Abstract
The chapter describes the reasoning that leads from the single-particle Schrödinger equation for an electron in a crystal to the mathematical model of semiconductor devices. The latter is a set of equations describing the evolution in space and time of a number of average quantities of interest: with reference to the electrons of the conduction band or holes of the valence band, such quantities are the concentration, average velocity, current density, average kinetic energy, and so on. The model of semiconductor devices has different levels of complexity depending on the trade-off between the information that one needs to acquire about the physical behavior of the device under investigation and the computational cost of the system of differential equations to be solved. In fact, the possible models are hierarchically ordered from the drift-diffusion model, which is the simplest one, to the hydrodynamic model, and so on. In essence, these models are different approaches to the problem of solving, in a more or less simplified form, the Boltzmann Transport Equation. Those described in this chapter are the most widely adopted in the commercial simulation programs used by semiconductor Companies. Other important methods, that are not addressed in this book, are the Monte Carlo method and the spherical-harmonics expansion.
The steps leading to the mathematical model of semiconductor devices start with a form of the single-particle Schrödinger equation based on the equivalent Hamiltonian operator, where it is assumed that the external potential energy is a small perturbation superimposed to the periodic potential energy of the nuclei; this leads to a description of the collisionless electron’s dynamics in terms of canonically-conjugate variables, that are the expectation values of the wave packet’s position and momentum. The dynamics of the Hamiltonian type makes it possible to introduce the statistical description of a many-electron system, leading to the semiclassical Boltzmann Transport Equation. After working out the collision operator, the perturbative approximation is considered; the simplified form of the transport equation thus found is tackled by means of the moments method, whence the hydrodynamic and drift-diffusion versions of the model are derived. A detailed analysis of the derivation of the electron and hole mobility in the parabolic-band approximation is provided. Then, the semiconductor model is coupled with Maxwell’s equation, and the applicability of the quasi-static approximation is discussed. The typical boundary conditions used in the analysis of semiconductor devices are shown, and an example of analytical solution of the one-dimensional Poisson equation is given.
The complements discuss the analogy between the equivalent Hamiltonian operator and the corresponding Hamiltonian function introduced in an earlier chapter, provide a detailed description of the closure conditions of the models, and illustrate the Matthiessen’s rule for the relaxation times. Finally, a short summary of the approximations leading to the derivation of the semiconductor model is given.
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- 1.
Like in Sect. 17.6.1, the periodic part of the Bloch function is indicated with \(\zeta_{i\mathbf{k}}\) to avoid confusion with the group velocity.
- 2.
This method of reconstructing the operator from the eigenvalues was anticipated in Sect. 17.6.8.
- 3.
For the sake of simplicity, suffix “ext” is dropped from the symbol of the external energy.
- 4.
The approach is the same as that used for treating the time-dependent perturbation theory (compare with 14.4).
- 5.
The field produced by non uniformities in the local charge density, which is present also in an equilibrium condition if the dopant distribution is not spatially constant (compare with Sect. {18.5}), is classified as “external” because it can be treated as a perturbation. Instead, rapid variations of the physical properties of the material, like those that typically occur at interfaces, can not be treated using the perturbative method and require the solution of the Schrödinger equation without approximations.
- 6.
- 7.
More comments about the analogy with the perturbation theory in the classical case are made in Sect. 19.6.1.
- 8.
For Si, Ge, and GaAs it is \(k_{ia} = 0\) (Sect. 17.6.5).
- 9.
For the sake of simplicity, spin is not considered here.
- 10.
Suffix μ is dropped to distinguish this distribution function from that of the classical case.
- 11.
- 12.
Here the extrema are the minima of the conduction band. The region near an extremum of a band is also called valley.
- 13.
The units of S are \([S] = \hbox{s}^{-1}\).
- 14.
- 15.
- 16.
- 17.
The semiconductor’s purification degree necessary for manufacturing integrated circuit is called electronic grade; it indicates that the ratio between the concentration of impurities and (different from dopants) that of the semiconductor atoms is smaller than \(10^{-9}\). Lower-quality materials, with a ratio smaller than \(10^{-6}\), are used in the fabrication of solar cells; in this case the purification degree is called solar grade.
- 18.
On the other hand, in a collisionless case it is \(S_0 \rightarrow 0\) whence, from ({19.43}), it follows \(\tau\rightarrow\infty\). In this situation there is no limit to the departure of f from f eq.
- 19.
Compare with the discussion carried out in Sect. 6.6.3.
- 20.
As indicated in Sect. {C.6}, term “moment” is specifically used when α is a polynomial in \(\mathbf{k}\). As the dependence of α on \(\mathbf{k}\) is not specified yet, it is implied that the form of α is such that the integrals in ({19.51}) converge.
- 21.
In the case of the conduction band of germanium, the minima are at the boundary (Sect. 17.6.5), which makes the hypothesis inconsistent as it stands; to perform the integration one must shift the origin of the \(\mathbf{k}\) space and exploit the periodicity of the band structure. The hypothesis that the distribution function vanishes at the boundary of the first Brillouin zone is made also in the application of the moments method to the holes of the valence band.
- 22.
The choice of the highest-order moment as the function to be approximated is reasonable in view of the analysis of the moments method carried out in Sect. {C.6}. In fact, as the moments are the coefficients of a converging Taylor series, they become smaller and smaller as the order increases; thus, the error due to approximating the highest-order coefficient is expected to be the smallest.
