public DoubleVector Get_EnergyLevels(ILayer[] layers, ref SpinResolved_Data charge_density, Band_Data chem_pot)
{
// interpolate the input charge density and chemical potential onto a reduced domain to simplify DFT solve
SpinResolved_Data dft_dens = new SpinResolved_Data(new Band_Data(new DoubleVector(nz)), new Band_Data(new DoubleVector(nz)));
Band_Data dft_pot = new Band_Data(new DoubleVector(nz));
Interpolate_DFT_Grid(ref dft_dens, ref dft_pot, charge_density, chem_pot);
Get_Potential(ref dft_pot, layers);
DoubleHermitianMatrix hamiltonian = Create_Hamiltonian(layers, dft_pot);
DoubleHermitianEigDecomp eig_decomp = new DoubleHermitianEigDecomp(hamiltonian);
System.IO.StreamWriter sw_pot = new System.IO.StreamWriter("tmp_pot.dat");
System.IO.StreamWriter sw_dens = new System.IO.StreamWriter("tmp_dens.dat");
Band_Data dft_dens_spin_summed = dft_dens.Spin_Summed_Data;
for (int i = 0; i < dft_pot.Length; i++)
{
sw_dens.WriteLine(dft_dens_spin_summed[i].ToString());
sw_pot.WriteLine(dft_pot[i].ToString());
}
sw_dens.Close();
sw_pot.Close();
return(eig_decomp.EigenValues);
}
DoubleHermitianMatrix Create_H_Using_SOIP(Band_Data dft_pot, double k)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(2 * nx * ny);
// create the SOIP parts of the matrix
result = Create_SOIP_Matrix(Direction.x, Direction.y, Direction.z, k, nx, ny);
result += Create_SOIP_Matrix(Direction.y, Direction.x, Direction.z, k, nx, ny);
// and add the scalar terms from the k-direction
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// add momentum and potential terms
result[i * ny + j, i *ny + j] += Physics_Base.hbar * Physics_Base.hbar * k * k / (2.0 * mass) + dft_pot.mat[i, j];
result[i * ny + j + nx * ny, i *ny + j + nx * ny] += Physics_Base.hbar * Physics_Base.hbar * k * k / (2.0 * mass) + dft_pot.mat[i, j];
// add spin-orbit terms
result[i * ny + j, i *ny + j + nx * ny] += (-1.0 * I * alpha * dV_x.mat[i, j] - alpha * dV_y.mat[i, j]) * k;
result[i * ny + j + nx * ny, i *ny + j] += (I * alpha * dV_x.mat[i, j] - alpha * dV_y.mat[i, j]) * k;
}
}
return(result);
}
DoubleHermitianMatrix Create_Hamiltonian(Band_Data dft_pot, double k)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(2 * nx * ny);
DoubleHermitianMatrix h_upup, h_updn, h_dnup, h_dndn;
// create sub-matrices with no spin orbit
h_upup = Create_NoSOI_Hamiltonian(dft_pot, k, nx, ny); // takes up to up (ie. H_11)
h_dndn = Create_NoSOI_Hamiltonian(dft_pot, k, nx, ny); // takes dn to dn (ie. H_22)
h_updn = new DoubleHermitianMatrix(nx * ny); // takes up to dn (ie. H_21)
h_dnup = new DoubleHermitianMatrix(nx * ny); // takes dn to up (ie. H_12)
// add spin-orbit
h_upup -= alpha * Create_SOI_Hamiltonian(Direction.y, Direction.x, k, nx, ny);
h_dndn -= -1.0 * alpha * Create_SOI_Hamiltonian(Direction.y, Direction.x, k, nx, ny);
h_updn -= alpha * (Create_SOI_Hamiltonian(Direction.z, Direction.y, nx, ny) - Create_SOI_Hamiltonian(Direction.y, Direction.z, nx, ny)
+ I * Create_SOI_Hamiltonian(Direction.z, Direction.x, k, nx, ny));
h_dnup -= alpha * (Create_SOI_Hamiltonian(Direction.z, Direction.y, nx, ny) - Create_SOI_Hamiltonian(Direction.y, Direction.z, nx, ny)
- I * Create_SOI_Hamiltonian(Direction.z, Direction.x, k, nx, ny));
// recombine matrices and output result
for (int i = 0; i < nx * ny; i++)
{
for (int j = 0; j < nx * ny; j++)
{
result[i, j] = h_upup[i, j];
result[i, j + nx * ny] = h_dnup[i, j];
result[i + nx * ny, j] = h_updn[i, j];
result[i + nx * ny, j + nx * ny] = h_dndn[i, j];
}
}
return(result);
}
DoubleHermitianMatrix Create_Hamiltonian(double[] V, double t, int N)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(N);
// set off diagonal elements
for (int i = 0; i < N; i++)
{
// coupling sites in the growth direction
if (i != 0)
{
result[i - 1, i] = t;
}
if (i != N - 1)
{
result[i + 1, i] = t;
}
}
// set diagonal elements
for (int i = 0; i < N; i++)
{
result[i, i] = -2.0 * t + V[i];
}
return(result);
}
DoubleHermitianMatrix Create_2DEG_Hamiltonian(double[,] V, double tx, double ty, int nx, int ny, bool periodic_in_x, bool periodic_in_y)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(nx * ny);
// set off diagonal elements
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// coupling sites in the transport direction
if (i != 0)
{
result[i * ny + j, i *ny + j - ny] = tx;
}
if (i != nx - 1)
{
result[i * ny + j, i *ny + j + ny] = tx;
}
// coupling sites in the transverse direction
if (j != 0)
{
result[i * ny + j, i *ny + j - 1] = ty;
}
if (j != ny - 1)
{
result[i * ny + j, i *ny + j + 1] = ty;
}
}
}
// set diagonal elements
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
result[i * ny + j, i *ny + j] = -2.0 * tx + -2.0 * ty + V[i, j];
}
}
