# Diffusive Transport in Quasi-2D and Quasi-1D Electron Systems

###### Abstract

Quantum-confined semiconductor structures are the cornerstone of modern-day electronics. Spatial confinement in these
structures leads to formation of discrete low-dimensional subbands. At room temperature, carriers transfer among different states due to efficient scattering with phonons, charged impurities, surface roughness and other electrons, so transport is scattering-limited (diffusive) and well described by the Boltzmann transport equation. In this review, we present
the theoretical framework used for the description and simulation of diffusive electron transport in quasi-two-dimensional
and quasi-one-dimensional semiconductor structures. Transport in silicon MOSFETs and nanowires is presented in detail.

Keywords: quantum confinement, nanostructures, 2DEG, nanowires, SiNW, scattering, diffusive transport, Boltzmann transport equation, Monte Carlo simulation, confined phonons

###### Contents

- I Introduction
- II Quasi-2D Electron Systems
- III Scattering in Quasi-2D Electron Systems
- IV Transport in Quasi-2D Systems
- V Quasi-1D Semiconductors (Nanowires)
- VI Scattering in Quasi-1D Electron Systems
- VII Transport in Quasi-1D Electron Systems
- VIII Concluding remarks

## I Introduction

Quantum-confined semiconductor structures, such as silicon MOSFETa or GaAs-based resonant-tunneling diodes, are the cornerstone of modern-day electronics. Ando et al. (1982); Ferry (1991); Ferry and Goodick (1997) Spatial confinement in these structures leads to formation of discrete low-dimensional subbands: energy levels are quantized in each direction of confinement, while the momentum remains a good (continuous) quantum number in the unconfined directions. If carriers are confined along one direction and free to move in the two-dimensional (2D) plane perpendicular to it, the structure is being referred to as a quasi-two-dimensional electron gas (Q2DEG). The structure is considered a quasi-one-dimensional electron gas (Q1DEG) if the carrier motion is unbound in one dimension (1D) as a result of 2D-confinement. At room temperature, transport within each subband and transitions among subbands can essentially be described semiclassically, using the Boltzmann transport equation. Its state-of-the-art solution is obtained via the ensemble Monte Carlo technique. Jacoboni and Reggiani (1983) At low temperatures, quantum-coherence effects become prominent, and transport ceases to be semiclassical in nature. In this regime, transport description is better achieved using the Wigner-function formalism Nedjalkov et al. (2006) or nonequilibrium Green’s functions, Lake and Datta (1992); Lake et al. (1997) as long as the single particle picture is valid. However, in the remainder of this text, we will be concerned with the semiclassical transport picture, appropriate for room-temperature operation of Q2DEG and Q1DEG electronic structures.

In structures such as the resonant tunneling diode Tsu and Esaki (1973) or the quantum cascade laser, Faist et al. (1994) the actual device operation is based on utilization of quantum confinement and tunneling. In contrast, in structures such as silicon MOSFETs, quantum-confinement features emerge as a result of miniaturization and are usually detrimental: tunneling through the gate oxide, Sheu and Jang (2000) source-to-drain tunneling and space-quantization effects are expected to be important in nanoscale MOSFETs and HEMT devices and require a solution of the 1D Schrödinger-Poisson problem. Solution of the 2D Schrödinger-Poisson problem is needed, for example, for describing the channel charge in narrow-width MOSFETs Vasileska and Ahmed (2005) and nanowires. Jin et al. (2007); Ramayya et al. (2008a)

Successful scaling of MOSFETs towards shorter channel lengths requires thinner gate oxides and higher doping levels to achieve high drive currents and minimized short-channel effects. Dennard et al. (1974); Brews et al. (1980) As the oxide thickness is scaled to below 10 nm, quantum confinement of inversion charge leads to an appreciable inversion layer capacitance Takagi and Toriumi (1995); Hareland et al. (1996) in series with the oxide, so the total gate capacitance is lowered. (Further modification of the gate oxide capacitance stems from the image and many-body exchange-correlation effects in the inversion layer, Vasileska et al. (1997) as well as poly-silicon gate depletion. Krisch et al. (1996))

The low-field electron mobility is an important quantity that determines the performance of semiconductor devices. Surface roughness scattering (SRS) is by far the most important cause of mobility degradation in conventional MOSFETs at high transverse fields. One would expect the SRS to be even more detrimental in silicon nanowires (SiNWs) than in conventional MOSFETs because SiNWs have four Si-SiO interfaces, as opposed to one such interface in conventional MOSFETs. This has indeed been confirmed recently on ultrathin cylindrical Jin et al. (2007) and rectangular nanowires. Ramayya et al. (2008a) In addition, confinement leads to a modification in the acoustic phonon spectrum in SiNWs, Ramayya et al. (2008a); Buin et al. (2008) which leads to increased electron-phonon scattering and a lowered electron mobility. Lü et al. (2003); Lü and Chu (2006); Chen et al. (2005)

In this review, we present the theoretical framework typically used for the description and simulation of diffusive electron transport in quasi-2D and quasi-1D semiconductor structures. In Sec. II, we overview the formation of the Q2DEG in Si inversion layers and GaAs-modulation doped heterostructures. We discuss the solution to the 1D Schrödinger-Poisson problem in the direction of confinement and the density of states calculation. In Sec. III, we overview the scattering mechanisms in semiconductors, their origin and model Hamiltonians, and give the scattering rates for Q2DEGs in the Born approximation. In Sec. IV, we present the ensemble Monte Carlo simulation of the electron mobility in the inversion layer Q2DEG formed near the Si/SiO interface of a nanoscale MOSFET. Section V introduces the Q1DEG in nanowires, and overviews phonon confinement and bandstructure modification in these structures. In Sec. VI, scattering rates for the Q1DEG are given, while the electron mobility in thin silicon nanowires, as obtained from detailed ensemble Monte Carlo simulations, is presented in Sec. VII.