- 23.
A similar reasoning is used to explain ({20.16}).
- 24.
In the equilibrium condition the product \(E_e \, f^{\rm eq}\) is even with respect to \(\mathbf{k}\). In turn, \(\mathbf{u} = (1/\hbar) \, {\mathop{\rm grad}}_{\mathbf{k}} E\) is odd, so that \(\mathbf{b}^{\rm eq} = 0\). Compare with the similar comment made about the average velocity in ({19.31}).
- 25.
Comprehensive reviews of the solution methods for the BTE are in [55], {[56]} as far as the Monte Carlo method is concerned, and in [49] for deterministic methods.
- 26.
A similar reasoning is used to treat the time derivative of the vector potential when the semiconductor equations are coupled with the Maxwell equations (Sect. 19.5.4).
- 27.
The adoption of the parabolic-band approximation may be avoided at the cost of redefining the carrier temperature and introducing more relaxation times [108].
- 28.
Typically the parameter used in this procedure is the electric field [57]. An expansion truncated to the first order is coherent with the first-order perturbation approach.
- 29.
The term “momentum” for \(\tau_{pi}\) derives from the observation that the continuity equation for the ith component of the average velocity v i of the electrons, ({19.67}), may also be thought of as the continuity equation for the ith component of the average momentum, \(m_{ia} \, v_i\). In turn, \(\tau_{bi}\) is also called heat-relaxation time.
- 30.
Also, the generation-recombination term W embeds non-linear dependencies on some of the unknowns, Chap. {20}.
- 31.
From this assumption and from ({19.100}) it also follows \(\mathbf{J} =-q\,\sum_{a=1}^{M_C} n_a \, \mathbf{v}_a = - q \, (n/6) \, \sum_{a=1}^{M_C} \mathbf{v}_a\), whence \(\mathbf{J} = -q \, n \, \mathbf{v}\) with \(\mathbf{v} = (1/6) \, \sum_{a=1}^{M_C} \mathbf{v}_a\).
- 32.
The relation derives from Einstein’s investigation on the Brownian motion [35] and has therefore a broader application. In a semiconductor it holds within the approximations of parabolic bands and non-degenerate conditions.
- 33.
Equation (18.53) is the definition of charge density in a semiconductor; as a consequence it holds in general, not only in the equilibrium condition considered in Sect. {18.5}. In fact, it can readily be extended to account for charges trapped in energy states different from those of the dopants (Sect. 20.2.2).
- 34.
A typical example is found when a semiconductor device or circuit is used as a magnetic-field sensor or in specific measurement setups, like in the Hall-voltage measurement (Sect. 25.4).
- 35.
A similar reasoning is used to treat the time derivative of v i , w, and b i in the derivation of the BTE’s moments of order larger than zero (Sect. 19.4.2).
- 36.
The progressive device scaling from one generation to the next is in general associated to an increase in the size of the chips. Due to this, the constraints on the circuit’s speed are rather imposed by the lines connecting the devices than by the devices themselves.
- 37.
The insulator’s permittivity is indicated with \(\varepsilon_{\rm ox}\) because, in the examples shown later, silicon dioxide (\(\mathrm{SiO_2}\)) is used as the reference insulator.
- 38.
The units are \([W_n,W_p] =\hbox{m}^{-3}~\hbox{s}^{-1}\).
- 39.
A two- or three-dimensional case is considered. In the one-dimensional case the boundary reduces to the two points enclosing the segments over which the equations are to be solved.
- 40.
- 41.
This outcome becomes immediately clear by applying a numerical-discretization method to the problem. In fact, the component of the current density normal to the contact depends on the electric potential of the contact itself; thus, the extra relation provided by the flux-conservation equation embeds the extra unknown ϕ c .
- 42.
By some authors, ϕ n and ϕ p are called Imref potentials, where “Imref” is “Fermi” read from right to left [103].
- 43.
A PDE of order s in the unknown ϕ is called quasi-linear if it is linear in the order-s derivatives of ϕ and its coefficients depend on the independent variables and the derivatives of ϕ of order m < s. A quasi-linear PDE where the coefficients of the order-s derivatives are functions of the independent variables alone, is called semi-linear. A PDE which is linear in the unknown function and all its derivatives, with coefficients depending on the independent variables alone, is called linear. PDE’s not belonging to the classes above are fully non-linear.
- 44.
As above, for this result to hold the parabolic-band approximation is not necessary.
- 45.
Mobility is traditionally expressed in cm2/(V s) instead of m2/(V s).
- 46.
The equilibrium concentrations are used in the estimates.
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19.1
In the expressions (19.115), (19.118) defining the hole mobility μ p , assume that \(\tau_{ph} \simeq \tau_{pl}\). Letting τ p be the common value, determine the value of the normalized effective mass \(\overline{m}_h/m_0\) to be used in \(\mu_p = q \, \tau_p/\overline{m}_h\) for silicon at room temperature. Also, determine the value of parameter a p in (19.122) in the same conditions.
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Rudan, M. (2015). Mathematical Model of Semiconductor Devices. In: Physics of Semiconductor Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1151-6_19
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DOI: https://doi.org/10.1007/978-1-4939-1151-6_19
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