// and finally, add periodic boundary conditions where appropriate
if (periodic_in_x)
{
for (int j = 0; j < ny; j++)
{
result[j, j + (nx - 1) * ny] = tx;
result[j + (nx - 1) * ny, j] = tx;
}
}
if (periodic_in_y)
{
for (int i = 0; i < nx; i++)
{
result[i * ny + 1, i *ny] = ty;
result[i * ny, i *ny + 1] = ty;
}
}
return(result);
}
/// <summary>
/// Calculates and prints the band structure
/// </summary>
void Print_Band_Structure(Band_Data dft_pot, ILayer[] layers, int Nk, double dk, string outfile, int max_eigval)
{
// set temperature high so that all wave functions will be calculated
double old_temperature = temperature;
temperature = 1000.0;
if (dV_x == null)
{
throw new Exception("Error - Band structure derivatives are null! Have you initiated this type properly by calling Get_SOI_parameters(Band_Data chem_pot)?");
}
// calculate the energies up to a given maximum energy of the lowest state
double k = 0;
double[][] energies = new double[Nk][];
for (int i = 0; i < Nk; i++)
{
k = i * dk;
Console.WriteLine(i.ToString() + ": Calculating for k = " + k.ToString());
// generate the Hamiltonian for this k value
DoubleHermitianMatrix hamiltonian = Create_H_Using_SOIP(dft_pot, k);
// and diagonalise it
int max_wavefunction;
DoubleHermitianEigDecomp eig_decomp = Diagonalise_Hamiltonian(hamiltonian, out max_wavefunction);
// add the calculated energies up to either the maximum required eigenvalue or
// to the maximum calculated wave function (which is 50*kb*T above the chemical potential)
double[] tmp_energies = new double[max_eigval];
for (int j = 0; j < max_eigval; j++)
{
if (j < max_wavefunction)
{
tmp_energies[j] = eig_decomp.EigenValues[j];// - 0.5 * Physics_Base.hbar * Physics_Base.hbar * k * k / Physics_Base.mass;
}
else
{
tmp_energies[j] = eig_decomp.EigenValues[max_wavefunction - 1];// -0.5 * Physics_Base.hbar * Physics_Base.hbar * k * k / Physics_Base.mass;
}
}
energies[i] = tmp_energies;
}
// output the data to file
StreamWriter sw = new StreamWriter(outfile);
for (int i = 0; i < Nk; i++)
{
for (int j = 0; j < max_eigval; j++)
{
sw.Write(energies[i][j].ToString() + "\t");
}
sw.WriteLine();
}
sw.Close();
// reset temperature
temperature = old_temperature;
}
/// <summary>
/// returns an eigenvector decomposition for the x-direction after having calculated the density in the
/// y-direction and storing it in "charge_density"
/// </summary>
/// <param name="dft_pot"></param>
/// <param name="charge_density"></param>
/// <returns></returns>
private DoubleHermitianEigDecomp Solve_Eigenvector_Problem(Band_Data dft_pot, ref SpinResolved_Data charge_density)
{
DoubleHermitianEigDecomp eig_decomp;
double[] x_energy = new double[nx];
DoubleMatrix dens_up = new DoubleMatrix(nx, ny, 0.0);
DoubleMatrix dens_down = new DoubleMatrix(nx, ny, 0.0);
// cycle over x-direction calculating the ground state energy and eigenstate in the y-direction
for (int i = 0; i < nx; i++)
{
// pull out the chemical potential for this slice
double[] y_dft_pot = new double[ny];
for (int j = 0; j < ny; j++)
{
y_dft_pot[j] = dft_pot.mat[i, j];
}
// calculate its eigendecomposition
DoubleHermitianMatrix h_y = Create_Hamiltonian(y_dft_pot, ty, ny);
eig_decomp = new DoubleHermitianEigDecomp(h_y);
// insert the eigenstate into density and record the local confinement energy
x_energy[i] = eig_decomp.EigenValue(0);
for (int j = 0; j < ny; j++)
{
double dens_tot = DoubleComplex.Norm(eig_decomp.EigenVector(0)[j]) * DoubleComplex.Norm(eig_decomp.EigenVector(0)[j]);
dens_up[i, j] = 0.5 * dens_tot;
dens_down[i, j] = 0.5 * dens_tot;
}
}
// check whether to enforce symmetry in the transverse direction
if (force_symmetry)
{
for (int i = nx / 2 + 1; i < nx; i++)
{
x_energy[i] = x_energy[nx - i - 1];
for (int j = 0; j < ny; j++)
{
dens_up[i, j] = dens_up[nx - i - 1, j];
dens_down[i, j] = dens_down[nx - i - 1, j];
}
}
}
// calculate the eigenstates in the x-direction
DoubleHermitianMatrix h_x = Create_Hamiltonian(x_energy, tx, nx);
eig_decomp = new DoubleHermitianEigDecomp(h_x);
energies = eig_decomp.EigenValues;
// put the calculated densities into charge_density
charge_density = new SpinResolved_Data(new Band_Data(dens_up), new Band_Data(dens_down));
return(eig_decomp);
}
public override void Get_ChargeDensity_Deriv(ILayer[] layers, ref SpinResolved_Data charge_density_deriv, Band_Data chem_pot)
{
Get_SOI_parameters(chem_pot);
// if (dV_x == null)
// throw new Exception("Error - Band structure derivatives are null! Have you initiated this type properly by calling Get_SOI_parameters(Band_Data chem_pot)?");
// convert the chemical potential into a quantum mechanical potential
Band_Data dft_pot = chem_pot.DeepenThisCopy();
Get_Potential(ref dft_pot, layers);
// reset charge density