## Ii Quasi-2D Electron Systems

### ii.1 Silicon Inversion Layers

The best known examples of the Q2DEG are silicon MOSFETs and GaAs/AlGaAs heterostructures. An integral part of any MOSFET device is the metal-oxide-semiconductor (MOS) capacitor. Regarding the MOS capacitors, the induced interface charge is closely linked to the shape of the electron energy bands of the semiconductor near the interface. At zero applied voltage, the bending of the energy bands is ideally determined by the difference in the work functions of the metal and the semiconductor. This band bending changes with the applied bias and the bands become flat when we apply the so-called flat-band voltage given by

(1) |

where and are the work functions of the metal and the semiconductor, respectively. The various energies involved are indicated in Fig. 1, where we show typical band diagrams of an MOS capacitor at zero bias. is the electron affinity for the semiconductor, is the energy of the conduction band edge, and is the Fermi level at zero applied voltage.

In stationary conditions, no net current flows in the direction perpendicular to the interface, owing to the very high resistance of the insulator layer (however, this does not apply to very thin oxides of a few nanometers, where tunneling becomes important). Hence, the Fermi level will remain constant inside the semiconductor, irrespective of the biasing conditions. However, between the semiconductor and the metal contact, the Fermi level is shifted by (see Figs. 2 and 3). Hence, we have a quasi-equilibrium situation in which the semiconductor can be treated as if in thermal equilibrium.

An MOS structure with a p-type semiconductor will enter the accumulation regime of operation when the voltage applied between the metal and the semiconductor is more negative than the flat-band voltage ( in Fig. 2). In the opposite case, when , the semiconductor-oxide interface first becomes depleted of holes and we enter the so-called depletion regime (Fig. 3, top panel). By increasing the applied voltage, the band bending becomes so large that the energy difference between the Fermi level and the bottom of the conduction band at the insulator-semiconductor interface becomes smaller than that between the Fermi level and the top of the valence band. This is the inversion regime of operation (Fig. 3, bottom panel).

Carrier statistics tells us that the electron concentration will then exceed the hole concentration near the interface and we enter the inversion regime. At a larger still applied voltage, we finally arrive at a situation in which the electron density at the interface exceeds the doping density in the semiconductor. This is the strong inversion case, in which we have a significant conducting sheet of inversion charge at the interface (Fig. 3, bottom panel). In the description that follows, symbol is used to denote the potential in the semiconductor measured relative to the potential at a position deep inside the semiconductor. Note that becomes positive when the bands bend down, as in the example of a p-type semiconductor shown in Figure 3. From equilibrium electron statistics, we find that the intrinsic Fermi level in the bulk corresponds to an energy separation from the actual Fermi level of the doped semiconductor,

(2) |

where is the thermal voltage, is the shallow acceptor density in the p-type semiconductor and is the intrinsic carrier density of silicon. According to the usual definition, strong inversion is reached when the total band bending equals , corresponding to the surface potential . Values of the surface potential such that correspond to the depletion and the weak inversion regimes, respectively, is the flat-band condition, and corresponds to the accumulation mode. Note that, deep inside the semiconductor, we have . Under the flat-band condition (), the surface charge is equal to zero. In accumulation (), the surface charge is positive, and in depletion and inversion ), the surface charge is negative.

In order to relate the semiconductor surface potential to the applied voltage , we have to investigate how this voltage is divided between the insulator and the semiconductor. Using the condition of continuity of the electric flux density at the semiconductor-insulator interface, we find , where and are the absolute permittivities (also known as dielectric constants) of the oxide layer and the semiconductor, respectively, while and are the respective electric fields in the two materials. Hence, for an insulator of thickness , the voltage drop across the insulator becomes . Accounting for the flat-band voltage, the applied voltage can be written as , where is the insulator capacitance per unit area.

The threshold voltage, , is the gate voltage corresponding to the onset of strong inversion. It is one of the most important parameters characterizing metal-insulator-semiconductor devices. As discussed above, strong inversion occurs when the surface potential, , becomes equal to . For this surface potential, the free charge induced at the insulator-semiconductor interface is still small compared to the charge in the depletion layer, and the threshold voltage is calculated using:

(3) |

Note that the threshold voltage may also be affected by the so-called fast surface states at the semiconductor-oxide interface and by fixed charges in the insulator layer. However, this is not a significant concern with modern-day fabrication technology.

