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Electron spectrum in doped
*n*-Si quantum wires is calculated by the Thomas-Fermi (TF) method under finite temperatures. The many-body exchange corrections are taken into account. The doping profile is arbitrary. At the first stage, the electron potential energy is calculated from a simple two-dimensional equation. The effective iteration scheme is proposed there that is valid for multidimensional problems. Then the energy levels and wave functions of this quantum well are simulated from the Schr
ödinger equations. The expansion by the full set of eigenfunctions of the linear harmonic oscillator is used. The quantum mechanical perturbation theory can be utilized to compute the energy levels. Generally, the perturbation theory for degenerate energy levels should be used.

Investigations of the electron spectra of low-dimensional and highly doped structures are important to many nanotechnology applications including nanoscale transistors, sensors and modern plasmonics [

The combined Schrödinger-Poisson method is widely used to compute spectra of low-dimensional structures [

Investigations of δ-doped quantum structures are possible with a simpler approach based on the statistical Thomas-Fermi (TF) method [

In this paper, it is used the TF method to calculate the electron spectrum in highly doped n-Si quantum wires under finite temperatures T, where the many-body effects like exchange-correlations are taken into account [

Consider a single δ-doped electron quantum wire of n-Si. The arbitrary high doping profile is considered in the plane x, y perpendicular to the axis of the wire OZ. The atomic units are used for distances a 0 * = ε ℚ 2 / ( m c e 2 ) and for energies R y * = e 2 / ( 2 ε a 0 * ) , where m c = ν 2 / 3 ( m ⊥ 2 m ∥ ) 1 / 3 ≈ 1.06 m e ≈ 10 − 27 g , ν = 6 is the number of the lowest electron valleys in Si [_{||}, m_{^}.

At the first stage the electrostatic electron energy V is calculated by the TF method. The single equation for δ-doped electron quantum wire with dimensionless variables is [

Δ ⊥ V ≡ ∂ 2 V ∂ x 2 + ∂ 2 V ∂ y 2 = 8 π { − n [ V ] + N d 0 × ( 2 exp ( μ − E d − V T ) + 1 ) − 1 + N 1 d ( x , y ) } ; V ( x , y → ± ∞ ) = 0 ; (1)

where

n [ V ] = 1 4 ( T π ) 3 / 2 Φ 1 / 2 ( μ − V − V x c T ) ; V x c ≈ − 2 n 1 / 3 ⋅ ( 1 + 0.025 n − 1 / 3 ⋅ log ( 1 + 35 n 1 / 3 ) ) ⋅ f ( n ) ; f ( n ) = { tanh ( ( n / n c ) − 1 ) , n > n c ; n c ≈ 10 17 cm − 3 0 , n ≤ n c ; Φ 1 / 2 ( v ) ≡ 2 π 1 / 2 ∫ 0 ∞ u 1 / 2 d u 1 + exp ( u − v ) ; N 1 d ( x , y ) = N 1 d 0 exp ( − ( x / x 0 ) 2 − ( y / y 0 ) 2 ) . (2)

Here n is the total electron concentration; N_{1d} and N_{d}_{0} are 1D and bulk donor concentrations respectively. Φ_{1/2}(v) is the Fermi integral that has been approximated from [_{1d0} ≥ 10^{20} cm^{−}^{3}; they are fully ionized. The 1D doping is localized at the distances x_{0}, y_{0} ≈ 1 - 5 nm and depends on two coordinates x, y. The value of V_{xc} is the many-body correction to the electron energy due to exchange-correlations [_{c} ≤ 10^{18} cm^{−3}.

Because in n-Si the achievable doping concentrations are N_{1d} ≤ 3 × 10^{21} cm^{−3}, the exchange term is dominating in the many-body corrections in n-Si. When neglecting the correlation term with the logarithm in (2), the error is < 10%. Therefore, below a simpler correction term due to the exchange only is used, V_{x} = −2n^{1/3} × f(n).

The position of the Fermi level μ has been obtained from the condition of the total neutrality [

n [ V = 0 ] = N d 0 ⋅ ( 2 exp ( μ − E d T ) + 1 ) − 1 . (3)

Here E_{d} is the donor energy with respect to the bottom of the conduction band.