charge_density_deriv = 0.0 * charge_density_deriv;
// integrate up density from k = 0 to k = k_F
int count = 0;
double k = 0;
// create initial hamiltonian and eigendecomposition
int max_wavefunction_old_p, max_wavefunction_old_m;
DoubleHermitianMatrix hamiltonian_p = Create_H_Using_SOIP(dft_pot, k);
DoubleHermitianEigDecomp eig_decomp_old_p = Diagonalise_Hamiltonian(hamiltonian_p, out max_wavefunction_old_p);
DoubleHermitianEigDecomp eig_decomp_old_m = Diagonalise_Hamiltonian(hamiltonian_p, out max_wavefunction_old_m);
while (k < kmax)
{
// increment the wavevector
k += delta_k;
count += 1;
// create new decompositions for forwards and backwards analysis
int max_wavefunction_p, max_wavefunction_m;
DoubleHermitianEigDecomp eig_decomp_p = Diagonalise_Hamiltonian(Create_H_Using_SOIP(dft_pot, k), out max_wavefunction_p);
DoubleHermitianEigDecomp eig_decomp_m = Diagonalise_Hamiltonian(Create_H_Using_SOIP(dft_pot, -1.0 * k), out max_wavefunction_m);
SpinResolved_Data new_charge_deriv = new SpinResolved_Data(nx, ny);
// cycle over each of the positive bands
for (int i = 0; i < max_wavefunction_p; i++)
{
new_charge_deriv += Calculate_Density_Derivative(eig_decomp_old_p, eig_decomp_p, i);
}
// and negative bands
for (int i = 0; i < max_wavefunction_m; i++)
{
new_charge_deriv += Calculate_Density_Derivative(eig_decomp_old_m, eig_decomp_m, i);
}
// set new eigenvalue decompositions and max_wavefunctions to the old ones
eig_decomp_old_m = eig_decomp_m; eig_decomp_old_p = eig_decomp_p;
max_wavefunction_old_m = max_wavefunction_m; max_wavefunction_old_p = max_wavefunction_p;
// and finally, add the charge density calculated
charge_density_deriv += new_charge_deriv;
}
}
/// <summary>
/// Calculate the charge density for this potential
/// NOTE!!! that the boundary potential is (essentially) set to infty by ringing the density with a set of zeros.
/// this prevents potential solvers from extrapolating any residual density at the edge of the eigenstate
/// solution out of the charge density calculation domain
/// </summary>
/// <param name="layers"></param>
/// <param name="charge_density"></param>
/// <param name="chem_pot"></param>
public override void Get_ChargeDensity(ILayer[] layers, ref SpinResolved_Data charge_density, Band_Data chem_pot)
{
// convert the chemical potential into a quantum mechanical potential
Band_Data dft_pot = chem_pot.DeepenThisCopy();
Get_Potential(ref dft_pot, layers);
DoubleHermitianMatrix hamiltonian = Create_Hamiltonian(layers, dft_pot);
DoubleHermitianEigDecompServer eig_server = new DoubleHermitianEigDecompServer();
eig_server.ComputeEigenValueRange(dft_pot.mat.Min(), no_kb_T * Physics_Base.kB * temperature);
eig_server.ComputeVectors = true;
DoubleHermitianEigDecomp eig_decomp = eig_server.Factor(hamiltonian);
int max_wavefunction = 0;
if (eig_decomp.EigenValues.Length != 0)
{
double min_eigval = eig_decomp.EigenValues.Min();
max_wavefunction = (from val in eig_decomp.EigenValues
where val < no_kb_T * Physics_Base.kB * temperature
select val).ToArray().Length;
}
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
double dens_val = 0.0;
// do not add anything to the density if on the edge of the domain
if (i == 0 || i == nx - 1 || j == 0 || j == ny - 1)
{
charge_density.Spin_Up.mat[i, j] = 0.0;
charge_density.Spin_Down.mat[i, j] = 0.0;
continue;
}
for (int k = 0; k < max_wavefunction; k++)
{
// and integrate the density of states at this position for this eigenvector from the minimum energy to
// (by default) 50 * k_b * T above mu = 0
dens_val += DoubleComplex.Norm(eig_decomp.EigenVector(k)[i * ny + j]) * DoubleComplex.Norm(eig_decomp.EigenVector(k)[i * ny + j]) * Get_OneD_DoS(eig_decomp.EigenValue(k), no_kb_T);
}
// just share the densities (there is no spin polarisation)
charge_density.Spin_Up.mat[i, j] = 0.5 * dens_val;
charge_density.Spin_Down.mat[i, j] = 0.5 * dens_val;
}
}
// and multiply the density by -e to get the charge density (as these are electrons)
charge_density = unit_charge * charge_density;
}
/// <summary>
/// returns the eigen-energies for the given potential and charge density
/// </summary>
public override DoubleVector Get_EnergyLevels(ILayer[] layers, Band_Data pot)
{
// convert the chemical potential into a quantum mechanical potential
Band_Data band_pot = pot.DeepenThisCopy();
Get_Potential(ref band_pot, layers);
DoubleHermitianMatrix hamiltonian = Create_Hamiltonian(layers, band_pot);
DoubleHermitianEigDecomp eig_decomp = new DoubleHermitianEigDecomp(hamiltonian);
return(eig_decomp.EigenValues);
}
/// <summary>
/// Calculate the charge density for this potential
/// NOTE!!! that the boundary potential is (essentially) set to infty by ringing the density with a set of zeros.