The threshold voltage separates the subthreshold regime, where the mobile carrier charge increases exponentially with increasing applied voltage , from the above-threshold regime, where the mobile carrier charge is linearly dependent on the applied voltage . However, there is no clear point of transition between the two regimes, so different definitions and experimental techniques have been used to determine . Well above threshold, the charge density of the mobile carriers in the inversion layer can be calculated using the parallel-plate charge control model. This model gives an adequate description for the strong inversion regime of the MOS capacitor, but fails for applied voltages near and below threshold (i.e., in the weak inversion and depletion regimes). Several expressions have been proposed for a unified charge control model (UCCM) that covers all regimes of operation.

Successful scaling of MOSFETs towards shorter channel lengths requires thinner gate oxides and higher doping levels to achieve high drive currents and minimized short-channel effects. Dennard et al. (1974); Brews et al. (1980) For these nanometer devices it was demonstrated that, as the oxide thickness is scaled to 10 nm and below, the total gate capacitance is lower than the oxide capacitance due to the comparable values of the oxide and the inversion layer capacitances. As a consequence, the device transconductance is degraded relative to the expectations of the scaling theory. Bacarani and Worderman (1982)

The two physical origins of the inversion layer capacitance, the finite density of states and the finite inversion layer thickness, were demonstrated experimentally by Takagi and Toriumi. Takagi and Toriumi (1995) A computationally efficient three-subband model, that predicts both the quantum-mechanical effects in the electron inversion layers and the electron distribution within the inversion layer, was proposed and implemented into the PISCES simulator. Hareland et al. (1996) The influence of the image and many-body exchange-correlation effects on the inversion layer and the total gate capacitance was studied by Vasileska et al. Vasileska et al. (1997) It was also pointed out that the depletion of the poly-silicon gates considerably affects the magnitude of the total gate capacitance. Krisch et al. (1996)

The inversion layer capacitance was also identified as being the main cause of the second-order thickness dependence of the MOSFET IV-characteristics. Liang et al. (1986) The finite inversion layer thickness was estimated experimentally by Hartstein and Albert. Hartstein and Albert (1988) The high levels of substrate doping were found responsible for the increased threshold voltage and decreased channel mobility. A simple analytical model that accounts for this effect was proposed by van Dort and co-workers van Dort et al. (1992, 1994) and confirmed by Vasileska and Ferry Vasileska and Ferry (1998) by investigating the doping dependence of the threshold voltage in MOS capacitors (Fig. 4).

### ii.2 1D Schrödinger-Poisson Problem for Silicon Inversion Layers

The periodic crystal potential in bulk semiconducting materials is such that, for a given energy in the conduction band, the allowed electron wavevectors trace out a surface in -space. In the effective-mass approximation for silicon, these constant energy surfaces can be visualized as six equivalent ellipsoids of revolution (Fig. 5), whose major and minor axes are inversely proportional to the effective masses. A collection of such ellipsoids for different energies is referred to as a valley.

In this framework, the bulk Hamiltonian for an electron, residing in one of these valleys is of the form

(4) | |||||

where , is the effective potential energy profile of the confining potential along the -direction, is the Hartree potential which is nothing more but a solution of the 1D Poisson equation introduced later in the text, is the exchange-correlation potential (also discussed later in the text), is the parallel part of (associated with the motion in the -plane, perpendicular to the confinement direction), and the transverse part is defined as

(5) |

The basis-states of the unperturbed Hamiltonian are assumed to be of the form

(6) |

where is a wavevector in the -plane and is the area of the sample interface. The subband wavefunctions satisfy the one-dimensional Schrödinger equation

(7) |

subject to the boundary conditions that are zero for and approach zero as . In Eq. (7), is the subband energy and is the corresponding wavefunction. In the parabolic band approximation, the total energy of an electron is given by

(8) |

where is the kinetic energy and is the density-of-states mass along the -plane.

An accurate description of the charge in the inversion layer of deep-submicrometer devices and, therefore, the magnitude of the total gate capacitance, , requires a self-consistent solution of the 1D Poisson equation

(9) |

and the 1D Schrödinger equation

(10) |

In Eqs. (9) and (10), is the electrostatic potential [the Hartree potential ], is the spatially dependent dielectric constant, and are the ionized donor and acceptor concentrations, and are the electron and hole densities, is the effective potential energy term that equals the sum of the Hartree and exchange-correlation corrections to the ground state energy of the system, is the effective mass normal to the semiconductor-oxide interface of the -th valley, while and are the energy level and the corresponding wavefunction of the electrons residing in the -th subband from the -th valley. The electron-density is calculated using

(11) |

where is the sheet electron concentration in the -th subband from the -th valley, given by

(12) |

where is the valley degeneracy factor and is the Fermi energy. When evaluating the exchange-correlation corrections to the chemical potential, we have relied on the validity of the density functional theory (DFT) of Hohenberg and Kohn, Hohenberg and Kohn (1964) and Kohn and Sham. Kohn and Sham (1965) According to DFT, the effects of exchange and correlation can be included through a one-particle exchange-correlation term , defined as a functional derivative of the exchange-correlation part of the ground-state energy of the system with respect to the electron density . In the local density approximation (LDA), one replaces the functional with a function , where is the exchange-correlation contribution to the chemical potential of a homogeneous electron gas of density , which is taken to be equal to the local electron density of the inhomogeneous system. In our model, we use the LDA and approximate the exchange-correlation potential energy term by an interpolation formula developed by Hedin and Lundqvist. Hedin and Lindquist (1969) Exchange and correlation effects tend to lower the total energy of the system, and, as discussed later, lead to a non-uniform shift of the energy levels and repopulation of the various subbands. The enhancement of the exchange-correlation contribution to the energy predominantly affects the ground subband of the occupied valley; the unoccupied subbands of the same valley are essentially unaffected. As a result, a noticeable increase in the energy of intersubband transitions can be observed at high electron densities (more on electron scattering can be found in Sec. III).