In [

∂ V ∂ u − Δ ⊥ V + 8 π { − n [ V ] + N d 0 × ( 2 exp ( μ − E d − V T ) + 1 ) − 1 + N 1 d ( x , y ) } = 0. (4)

The solving of Equation (4) should be realized till establishing stationary solution. Here the iterations have been applied based on the factorization, or Douglas-Rachford, method [

∂ V ∂ u ≈ V ( s + 1 ) − V ( s ) u ≡ χ , χ − u Δ ⊥ χ + u Q ( s ) χ = 8 π { − n [ V ( s ) ] + N d 0 × ( 2 exp ( μ − E d − V ( s ) T ) + 1 ) − 1 + N 1 d ( x , y ) } ≡ F ( s ) ; Q ( s ) ≡ 8 π ∂ ∂ V { − n [ V ] + N d 0 ⋅ ( 2 exp ( μ − E d − V T ) + 1 ) − 1 } , (5)

where s is the number of the iteration.

The iteration algorithm with two fractional steps is used to compute χ and then V^{(s+1)}:

χ ( s + 1 / 2 ) − u ∂ 2 χ ( s + 1 / 2 ) ∂ x 2 + u Q ( s ) χ ( s + 1 / 2 ) = F ( s ) χ ( s + 1 ) − u ∂ 2 χ ( s + 1 ) ∂ y 2 = χ ( s + 1 / 2 ) ; V ( s + 1 ) = V ( s ) + u χ ( s + 1 ) . (6)

The iteration scheme Equation (6) approximates Equation (4) at all the fractional steps. The rapid convergence of the method has been demonstrated even when the exchange energy has been taken into account. To obtain the accuracy < 0.1%, the number of iterations should be ~50, when the iteration parameter u is u = 0.1 - 1. Note that a similar approach based on the Douglas-Rachford, or factorization, method can be applied for general multidimensional structures, whereas in 2D case also the iteration scheme based on the Peaceman-Rachford, or alternate directions, algorithm can be used. While the Peaceman-Rachford method sometimes results in a more rapid convergence, it is not applicable to 3D problems [

After calculations of the electron potential energy V(x,y) and the exchange energy V_{x}(x,y), the energy levels E_{j}, the wave functions Ψ_{j}(x,y) of the discrete spectrum of the quantum well, and the electron concentration n in each electron level have been computed from the Schrödinger equations [

1 M 1 ( ∂ 2 Ψ j ( 1 ) ∂ x 2 + ∂ 2 Ψ j ( 1 ) ∂ y 2 ) − [ W ( x , y ) − E j ( 1 ) ] Ψ j ( 1 ) = 0 ; W ( x , y ) ≡ V ( x , y ) + V x ( x , y ) ; (7a)

1 M 1 ∂ 2 Ψ j ( 2 ) ∂ x 2 + 1 M 2 ∂ 2 Ψ j ( 2 ) ∂ y 2 − [ W ( x , y ) − E j ( 2 ) ] Ψ j ( 2 ) = 0 ; M 1 ≡ m ⊥ m c , M 2 ≡ m | | m c ; ∫ − ∞ + ∞ ∫ − ∞ + ∞ | Ψ j | 2 d x d y = 1 . (7b)

There are two different orientations of electron valleys, as seen from Equations (7).

The main attention is paid here to the solving of more complicated Equation (7b). The preference of the proposed method is in the sequential realization of the finding of the electron potential energy and only then the solving of the Schrödinger equations. The second problem can be solved separately for instance by the standard finite element approach using COMSOL Multiphysics [

Below a simple approximate solution method is realized, which is based on the parabolic approximation of the total potential energy with the exchange correction and the expansion of the wave function by the full set of the eigenfunctions of the linear harmonic oscillator. Then the standard quantum mechanical perturbation theory is applied. The discrete Fourier transform is unsatisfactory, because as the result a non-sparse matrix is formed, as our investigations have been demonstrated. Moreover, there is a problem how to select the functions of the discrete spectrum there; those functions should tend to zero at the infinity.