/// this prevents potential solvers from extrapolating any residual density at the edge of the eigenstate
/// solution out of the charge density calculation domain
/// </summary>
/// <param name="layers"></param>
/// <param name="charge_density_deriv"></param>
/// <param name="chem_pot"></param>
public void Get_ChargeDensity_Deriv(ILayer[] layers, ref SpinResolved_Data charge_density_deriv, Band_Data chem_pot)
{
// interpolate the input charge density and chemical potential onto a reduced domain to simplify DFT solve
SpinResolved_Data dft_dens_deriv = new SpinResolved_Data(nz);
Get_ChargeDensity(layers, ref charge_density_deriv, chem_pot);
Band_Data dft_pot = new Band_Data(new DoubleVector(nz));
Interpolate_DFT_Grid(ref dft_dens_deriv, ref dft_pot, charge_density_deriv, chem_pot);
Get_Potential(ref dft_pot, layers);
// // set dft_dens to zero so that the hamiltonian doesn't include the XC term
// dft_dens = 0.0 * dft_dens;
DoubleHermitianMatrix hamiltonian = Create_Hamiltonian(layers, dft_pot);
DoubleHermitianEigDecomp eig_decomp = new DoubleHermitianEigDecomp(hamiltonian);
double min_eigval = eig_decomp.EigenValues.Min();
max_wavefunction = (from val in eig_decomp.EigenValues
where val < no_kb_T * Physics_Base.kB * temperature
select val).ToArray().Length;
DoubleVector dens_up_deriv = new DoubleVector(nz, 0.0);
DoubleVector dens_down_deriv = new DoubleVector(nz, 0.0);
for (int j = 0; j < nz; j++)
{
double dens_val = 0.0;
for (int i = 0; i < max_wavefunction; i++)
{
// and integrate the density of states at this position for this eigenvector from the minimum energy to
// (by default) 50 * k_b * T above mu = 0
//dens_val += dens_of_states.Integrate(min_eigval, no_kb_T * Physics_Base.kB * temperature);
dens_val += DoubleComplex.Norm(eig_decomp.EigenVector(i)[j]) * DoubleComplex.Norm(eig_decomp.EigenVector(i)[j]) * Get_TwoD_DoS_Deriv(eig_decomp.EigenValue(i), no_kb_T);// *mass / (2.0 * Math.PI * Physics_Base.hbar * Physics_Base.hbar);
}
// just share the densities (there is no spin polarisation)
dens_up_deriv[j] = 0.5 * dens_val;
dens_down_deriv[j] = 0.5 * dens_val;
}
// and multiply the density derivative by e to get the charge density and by e to convert it to d/dphi (as increasing phi decreases the charge: dn/dphi*-e^2 )
dft_dens_deriv = -1.0 * unit_charge * unit_charge * new SpinResolved_Data(new Band_Data(dens_up_deriv), new Band_Data(dens_down_deriv));
Insert_DFT_Charge(ref charge_density_deriv, dft_dens_deriv);
}
/// <summary>
/// returns an eigenvector decomposition for the xy-plane after having calculated the density in the
/// z-direction and storing it in "charge_density"
/// </summary>
/// <param name="dft_pot"></param>
/// <param name="charge_density"></param>
/// <returns></returns>
private double[,] Solve_Eigenvector_Problem(Band_Data dft_pot, ref SpinResolved_Data charge_density)
{
DoubleHermitianEigDecomp eig_decomp;
double[,] xy_energy = new double[nx, ny];
Band_Data dens_up = new Band_Data(nx, ny, nz, 0.0);
Band_Data dens_down = new Band_Data(nx, ny, nz, 0.0);
// cycle over xy-plane calculating the ground state energy and eigenstate in the growth direction
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// pull out the chemical potential for this slice
double[] z_dft_pot = new double[nz];
for (int k = 0; k < nz; k++)
{
z_dft_pot[k] = dft_pot.vol[k][i, j];
}
// calculate its eigendecomposition
DoubleHermitianMatrix h_z = Create_Hamiltonian(z_dft_pot, tz, nz);
eig_decomp = new DoubleHermitianEigDecomp(h_z);
// insert the eigenstate into density and record the local confinement energy
xy_energy[i, j] = eig_decomp.EigenValue(0);
for (int k = 0; k < nz; k++)
{
double dens_tot = DoubleComplex.Norm(eig_decomp.EigenVector(0)[k]) * DoubleComplex.Norm(eig_decomp.EigenVector(0)[k]);
dens_up.vol[k][i, j] = 0.5 * dens_tot;
dens_down.vol[k][i, j] = 0.5 * dens_tot;
}
}
}
// calculate the eigenstates in the xy-plane
/*DoubleHermitianMatrix h_xy = Create_2DEG_Hamiltonian(xy_energy, tx, ty, nx, ny, true, false);
* eig_decomp = new DoubleHermitianEigDecomp(h_xy);
* energies = eig_decomp.EigenValues;*/
// put the calculated densities into charge_density
charge_density = new SpinResolved_Data(dens_up, dens_down);
return(xy_energy);
}
public void Write_Out_Hamiltonian(DoubleHermitianMatrix h)
{
System.IO.StreamWriter sw = new System.IO.StreamWriter("hamiltonian.dat");
for (int i = 0; i < h.Cols; i++)
{
for (int j = 0; j < h.Rows; j++)
{
sw.Write(h[i, j].Real.ToString() + '\t');
if (j == h.Rows - 1)
{
sw.WriteLine();
}
}
}
sw.Close();
}
/// <summary>
/// Calculate the charge density for this potential
/// NOTE!!! that the boundary potential is (essentially) set to infty by ringing the density with a set of zeros.