### ii.3 GaAs/AlGaAs Heterostructures. Effective Mass Schrödinger Equation for Heterostructures

While our primary focus in this paper will remain on Si-based electron system, in this section we will briefly discuss another important Q2DEG: the modulation-doped GaAs-AlGaAs heterostructure. Ferry (1991) The bandgap in AlGaAs is wider than in GaAs. By variation of doping it is possible to move the Fermi level inside the forbidden gap. When the materials are put together, a unified chemical potential is established, and an inversion layer is formed at the interface (see Fig. 6).

The Q2DEG created by modulation doping can be squeezed into narrow channels by selective depletion in spatially separated regions. The simplest lateral confinement technique is to create split metallic gates in a way shown in Fig. 7. We refer the reader to extensive literature on transport properties of GaAs/AlGaAs and other modulation doped heterostructures (for instance, see Ref. Ferry and Goodick, 1997).

In semiconductors, some of the most interesting applications of the Schrödinger equation in the effective mass approximation involve spatially varying material compositions and heterojunctions. The effective mass approximation can still be used with some caution. Since the effective mass is a property of the bulk, it is not well defined in the neighborhood of a sharp material transition. In the hypothesis of slow material composition variations in space, one can adopt the Schrödinger equation with a spatially varying effective mass, taken to be the mass of the bulk with the local material properties. However, it can be shown that the Hamiltonian operator is no longer Hermitian for varying mass. A widely used Hermitian form brings the effective mass inside the differential operator as . This approach is extended to abrupt heterojunctions, as long as the materials on the two sides have similar properties and bandstructure, as in the case of the GaAs/AlGaAs system with less than 45 Al. One has to keep in mind that very close to the heterojunctions the effective mass Schrödinger equation provides a reasonable mathematical connection between the two regions, but the physical quantities are not necessarily well defined. For instance, in the case of a narrow potential barrier obtained by using a thin layer of AlGaAs surrounded by GaAs, it is not clear at all what effective mass should be used for the AlGaAs, since such a region can certainly not be approximated by the bulk. It is even more difficult to treat the case when there is a transition between direct and indirect bandgap materials (for example, GaAs and AlGaAs with over 45 Al).

Assuming a uniform mesh size , the Hamiltonian of the Schrödinger equation can be discretized in 1D by introducing midpoints in the mesh on both sides of a generic grid point . First, we evaluate the outer derivative at point with centered finite differences, using quantities defined at points and

and then the derivatives defined on the midpoints are also evaluated with centered differences using quantities on the grid points

(14) |

The effective mass is the only quantity which must be known at the midpoints. If an abrupt heterojunction is located at point , the abrupt change in the effective mass is treated without ambiguity.

### ii.4 Density of States (DOS) for Low-Dimensional Systems

An important quantity characterizing a quantum-mechanical system is the density of states (DOS) function. The density of states is defined as the number of states per energy interval . It is clear that

(15) |

where is the set of quantum numbers characterizing the states. In the present case, it includes the subband quantum number , spin quantum number , valley quantum number and the in-plane quasi-momentum . If the spectrum is degenerate with respect to spin and valleys, one can define the spin degeneracy and the valley degeneracy to get

(16) |

Here we calculate the number of states per unit volume, being the dimensionality of the space. For a 2D case, we obtain

(17) |

Within a given subband, the 2D density of states function is energy-independent. Since there can exist several subbands in the confining potential, the total density of states can be represented as a set of steps, as shown in Fig. 9. At a low temperature (), all the states are filled up to the Fermi level. Because of the energy-independent density of states, the sheet electron density is linear in the Fermi energy, namely

(18) |

The Fermi momentum in each subband can be determined as

(19) |

In Eq. (18), is the number of transverse modes having the edge below the Fermi energy.

The situation is more complicated if the gas is further confined in a narrow channel, say, along the -axis. The in-plane wave function can be decoupled as a product

(20) |

the corresponding energy being

(21) |

In the last equation, characterizes the energy level in the potential confined in both ( and ) directions. For square-box confinement, the terms are

(22) |

where is the channel width, while for parabolic confinement (typical for split-gate structures), we have

(23) |

For these systems, confined in 2D, the total density of states is

(24) |

The energy dependence of the density of states is shown in Fig. 9.

## Iii Scattering in Quasi-2D Electron Systems

Charge transport in the diffusive regime is governed by carrier scattering from lattice vibrations, charged impurities, defects, interface roughness, as well as other electrons. Calculation of the scattering rates for confined carriers proceeds in a similar manner as in the 3D case, Ferry (1991); Lundstrom (1992) but proper wavefunctions for 2D carriers must be used. Before we go into the details of the calculation of the matrix elements of some of the most important scattering mechanisms listed in Fig. 10, we will derive a few expressions.