To solve Equation (7b), the following transformations of variables x, y have been applied:

x ˜ = x x c , y ˜ = x y c . (8)

As a result, Equation (7b) takes the form:

− ( α ∂ 2 Ψ ∂ x ˜ 2 + β ∂ 2 Ψ ∂ y ˜ 2 ) + W ( x , y ) Ψ = E Ψ ; α ≡ 1 M 1 x c 2 , β ≡ 1 M 2 y c 2 (9)

A solution of Equation (9) is searched as a series [

Ψ = ∑ m , n A m n φ m ( x ˜ ) φ n ( y ˜ ) . (10)

An arbitrary complete set of the functions can be used. Here an expansion with the orthonormal Hermite functions φ_{m}(x) is applied [

− d 2 φ m ( x ) d x 2 + x 2 φ m ( x ) = ε m φ m ( x ) , ε m = 2 m + 1 , m = 0 , 1 , 2 , ⋯ (11)

φ m ( x ) = H m ( x ) exp ( − x 2 2 ) , ∫ − ∞ + ∞ φ m ( x ) φ k ( x ) d x = δ m k .

H_{m}(x) are the Hermite polynomials [

Below the following notation is used:

P ( x ˜ , y ˜ ) ≡ W ( x , y ) − α x ˜ 2 − β y ˜ 2 − W 0 . (12)

For lowest energy levels it is rather better to use the values of x_{c}, y_{c} that result in P ≈ 0 near the minimum point x = y = 0, see

The approximate expression for the total electron potential energy near the minimum x = y = 0 is, see

W ( x , y ) ≈ W 0 + 1 2 ∂ 2 W ∂ x 2 ( x = 0 , y = 0 ) x 2 + 1 2 ∂ 2 W ∂ y 2 ( x = 0 , y = 0 ) y 2 , W 0 < 0. (13)

Therefore, to compute the minimum energy levels, the values of x_{c}, y_{c} are suitable:

x c = ( 2 M 1 1 ∂ 2 W ∂ x 2 ) 1 / 4 , y c = ( 2 M 2 1 ∂ 2 W ∂ y 2 ) 1 / 4 ; (14)

To get higher energy levels more accurately, it is possible to increase the values of x_{c}, y_{c}, to get a better approximation of the electron potential energy specifically near these energy values, see _{c} are bigger than those in Equation (14).

Equation (9) is equivalent to the following matrix equation:

( Ε − α ε m − β ε n ) A m n − ∑ k , l P m n , k l A k l = 0 , Ε ≡ E − W 0 . (15)

Here Ε > 0 is the electron energy measured from the bottom of the electron potential energy with the exchange correction. The corresponding matrix elements are:

P m n , k l = ∫ − ∞ + ∞ ∫ − ∞ + ∞ φ m ( x ˜ ) φ k ( x ˜ ) φ n ( y ˜ ) φ l ( y ˜ ) P ( x ˜ , y ˜ ) d x ˜ d y ˜ . (16)

The numerical integration has been done by means of the Gauss quadrature formula [

∫ − ∞ + ∞ exp ( − x 2 ) F ( x ) d x ≈ ∑ j = 1 N w j F ( x j ) . (17)

When it has been used the renumbering (m,n) ? p, Equation (15) can be rewritten:

( Ε − Ε 0 p − P p p ) A p − ∑ q ≠ p P p q A q = 0 , Ε 0 p ≡ α ε m + β ε n . (18)

The full spectrum of the quantum well can be obtained from Equation (18). The eigenvalues can be obtained directly or with using the iterations for the inverse matrix [

Because 2D problem is investigated, a possible degeneration of energy levels may occur. Consider a situation when there are several close levels [

a few 2 - 5 close energy levels, or resonant ones, take place. Let us rewrite Equation (18) for these resonant levels:

( Ε − Ε 0 p j − P p j p j ) A p j − ∑ k ≠ j P p j p k A p k = ∑ q ≠ p P p j q A q . (19)

For another, or nonresonant, levels E_{0q} Equation (18) is simplified:

( Ε − Ε 0 q − P q q ) A q ≈ ∑ p k P q p k A p k . (20)

A q ≈ ∑ p k P q p k A p k ( Ε 0 p k + P p k p k − Ε 0 q − P q q ) . (20a)

As a result, the following secular equation has been derived:

∑ p k M p j p k A p k = Ε ⋅ A p j . , (21)

Here the matrix elements are

M p j p k = Ε 0 p j δ p j p k + P p j p k + ∑ p k P p j q P q p k A p k ( Ε 0 p k + P p k p k − Ε 0 q − P q q ) . (22)

Thus, to find the eigenvalues and eigenfunctions, the problem of seeking the eigenvalues of the relatively small matrix M_{pjpk} should be solved. Really for higher energy levels it is no more than 6 × 6.