/// this prevents potential solvers from extrapolating any residual density at the edge of the eigenstate
/// solution out of the charge density calculation domain
/// </summary>
/// <param name="layers"></param>
/// <param name="charge_density_deriv"></param>
/// <param name="chem_pot"></param>
public override void Get_ChargeDensity_Deriv(ILayer[] layers, ref SpinResolved_Data charge_density_deriv, Band_Data chem_pot)
{
// convert the chemical potential into a quantum mechanical potential
Band_Data dft_pot = chem_pot.DeepenThisCopy();
Get_Potential(ref dft_pot, layers);
DoubleHermitianMatrix hamiltonian = Create_Hamiltonian(layers, dft_pot);
DoubleHermitianEigDecomp eig_decomp = new DoubleHermitianEigDecomp(hamiltonian);
double min_eigval = eig_decomp.EigenValues.Min();
int max_wavefunction = (from val in eig_decomp.EigenValues
where val < no_kb_T * Physics_Base.kB * temperature
select val).ToArray().Length;
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
double dens_val_deriv = 0.0;
// do not add anything to the density if on the edge of the domain
if (i == 0 || i == nx - 1 || j == 0 || j == ny - 1)
{
charge_density_deriv.Spin_Up.mat[i, j] = 0.0;
charge_density_deriv.Spin_Down.mat[i, j] = 0.0;
continue;
}
for (int k = 0; k < max_wavefunction; k++)
{
// and integrate the density of states at this position for this eigenvector from the minimum energy to
// (by default) 50 * k_b * T above mu = 0
dens_val_deriv += DoubleComplex.Norm(eig_decomp.EigenVector(k)[i * ny + j]) * DoubleComplex.Norm(eig_decomp.EigenVector(k)[i * ny + j]) * Get_OneD_DoS_Deriv(eig_decomp.EigenValue(k), no_kb_T);
}
// just share the densities (there is no spin polarisation)
charge_density_deriv.Spin_Up.mat[i, j] = 0.5 * dens_val_deriv;
charge_density_deriv.Spin_Down.mat[i, j] = 0.5 * dens_val_deriv;
}
}
// and multiply the density derivative by e to get the charge density and by e to convert it to d/dphi (as increasing phi decreases the charge: dn/dphi*-e^2 )
charge_density_deriv = unit_charge * unit_charge * charge_density_deriv;
}
DoubleHermitianMatrix Create_Hamiltonian(ILayer[] layers, Band_Data pot)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(nz);
// set off diagonal elements
for (int i = 0; i < nz - 1; i++)
{
result[i + 1, i] = t; result[i, i + 1] = t;
}
// set diagonal elements
for (int i = 0; i < nz; i++)
{
result[i, i] = -2.0 * t + pot[i];
}
return(result);
}
DoubleHermitianMatrix Create_NoSOI_Hamiltonian(Band_Data dft_pot, double k, int nx, int ny)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(nx * ny);
// set off diagonal elements
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// coupling sites in the growth direction
if (i != 0)
{
result[i * ny + j, i *ny + j - ny] = tx;
}
if (i != nx - 1)
{
result[i * ny + j, i *ny + j + ny] = tx;
}
// coupling sites in the transverse direction
if (j != 0)
{
result[i * ny + j, i *ny + j - 1] = ty;
}
if (j != ny - 1)
{
result[i * ny + j, i *ny + j + 1] = ty;
}
}
}
double[,] potential = new double[nx, ny];
// set diagonal elements
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
potential[i, j] = dft_pot.mat[i, j];
potential[i, j] += Physics_Base.hbar * Physics_Base.hbar * k * k / (2.0 * mass);
result[i * ny + j, i *ny + j] = -2.0 * tx + -2.0 * ty + potential[i, j];
}
}
return(result);
}
public override void Get_ChargeDensity(ILayer[] layers, ref SpinResolved_Data charge_density, Band_Data chem_pot)
{
// interpolate the input charge density and chemical potential onto a reduced domain to simplify DFT solve
SpinResolved_Data dft_dens = new SpinResolved_Data(nz);
Band_Data dft_pot = new Band_Data(new DoubleVector(nz));
Interpolate_DFT_Grid(ref dft_dens, ref dft_pot, charge_density, chem_pot);
Get_Potential(ref dft_pot, layers);
DoubleHermitianMatrix hamiltonian = Create_Hamiltonian(layers, dft_pot);
DoubleHermitianEigDecomp eig_decomp = new DoubleHermitianEigDecomp(hamiltonian);
double min_eigval = eig_decomp.EigenValues.Min();
max_wavefunction = (from val in eig_decomp.EigenValues
where val < no_kb_T * Physics_Base.kB * temperature
select val).ToArray().Length;
DoubleVector dens_up = new DoubleVector(nz, 0.0);
DoubleVector dens_down = new DoubleVector(nz, 0.0);
for (int j = 0; j < nz; j++)
{
double dens_val = 0.0;
for (int i = 0; i < max_wavefunction; i++)
{
// and integrate the density of states at this position for this eigenvector from the minimum energy to
// (by default) 50 * k_b * T above mu = 0
//dens_val += dens_of_states.Integrate(min_eigval, no_kb_T * Physics_Base.kB * temperature);
dens_val += DoubleComplex.Norm(eig_decomp.EigenVector(i)[j]) * DoubleComplex.Norm(eig_decomp.EigenVector(i)[j]) * Get_TwoD_DoS(eig_decomp.EigenValue(i), no_kb_T);
}
// just share the densities (there is no spin polarisation)
dens_up[j] = 0.5 * dens_val;
dens_down[j] = 0.5 * dens_val;
}