Suppose we want to calculate the scattering rate out of some state in subband . For that purpose we will use Fermi’s golden rule, which gives us the transition rate from state in subband into state belonging to subband by means of emission of an energetic particle (e.g. a phonon or a photon) with energy :

(25) |

Assuming a plane-wave basis for the wavefunctions in the unconfined direction (-plane), the total wavefunctions of the initial and the final states are of the following general form (for a Q2DEG):

(26) |

where is the area of the sample, is the position vector in the -plane and is a 3D position vector. The matrix element for scattering between states and in subbands and , respectively, is then given by

(27) |

where is the interaction potential and the form of the integral with respect to depends upon the type of the scattering dynamics considered. Ferry (1991); Lundstrom (1992); Vasileska (1995) In low-dimensional systems, since the momentum is quantized in one ore two directions to form subbands, it is important to note that we now have additional intrasubband and intersubband transitions, which significantly complicates the generation of the scattering tables and choosing the final state after scattering. Below, we give the matrix elements for some of the most important scattering mechanisms that are present in silicon inversion layers and GaAs/AlGaAs heterostructures.

### iii.1 Electron-phonon scattering

Phonon scattering can cause three different types of electronic transitions in the Si inversion layer: transitions between states within a single valley via acoustic phonons (called intravalley acoustic-phonon scattering) and nonpolar optical phonons (called intravalley optical phonon scattering), and transitions between different valleys mediated by high-momentum acoustic or nonpolar optical phonons (called together intervalley scattering). Price (1981); Ridley (1991); Roychoudhury and Basu (1980); Hao et al. (1985); Goodnick and Lugli (1988); Imanaga and Hayafuji (1991); Magnus (1993); Yamada et al. (1994) Intravalley acoustic-phonon scattering involves phonons with low energies and is an almost elastic process. The intravalley optical-phonon scattering is induced by optical phonons of low momentum and high energy. Intervalley scattering can be induced by the emission and absorption of high-momentum, high-energy phonons, which can be of either acoustic- or optical-mode variety. Intervalley scattering can therefore be important only for temperatures high enough that an appreciable number of suitable phonons is excited or for hot electrons that can emit high energy phonons. Ferry (1976a) In order to evaluate the scattering potential that describes the electron-phonon interaction, we need a Hamiltonian that describes the coupled electron-phonon system. The total Hamiltonian of the system is given by Mahan (1981); Ziman (1960)

(28) |

where is the electronic part, is the atomic part that describes the normal modes of vibration of the solid, and is the electron-ion interaction term of the form

(29) |

In general, each ion is at a position , which is a sum of the equilibrium position and the displacement . Under the assumption of small displacements, one can expand in a Taylor series

(30) |

The zero-order term is the potential function for the electrons when the atoms are in their equilibrium positions, which forms a periodic potential in the crystal. The solution of the Hamiltonian for electron motion in this periodic potential gives the Bloch states of the solid. Since the first order term is much smaller than the zero-order term, the electron-phonon interaction can be treated perturbatively. Therefore, the lowest order term for the electron-phonon interaction is of the form

(31) |

It is obvious that this interaction Hamiltonian does not act on the spin variables in this approximation. The Fourier transform of can be written as

(32) |

where is the number of primitive cells, and wavevector spans the whole -space. Ionic displacement may be decomposed into normal-mode representation and it is customary to write

where is the unit polarization vector that obeys the standard orthonormality and completeness relations, is the phonon frequency of phonon branch for wavevector (running over the whole Brillouin zone), () are the phonon annihilation (creation) operators. In acoustic waves, refers to the relative displacement of the unit cell as a whole with respect to adjacent unit cells; in optical waves it refers to the relative displacement of the basis atoms within the unit cell. Thus

where , and is the density of the solid. The summation over represents summation over all reciprocal lattice vectors of the solid. If one defines a function

(35) |

then the Hamiltonian for the electron-phonon interaction becomes

(36) |

The exact form of the matrix elements for acoustic and nonpolar-optical phonon scattering (zero- and first-order terms) are given below. Since we will need to make a clear distinction between 3D and 2D vectors, in what follows we are going to use the following notation: capital bold letters will refer to three-dimensional vectors, whereas small bold letters will be used for two-dimensional wavevectors that lie in the xy-plane.

#### iii.1.1 Deformation potential scattering

In general, the application of mechanical stress alters the band structure by shifting energies, and, where it destroys symmetry, by removing degeneracies. It is usually assumed that the mechanical stress does not change the band curvature, and therefore does not change the effective masses, but introduces ashift in the energy states that are close to the band extremum. Ridley (1993); Seitz (1948); Harrison (1956); Herring and Vogt (1956) For isotropic elastic continuum, the matrix element for deformation potential scattering (acoustic phonons) can be obtained by taking the long-wavelength limit of Eq. (35). Ferry and Jacoboni () For small values of , the summation over reciprocal lattice vectors can be neglected, except for the term . The screened electron-ion interaction becomes a constant which is usually denoted as (it gives the shift of the band edge per unit elastic strain). Under these assumptions, simplifies to

(37) |

where , is the sound velocity, is the phonon frequency. Long wavelength acoustic phonons (LA mode) have which makes the matrix element non-zero. TA phonons have which makes the matrix element vanish. Therefore, the deformation potential mainly couples electrons to LA phonons.