In the nondegenerate case for lower energy levels, the standard perturbation method for the eigenvalue of the energy with the number p_{0} is:

Ε ≈ Ε 0 p 0 + P p 0 p 0 + ∑ q ≠ p 0 P p 0 q 2 Ε 0 p 0 + P p 0 p 0 − Ε 0 q − P q q . (23)

Also the variation methods can be used to estimate the eigenvalues [

The following parameters are used: the uniform doping concentration is N_{d}_{0} = 10^{16} cm^{−3}, the maximum values of 1D doping are N_{1d0} = 5 × 10^{20} - 3 × 10^{21} cm^{−3}, the temperature interval is T = 10 - 300 K. The donor energy is E_{d} = −0.045 eV. Thus, at lower temperatures not all the donor levels are ionized. The results of simulations of the total electron potential energy W º V(x,y) + V_{x}(x,y), the exchange energy V_{x}(x,y), and the total electron concentration n(x,y) are presented in Figures 2-4. The energy unit is Ry^{*} ≈ 0.12 eV, the unit for distances x, y is

a 0 ∗ = 0.52 nm. The exchange correction is important for the doping levels N_{1d} ≥ 10^{21} cm^{−3}. But the total electron potential energy W and the electron concentration n is practically the same as without this many-body correction.

The potential energy depends on temperature T, as seen in Figures 2-4, right columns. This result takes place due to the partial ionization of the volume donors at low temperatures, as seen from Equation (3). But the total electron

concentration n does not depend practically on T, see Figures 2-4, left columns, as well as the exchange energy correction, central columns.

After calculating the electron potential energy it is possible to simulate the electron levels of the well and the wave functions from Equation (15). The profiles of three lowest wave functions symmetrical with respect to x and y are presented in

level can be obtained from the perturbation theory without degeneration, see Equation (23). But the general case of the perturbation theory with a possible degeneration should be used to compute higher energy levels, Equations (21), (22). A direct using of the perturbation theory without an account of a possible degeneration may lead to essential errors. An additional possibility to decrease the errors is the using of optimal values of approximation parameters x_{c}, y_{c}, see

A preference of the Thomas-Fermi method is a possible choice of the functions of a preferred symmetry only. In the Schrödinger-Poisson method all the wave functions should be simulated simultaneously.

The dependencies of the corresponding values of the energy on temperature are given in

Also the simulations are presented in

The Thomas-Fermi method with many-body corrections has been applied to calculate the electron spectrum in highly doped n-Si quantum wires of an arbitrary doping profile under finite temperatures T. An effective iteration algorithm is proposed for solving the two-dimensional equation for the electron potential energy. This iteration method can be used for arbitrary orientations of the axes of the quantum wire and for other multidimensional structures like quantum dots. The preference of the proposed method is in the sequential realization of

the finding of the electron potential energy and only then the solving of the Schrödinger equations for electron energy levels and wave functions. The electron eigenvalues and eigenfunctions are calculated then with using an expansion by the full set of the linear harmonic oscillator functions. The perturbation theory with possible degeneration can be applied to compute the eigenvalues and eigenfunctions.

The authors thank to SEP-CONACyT (Mexico) for a partial support of our work.

The authors declare no conflicts of interest regarding the publication of this paper.

Grimalsky, V., Koshevaya, S., Escobedo-Alatorre, J. and Moroz, I. (2018) Simulations of the Electron Spectrum of Quantum Wires in n-Si of Arbitrarily Doping Profile by Thomas-Fermi Method. Journal of Electromagnetic Analysis and Applications, 10, 143-156. https://doi.org/10.4236/jemaa.2018.108011