// and multiply the density by -e to get the charge density (as these are electrons)
dft_dens = unit_charge * new SpinResolved_Data(new Band_Data(dens_up), new Band_Data(dens_down));
Insert_DFT_Charge(ref charge_density, dft_dens);
}
DoubleHermitianMatrix Create_Hamiltonian(ILayer[] layers, Band_Data pot)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(nx * ny);
// set off diagonal elements
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// coupling sites in the transverse direction
if (i != 0)
{
result[i * ny + j, i *ny + j - ny] = tx;
}
if (i != nx - 1)
{
result[i * ny + j, i *ny + j + ny] = tx;
}
// coupling sites in the growth direction
if (j != 0)
{
result[i * ny + j, i *ny + j - 1] = ty;
}
if (j != ny - 1)
{
result[i * ny + j, i *ny + j + 1] = ty;
}
}
}
double[,] potential = new double[nx, ny];
// set diagonal elements
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
result[i * ny + j, i *ny + j] = -2.0 * tx + -2.0 * ty + pot.mat[i, j];
}
}
return(result);
}
DoubleHermitianEigDecomp Diagonalise_Hamiltonian(DoubleHermitianMatrix hamiltonian, out int max_wavefunction)
{
DoubleHermitianEigDecomp eig_decomp;
DoubleHermitianEigDecompServer eig_server = new DoubleHermitianEigDecompServer();
eig_server.ComputeEigenValueRange(E_min, no_kb_T * Physics_Base.kB * temperature);
eig_server.ComputeVectors = true;
eig_decomp = eig_server.Factor(hamiltonian);
max_wavefunction = 0;
if (eig_decomp.EigenValues.Length != 0)
{
double min_eigval = eig_decomp.EigenValues.Min();
max_wavefunction = (from val in eig_decomp.EigenValues
where val < no_kb_T * Physics_Base.kB * temperature
select val).ToArray().Length;
}
return(eig_decomp);
}
DoubleComplexMatrix Product(DoubleHermitianMatrix A, DoubleComplexMatrix B)
{
return new DoubleComplexMatrix(NMathFunctions.Product(MatrixFunctions.ToGeneralMatrix(A), B));
}
DoubleHermitianMatrix Create_SOI_Hamiltonian(Direction dV, Direction p, double k, int nx, int ny)
{
DoubleHermitianMatrix result = 0.0 * new DoubleHermitianMatrix(nx * ny);
Band_Data dV_data;
if (dV == Direction.x)
{
return(result);
}
else if (dV == Direction.y)
{
dV_data = dV_x;
}
else if (dV == Direction.z)
{
dV_data = dV_y;
}
else
{
throw new Exception("Error - It's completely impossible to get here");
}
if (p == Direction.x)
{
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
result[i * ny + j, i *ny + j] = Physics_Base.hbar * k * dV_data.mat[i, j];
}
}
}
else if (p == Direction.y)
{
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
if (i != nx - 1)
{
result[i * ny + j, i *ny + j + ny] = -0.5 * I * Physics_Base.hbar * dV_data.mat[i, j] / dx;
}
if (i != 0)
{
result[i * ny + j, i *ny + j - ny] = 0.5 * I * Physics_Base.hbar * dV_data.mat[i, j] / dx;
}
}
}
}
else if (p == Direction.z)
{
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
if (i != nx - 1)
{
result[i * ny + j, i *ny + j + 1] = -0.5 * I * Physics_Base.hbar * dV_data.mat[i, j] / dy;
}
if (i != 0)
{
result[i * ny + j, i *ny + j - 1] = 0.5 * I * Physics_Base.hbar * dV_data.mat[i, j] / dy;
}
}
}
}
else
{
throw new Exception("Error - It's completely impossible to get here");
}
return(result);
}
DoubleComplexMatrix Product(DoubleHermitianMatrix A, DoubleComplexMatrix B)
{
return(new DoubleComplexMatrix(NMathFunctions.Product(MatrixFunctions.ToGeneralMatrix(A), B)));
}
private DoubleHermitianMatrix Create_SOIP_Matrix(Direction p, Direction dV, Direction sigma, double k, int nx, int ny)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(2 * nx * ny);
double theta;
Band_Data dV_data;
if (dV == Direction.x)
{
dV_data = dV_x;
theta = theta_x;
}
else if (dV == Direction.y)
{
dV_data = dV_y;
theta = theta_y;
}
else if (dV == Direction.z)
{
dV_data = new Band_Data(nx, ny, 0.0);
theta = 0.0;
}
else
{
throw new Exception("Error - It's completely impossible to get here");
}
if (p == Direction.x)
{
if (sigma == Direction.x)
{
throw new InvalidArgumentException("Error - no spin-orbit interaction between p_x and sigma_x");
}
else if (sigma == Direction.y)
{
throw new NotImplementedException();
}
else if (sigma == Direction.z)
{
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// off-diagonal couplings in x backward
if (i != 0)
{
// for up -> up
result[i * ny + j, i *ny + j - ny] = NMathFunctions.Exp(-1.0 * I * theta * dV_data.mat[i, j]) * tx;
// for down -> down
result[i * ny + j + nx * ny, i *ny + j - ny + nx * ny] = NMathFunctions.Exp(I * theta * dV_data.mat[i, j]) * tx;
}
// off-diagonal couplings in x forward
if (i != nx - 1)
{
// for up -> up
result[i * ny + j, i *ny + j + ny] = NMathFunctions.Exp(-1.0 * I * theta * dV_data.mat[i, j]) * tx;
// for down -> down
result[i * ny + j + nx * ny, i *ny + j + ny + nx * ny] = NMathFunctions.Exp(I * theta * dV_data.mat[i, j]) * tx;
}
// and diagonal couplings
result[i * ny + j, i *ny + j] = -2.0 * Math.Cos(theta * dV_data.mat[i, j]) * tx;