For anisotropic elastic continuum such as silicon, the deformation potential constant becomes a tensor. The anisotropy of the intravalley deformation potential in the ellipsoidal valleys in silicon has been extensively studied by Herring and Vogt. Herring and Vogt (1956) Expanding the electron-phonon matrix element over spherical harmonics and retaining only the leading terms, they have expressed the anisotropy of the interaction in terms of the angle between wavevector of the emitted (absorbed) phonon and the longitudinal axis of the valley. They have shown that the matrix element is proportional to via the deformation potential (LA or TA) given by Meyer (1958); de Cerdeira (1973); Pintschovius et al. (1982)

(38) |

and

(39) |

Equation (39) accounts for the contribution of both TA branches. Therefore, the acoustic mode scattering is characterized by two constants: (uniaxial shear potential) and (dilatation potential) that is believed to have values of approximately 9.0 eV and -11.7 eV, respectively. In bulk silicon, this anisotropy is usually ignored by using an effective deformation potential constant for the interaction with longitudinal modes, and ignoring the role of the lower-energy TA modes. This approximation can be justified due to the following reasons: The acoustic modes are most effective at low energy. In this regime and in the usual elastic and equipartition approximation (described later), due to the linear dependence on , scattering of electrons at some energy samples almost uniformly the constant energy ellipsoid, so that one can take the average values of over the ellipsoid. Since there is nothing to fix the energy scale in the problem, this averaging procedure is independent of the electron energy. Moving to the two-dimensional situation, one cannot follow a parallel path to arrive at an isotropic, energy independent effective deformation potential, which complicates the treatment of this scattering process.

Since the wavefunctions of the initial and final states are usually expressed as a product of a one-electron Bloch wavefunction and a harmonic oscillator wavefunction, after the averaging over the phonon states is performed, the terms inside the brackets of Eq. (36) that represent phonon absorption (term ) and phonon emission (term ) processes reduce to and , respectively. In thermal equilibrium with a lattice at temperature , the phonon occupation number is given by the Bose-Einstein statistics

(40) |

where is the Boltzmann constant. At high enough temperatures, the acoustic phonon energies are much smaller than the thermal energy of electrons. Therefore, one can expand the exponent in the denominator of Eq. (40) into a series and, in the equipartition approximation, appropriate at high temperatures, we have

(41) |

Incorporating these terms as well as the exponential term into the definition of , after a straightforward calculation one finds that the matrix element squared for scattering between subbands and due to acoustic phonons (for both absorption and emission processes together), and after the averaging over is performed, reduces to

(42) |

where

(43) |

The effective deformation potential constant is calculated from

(44) |

where

(45) |

The form-factor introduces an energy scale in the problem by fixing the fuzzy component, the wavevector . This result is an expected one and follows immediately from the uncertainty principle . Since the electrons are frozen into their wavefunctions, and cannot oscillate in the quantized direction, the uncertainty in the particle’s location along the z-axis has been reduced. Therefore, there must be a corresponding increase in the uncertainty in the particle’s z-directed momentum. Lundstrom (1992)

#### iii.1.2 Nonpolar optical phonon scattering

The scattering of electrons by zone-center optical and intervalley phonons in semiconductor crystals has been treated rather extensively by Ferry. Ferry (1976b, c, a) The nonpolar optical interaction is important for intrasubband scattering as well as for scattering of electrons (and holes) between different minima of the conduction (or valence) band. The latter interaction is important for scattering of carriers in semiconductors with many-valley band structure, such as Si and Ge, and in the Gunn effect, where scattering occurs between different sets of equivalent minima. Harrison Harrison (1956) pointed out that the nonpolar optical matrix element may be either of zero or higher order in the phonon wavevector. In subsequent treatments of electron transport in which the nonpolar interaction is important, only the zero-order term was considered, generally owing to the impression that the higher order terms are much smaller. Although this is usually the case, there arise many cases in which the zero-order term is forbidden by the symmetry of the state involved. In these cases, the first order term becomes the leading term, and can become significant in many instances. For example, the first-order intervalley scattering plays an important role in hot-electron transport in the n-type inversion layer in Si. Ignoring this scattering process means that there will be no saturation of the drift velocity at high electric fields, because the zero-order intervalley scattering rate is weakly dependent on the electron energy of high-energy electrons, while the first-order intervalley scattering rate increases as the electron energy increases. The matrix element for nonpolar optical phonon scattering is generally found from a deformable ion model explained in the introduction part of this section. If one thinks of an optical phonon as occurring at finite , then the dependence is unimportant, so that the entire matrix element becomes constant and we have

(46) |

where is the deformation field (usually given in eV/cm) and is the frequency of the relevant phonon mode which is usually taken to be independent of the phonon wave-vector for optical and intervalley processes. Fourier transforming back to real space, a constant in -space produces a delta-function in real space. Therefore, this zero-order term represents a short-ranged interaction. A local dilatation or compression of the lattice produces a local fluctuation in the energy of the electron or hole. Incorporating the exponential term into the definition of , after averaging over we find

