channels. Therefore, experiments yielding a value of $$v_sat lower than the bulk value are most likely to be more credible. Yet, neither DAMOCLES not any other calculation performed to date has succeeded in obtaining saturated velocities different from the 'universal' value of $107cm/s$. More recent experimental results[22] even suggest a dependence of $$v_sat on carrier density. Perhaps Coulomb scattering among electrons, not yet included in the simulation of the 2D electron gas by DAMOCLES, may explain these findings. For now, this remains one of the nagging problems DAMOCLES is unable to tackle.

References

1. T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).
2. C. Sun and J. D. Plummer, IEEE Trans. Electron Devices ED-27, 1497 (1980).
3. A. Cooper and D. F. Nelson, J. Appl. Phys. 54, 1445 (1983).
4. S. Manzini, J. Appl. Phys. 57 441 (1985).
5. H. Ezawa, Ann. Phys. 67, 438 (1971)
6. H. Ezawa, S. Kawaji, and K. Nakamura, Jpn. J. Appl. Phys. 13, 126 (1974); H. Ezawa, Surf. Sci. 58, 25 (1976).
7. D. Roychoudhury and P. K. Basu, Phys. Rev. B 22, 6325 (1980).
8. K. Masaki, C. Hamaguchi, K. Taniguchi, and M. Iwase, Jpn. J. Appl. Phys. 28, 1856 (1989); K. Masaki, K. Taniguchi, and C. Hamaguchi, Semicond. Sci. Technol. 7, B573 (1992).
9. D. K. Ferry, Phys. Rev. B 14, 5364 (1976).
10. M. V. Fischetti and S. E. Laux, Phys. Rev. B 48, 2244 (1993).
11. M. V. Fischetti and S. E. Laux, J. Appl. Phys. 80, 2234 (1996). (Abstract and post-script version available from the IBM Research Division CyberDigest).
12. C. Herring and E. Vogt, Phys. Rev. 101, 944 (1956).
13. C. G. Van de Walle, Phys. Rev. B 39, 1871 (1993).
14. P. D. Yoder, V. D. Natoli, and R. M. Martin, J. Appl. Phys. 73, 4378 (1993); P. D. Yoder, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 1994.
15. K. Ismail, B. S. Meyerson, and P. J. Wang, Appl. Phys. Lett. 58, 2117 (1991); S. F. Nelson, K. Ismail, J. O. Chu, and B. S. Meyerson, Appl. Phys. Lett. 63, 367 (1993); K. Ismail, S. F. Nelson, J. O. Chu, and B. S. Meyerson, Appl. Phys. Lett. 63, 660 (1993).
16. C. Canali, C. Jacoboni, F. Nava, G. Ottaviani, and A. Alberigi-Quaranta, Phys. Rev. B 12, 2265 (1975).
17. R. Brunetti, C. Jacoboni, F. Nava, L. Reggiani, G. Bosman, and R. J. J. Zijlstra, J. Appl. Phys. 52, 6713 (1981).
18. F. Fang and A. B. Fowler, J. Appl. Phys. 41, 1825 (1970).
19. W. Müller ans I. Isele, Solid-State Commun. 34, 447 (1980).
20. A. Modelli and S. Manzini, Solid-State Electron. 21, 99 (1988).
21. R. W. Cohen and R. S. Müller, Solid-State Electron. 23, 35 (1980).
22. C. Hamaguchi, Physica 134B, 87 (1985).

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APPENDIX: List of mathematical symbols

	damoclesNO-SPAM@watson.ibm.com (last updated: January 26, 1999)