result[i * ny + j + nx * ny, i *ny + j + nx * ny] = -2.0 * Math.Cos(theta * dV_data.mat[i, j]) * tx;
}
}
}
else
{
throw new Exception("Error - It's completely impossible to get here");
}
}
else if (p == Direction.y)
{
if (sigma == Direction.x)
{
throw new NotImplementedException();
}
else if (sigma == Direction.y)
{
throw new InvalidArgumentException("Error - no spin-orbit interaction between p_y and sigma_y");
}
else if (sigma == Direction.z)
{
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// off-diagonal couplings in y backward
if (j != 0)
{
// for up -> up
result[i * ny + j, i *ny + j - 1] = NMathFunctions.Exp(-1.0 * I * theta * dV_data.mat[i, j]) * ty;
// for down -> down
result[i * ny + j + nx * ny, i *ny + j - 1 + nx * ny] = NMathFunctions.Exp(I * theta * dV_data.mat[i, j]) * ty;
}
// off-diagonal couplings in y forward
if (j != ny - 1)
{
// for up -> up
result[i * ny + j, i *ny + j + 1] = NMathFunctions.Exp(-1.0 * I * theta * dV_data.mat[i, j]) * ty;
// for down -> down
result[i * ny + j + nx * ny, i *ny + j + 1 + nx * ny] = NMathFunctions.Exp(I * theta * dV_data.mat[i, j]) * ty;
}
// and diagonal couplings
result[i * ny + j, i *ny + j] = -2.0 * Math.Cos(theta * dV_data.mat[i, j]) * ty;
result[i * ny + j + nx * ny, i *ny + j + nx * ny] = -2.0 * Math.Cos(theta * dV_data.mat[i, j]) * ty;
}
}
}
else
{
throw new Exception("Error - It's completely impossible to get here");
}
}
else if (p == Direction.z)
{
throw new NotImplementedException("Error - at the moment, when momentum is in the direction of the wire, you should use a different method");
}
else
{
throw new Exception("Error - It's completely impossible to get here");
}
return(result);
}
public void Write_Out_Hamiltonian(DoubleHermitianMatrix h)
{
System.IO.StreamWriter sw = new System.IO.StreamWriter("hamiltonian.dat");
for (int i = 0; i < h.Cols; i++)
for (int j = 0; j < h.Rows; j++)
{
sw.Write(h[i, j].Real.ToString() + '\t');
if (j == h.Rows - 1)
sw.WriteLine();
}
sw.Close();
}
DoubleHermitianEigDecomp Diagonalise_Hamiltonian(DoubleHermitianMatrix hamiltonian, out int max_wavefunction)
{
DoubleHermitianEigDecomp eig_decomp;
DoubleHermitianEigDecompServer eig_server = new DoubleHermitianEigDecompServer();
eig_server.ComputeEigenValueRange(E_min, no_kb_T * Physics_Base.kB * temperature);
eig_server.ComputeVectors = true;
eig_decomp = eig_server.Factor(hamiltonian);
max_wavefunction = 0;
if (eig_decomp.EigenValues.Length != 0)
{
double min_eigval = eig_decomp.EigenValues.Min();
max_wavefunction = (from val in eig_decomp.EigenValues
where val < no_kb_T * Physics_Base.kB * temperature
select val).ToArray().Length;
}
return eig_decomp;
}
private DoubleHermitianMatrix Create_SOIP_Matrix(Direction p, Direction dV, Direction sigma, double k, int nx, int ny)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(2 * nx * ny);
double theta;
Band_Data dV_data;
if (dV == Direction.x)
{
dV_data = dV_x;
theta = theta_x;
}
else if (dV == Direction.y)
{
dV_data = dV_y;
theta = theta_y;
}
else if (dV == Direction.z)
{
dV_data = new Band_Data(nx, ny, 0.0);
theta = 0.0;
}
else
throw new Exception("Error - It's completely impossible to get here");
if (p == Direction.x)
if (sigma == Direction.x)
throw new InvalidArgumentException("Error - no spin-orbit interaction between p_x and sigma_x");
else if (sigma == Direction.y)
throw new NotImplementedException();
else if (sigma == Direction.z)
{
for (int i = 0; i < nx; i++)
for(int j = 0; j < ny; j++)
{
// off-diagonal couplings in x backward
if (i != 0)
{
// for up -> up
result[i * ny + j, i * ny + j - ny] = NMathFunctions.Exp(-1.0 * I * theta * dV_data.mat[i, j]) * tx;
// for down -> down
result[i * ny + j + nx * ny, i * ny + j - ny + nx * ny] = NMathFunctions.Exp(I * theta * dV_data.mat[i, j]) * tx;
}
// off-diagonal couplings in x forward
if (i != nx - 1)
{
// for up -> up
result[i * ny + j, i * ny + j + ny] = NMathFunctions.Exp(-1.0 * I * theta * dV_data.mat[i, j]) * tx;
// for down -> down
result[i * ny + j + nx * ny, i * ny + j + ny + nx * ny] = NMathFunctions.Exp(I * theta * dV_data.mat[i, j]) * tx;
}
// and diagonal couplings
result[i * ny + j, i * ny + j] = -2.0 * Math.Cos(theta * dV_data.mat[i, j]) * tx;
result[i * ny + j + nx * ny, i * ny + j + nx * ny] = -2.0 * Math.Cos(theta * dV_data.mat[i, j]) * tx;
}
}
else
throw new Exception("Error - It's completely impossible to get here");
else if (p == Direction.y)
if (sigma == Direction.x)
throw new NotImplementedException();
else if (sigma == Direction.y)
throw new InvalidArgumentException("Error - no spin-orbit interaction between p_y and sigma_y");
else if (sigma == Direction.z)
{
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
{
// off-diagonal couplings in y backward
if (j != 0)
{
// for up -> up
result[i * ny + j, i * ny + j - 1] = NMathFunctions.Exp(-1.0 * I * theta * dV_data.mat[i, j]) * ty;
// for down -> down