(47) |

for the squared matrix element for scattering between subbands and that belong to the and valleys, respectively. When the zero-order matrix element for the optical or intervalley interaction vanishes, is identically zero. In this case, one has to consider the first-order term of the interaction whose matrix element is

(48) |

In this context, a first order process means a process similar to acoustic phonon scattering. Following the previously explained procedure, we find that the matrix element squared for scattering between subbands (-valley) and (-valley) is given by

(49) |

where

(50) |

The constant term is a small correction term. Vasileska (1995)

In the scattering among equivalent valleys, there are two types of phonons that might be involved in the process (see Fig. 12). The first type, the so-called g-phonon couples the two valleys along opposite ends of the same axis, i.e. to . This is an umklapp process and has a net phonon wavevector . The f-phonons couple a valley with , , etc. The reciprocal lattice vector involved in the g-process is and that for an f-process is . Degeneracy factors () for transition between unprimed () and primed () set of subbands, for both g- () and f-phonons () are summarized in Table 1.

; | ; | |

; | ; |

Within a three-subband approximation (subband in a [100] valley and in , and a generic subband in one of the other four valleys – ). Scattering between the and subbands in the two valleys along the same axis involves only g-type phonons. Scattering between these two minima is usually treated by using a high-energy phonon of 750 K activation temperature (treated as zero-order interaction) and 134-K phonon treated via first order interaction. Scattering between or and the four subbands involves f-phonons with activation temperatures of 630 K and 230 K, treated via zero-order and first-order interaction, respectively. Scattering among subbands involves both g- and f-phonons with activation temperatures of 630 K (zero-order interaction) and 190 K (first-order interaction). All of the high energy phonons are assumed to be coupled with a value of eV/cm and all of the first-order phonons are assumed to be coupled with eV. (This value is consistent with the results given in Ref. Zollner et al., 1990). The first Born approximation result for the total electron-bulk phonon scattering rate for -type silicon with cm, cm, and T=300 K, with (thick line) and without (thin line) the inclusion of the correction term for the first order process, and for the lowest subband of the unprimed ladder of subbands, is given in Figure 13. We see that, throughout the whole energy range, there is an increase of approximately 10 of the total electron-bulk-phonon scattering rate due to the correction term introduced previously that could lead to mobility reduction. The same trend was also observed for the higher lying subbands.

#### iii.1.3 Polar optical phonon scattering

Polar optical phonon (POP) scattering is a very strong scattering mechanism in polar semiconductors, such as GaAs. (It is absent in nonpolar materials, such as Si.) Since our focus here is on Si-based structures, we will not discuss POP scattering in detail, just note that the strength of this mechanism lies in its electrostatic nature. Namely, an optical phonon in a polar binary semiconductor such as GaAs displaces the two atoms in the unit-cell basis with respect to one another, and lead to a modification of the dipole moment associated with the unit cell. Overall, the displacement field associated with the propagation of an optical phonon gibes rise to an electric field, which is what electrons scatter from. Detailed derivations of the POP scattering rates can be found in many texts. Ferry (1991); Lundstrom (1992). Here we will just give the matrix element squared for POP scattering:

(51) | |||

where is given by Eq. (45), is the longitudinal polar optical phonon frequency and and are the high frequency and the low frequency dielectric constants, respectively. The scattering rate is then found via integration over the final states and the final expression can be found in Refs. Ferry, 1991 and Lundstrom, 1992.

### iii.2 Scattering by Electrostatic Interactions

#### iii.2.1 Coulomb Scattering

Scattering associated with charged Coulomb centers near the plane of the 2D electron gas in MOS devices can be separated into contributions from the depletion layer, the interface charge and the oxide charge. An extensive discussion of the role of multiple-scattering contributions to the electron mobility of a doped semiconductor and the apparent difficulties with the impurity averaging is presented in the papers by Moore, Moore (1967) and Kohn and Luttinger. Kohn and Luttinger (1957); Luttinger (1967) The expressions for the potential due to a single charge located in the region of interest, in which the image term is properly included, are given by Stern and Howard. Stern and Howard (1967) Here, we will discuss it in a way that is suitable for a many-subband treatment. Using the usual method of images, Jackson (1975) one easily finds that, in the presence of dielectric medium, the potential due to a charged center located at equals

(52) |

for , and

(53) | |||||

for . In Eqs. (52) and (53), is the average dielectric constant at the interface. The depletion charge scattering occurs due to the ionized charges in the depletion layer. Using Eq. (53), we find that the matrix element squared for scattering between subbands and due to the depletion charge is equal to

(54) | |||

where is the depletion charge density, is the location of an arbitrary charge center in the depletion region, and and are the form factors due to the finite extension of the electron gas in the quantization direction, of the form

(55) |

and

(56) |

respectively, where

(57) |

In the above expressions, is a wavevector in the plane parallel to the interface (-plane in our case).

Near the Si/SiO interface, there are always many Coulomb centers, due to the disorder and defects in the crystalline structure in the neighborhood of the interface. They are associated with the dangling bonds and act as charge-trapping centers which scatter the free carriers through the Coulomb interaction. Using Eq. (52), we find that the matrix element squared for the scattering from a sheet of charge with charge density located in the oxide, at distance () from the interface is

For interface-trap scattering . By similar arguments, one finds that the matrix element for scattering from the oxide charge, with charge density , is

where is the oxide thickness.

For GaAs/AlGaAs heterostructures, and remote Coulomb scattering, in addition to polar optical phonon scattering, dominates the low-field electron mobility. Since there are no charges in the depletion layer in this material system (GaAs is intentionally left undoped), direct Coulomb scattering mechanism can safely be ignored. In addition to the smaller effective mass of electrons in the GaAs system, the absence of direct Coulomb scattering is a one of the main reasons for the observation of the very high mobility in GaAs modulation doped heterostructures.

#### iii.2.2 Surface-roughness scattering

This scattering mechanism is associated with the interfacial disorder and depends upon the oxidation temperature and ambient as well as post-oxidation anneal and removal of the wafer from the furnace. Early theories of surface roughness were based on the Boltzmann equation in which the surface is incorporated via boundary conditions into the electron distribution function. Thomson (1901); Fuchs (1938); Sondheimer (1952) The first quantum-mechanical treatment of the problem was given by Prange and Nee. Prange and Nee (1968) Subsequently, the theory followed two different paths. The basic idea of the first approach is to incorporate the variations in the confining potential of the rough surface as a boundary condition on the Hamiltonian of the system. Since there is no simple perturbation theory to treat arbitrary changes in the boundary conditions, the problem of a free-electron Hamiltonian with complicated boundary conditions is then transformed by an appropriate coordinate transformation into a problem with simpler boundary conditions (i.e. into a problem where we have flat surfaces). This coordinate transformation technique has been proposed by Tesanovic et al. Tesanovic et al. (1986) and was later used by Trivedi and Ashcroft. Trivedi and Ashcroft (1988) As a consequence of this transformation, the Hamiltonian of the system now has additional terms that play the role of potential interaction terms. These additional terms are treated by perturbative techniques, which are valid when the roughness of the surface is small compared to the thickness of the well.

In the second approach, Ferry (1991) the effect of the surface roughness is taken into account through a random local potential term

(60) |

which is then treated perturbatively ( is the step function). The random function is a measure of the roughness and is most conveniently expressed in terms of the autocovariance function of . The power spectrum is the two-dimensional Fourier transform of the autocovariance function of . For the Gaussian correlated roughness that is usually assumed, Goodnick et al. (1982); Gold (1986, 1987); Fishman and Calecki (1989); Sakaki et al. (1987) the power spectrum is given by

(61) |

Parameters and characterize the r.m.s. height of the bumps on the surface and the roughness correlation length, respectively. Goodnick et al. Goodnick et al. (1985a) have made an extensive analysis of high-resolution transmission electron microscopy (HRTEM) measurements to test the assumption of Gaussian correlation. They found that exponential correlation describes roughness much better than Gaussian correlation irrespective of growth conditions. Roughly speaking, it means that the interface may be regarded as consisting of terraces of a few nanometers in size separated by atomic steps of a few tenths of nanometers, as shown in Fig. 14. This result was later confirmed by Atomic Force Microscope (AFM) measurements. Feenstra (1994) The power spectrum for the exponential correlation is given by

(62) |

A generalization of the result given in Eq. (62) is a self-affine roughness correlation function, which in 2D takes the form

(63) |

where is an exponent describing the high- fall-off of the distribution. It reduces to exponential correlation for .

For identical roughness parameters, the Gaussian spectrum decays slower for small wavevectors and then falls to zero rapidly for large wavevectors. The exponential model also leads to a rougher interface due to the tails in the spectrum, which allows for short-range fluctuations to be considered as well. For and small values of , the power spectrum of the self-affine model decays faster compared to the previous two models, but then falls slowly for large wavevectors. This essentially means that, in this regime, it also allows for short-range fluctuations to be considered. For large exponents, the power spectral density of the self-affine model approaches the one for the Gaussian model. In general, the matrix element for scattering between subbands and for this scattering mechanism is of the form

(64) |

For large , matrix element reduces to

The last expression is the result obtained by Prange and Nee. Prange and Nee (1968) Matsumoto and Uemura Matsumoto and Uemura (1974) calculated that in the electronic quantum limit, , where ( and are the inversion layer and the depletion region sheet charge densities, respectively).

The change in the potential energy of the system due to surface-roughness was corrected by Ando, Ando (1977) by considering the change in the electron density distribution and the effective dipole moment of the deformed Si-SiO surface. The later scattering rate becomes

where is given by Eq. (III.2.2). Due to the presence of the dielectric medium, one needs to correct the expression given in Eq. (III.2.2) with the contribution of the image term

where and are the modified Bessel functions. An additional complication associated with the finite oxide thickness, which may further reduce the mobility via scattering with remote roughness, will be ignored in the present treatment.

#### iii.2.3 Electron-Electron Interaction

We consider the Coulomb interaction between an electron with wave vector k in subband and a second electron with wave vector in subband . The final states of these two electrons are