result[i * ny + j + nx * ny, i * ny + j - 1 + nx * ny] = NMathFunctions.Exp(I * theta * dV_data.mat[i, j]) * ty;
}
// off-diagonal couplings in y forward
if (j != ny - 1)
{
// for up -> up
result[i * ny + j, i * ny + j + 1] = NMathFunctions.Exp(-1.0 * I * theta * dV_data.mat[i, j]) * ty;
// for down -> down
result[i * ny + j + nx * ny, i * ny + j + 1 + nx * ny] = NMathFunctions.Exp(I * theta * dV_data.mat[i, j]) * ty;
}
// and diagonal couplings
result[i * ny + j, i * ny + j] = -2.0 * Math.Cos(theta * dV_data.mat[i, j]) * ty;
result[i * ny + j + nx * ny, i * ny + j + nx * ny] = -2.0 * Math.Cos(theta * dV_data.mat[i, j]) * ty;
}
}
else
throw new Exception("Error - It's completely impossible to get here");
else if (p == Direction.z)
throw new NotImplementedException("Error - at the moment, when momentum is in the direction of the wire, you should use a different method");
else
throw new Exception("Error - It's completely impossible to get here");
return result;
}
DoubleHermitianMatrix Create_NoSOI_Hamiltonian(Band_Data dft_pot, double k, int nx, int ny)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(nx * ny);
// set off diagonal elements
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
{
// coupling sites in the growth direction
if (i != 0)
result[i * ny + j, i * ny + j - ny] = tx;
if (i != nx - 1)
result[i * ny + j, i * ny + j + ny] = tx;
// coupling sites in the transverse direction
if (j != 0)
result[i * ny + j, i * ny + j - 1] = ty;
if (j != ny - 1)
result[i * ny + j, i * ny + j + 1] = ty;
}
double[,] potential = new double[nx, ny];
// set diagonal elements
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
{
potential[i, j] = dft_pot.mat[i, j];
potential[i, j] += Physics_Base.hbar * Physics_Base.hbar * k * k / (2.0 * mass);
result[i * ny + j, i * ny + j] = -2.0 * tx + -2.0 * ty + potential[i, j];
}
return result;
}
DoubleHermitianMatrix Create_H_Using_SOIP(Band_Data dft_pot, double k)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(2 * nx * ny);
// create the SOIP parts of the matrix
result = Create_SOIP_Matrix(Direction.x, Direction.y, Direction.z, k, nx, ny);
result += Create_SOIP_Matrix(Direction.y, Direction.x, Direction.z, k, nx, ny);
// and add the scalar terms from the k-direction
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
{
// add momentum and potential terms
result[i * ny + j, i * ny + j] += Physics_Base.hbar * Physics_Base.hbar * k * k / (2.0 * mass) + dft_pot.mat[i, j];
result[i * ny + j + nx * ny, i * ny + j + nx * ny] += Physics_Base.hbar * Physics_Base.hbar * k * k / (2.0 * mass) + dft_pot.mat[i, j];
// add spin-orbit terms
result[i * ny + j, i * ny + j + nx * ny] += (-1.0 * I * alpha * dV_x.mat[i, j] - alpha * dV_y.mat[i, j]) * k;
result[i * ny + j + nx * ny, i * ny + j] += (I * alpha * dV_x.mat[i, j] - alpha * dV_y.mat[i, j]) * k;
}
return result;
}
DoubleHermitianMatrix Create_Hamiltonian(Band_Data dft_pot, double k)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(2 * nx * ny);
DoubleHermitianMatrix h_upup, h_updn, h_dnup, h_dndn;
// create sub-matrices with no spin orbit
h_upup = Create_NoSOI_Hamiltonian(dft_pot, k, nx, ny); // takes up to up (ie. H_11)
h_dndn = Create_NoSOI_Hamiltonian(dft_pot, k, nx, ny); // takes dn to dn (ie. H_22)
h_updn = new DoubleHermitianMatrix(nx * ny); // takes up to dn (ie. H_21)
h_dnup = new DoubleHermitianMatrix(nx * ny); // takes dn to up (ie. H_12)
// add spin-orbit
h_upup -= alpha * Create_SOI_Hamiltonian(Direction.y, Direction.x, k, nx, ny);
h_dndn -= -1.0 * alpha * Create_SOI_Hamiltonian(Direction.y, Direction.x, k, nx, ny);
h_updn -= alpha * (Create_SOI_Hamiltonian(Direction.z, Direction.y, nx, ny) - Create_SOI_Hamiltonian(Direction.y, Direction.z, nx, ny)
+ I * Create_SOI_Hamiltonian(Direction.z, Direction.x, k, nx, ny));
h_dnup -= alpha * (Create_SOI_Hamiltonian(Direction.z, Direction.y, nx, ny) - Create_SOI_Hamiltonian(Direction.y, Direction.z, nx, ny)
- I * Create_SOI_Hamiltonian(Direction.z, Direction.x, k, nx, ny));
// recombine matrices and output result
for (int i = 0; i < nx * ny; i++)
for (int j = 0; j < nx * ny; j++)
{
result[i, j] = h_upup[i, j];
result[i, j + nx * ny] = h_dnup[i, j];
result[i + nx * ny, j] = h_updn[i, j];
result[i + nx * ny, j + nx * ny] = h_dndn[i, j];
}
return result;
}
DoubleHermitianMatrix Create_2DEG_Hamiltonian(double[,] V, double tx, double ty, int nx, int ny, bool periodic_in_x, bool periodic_in_y)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(nx * ny);
// set off diagonal elements
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
{
// coupling sites in the transport direction
if (i != 0)
result[i * ny + j, i * ny + j - ny] = tx;
if (i != nx - 1)
result[i * ny + j, i * ny + j + ny] = tx;
// coupling sites in the transverse direction
if (j != 0)
result[i * ny + j, i * ny + j - 1] = ty;
if (j != ny - 1)
result[i * ny + j, i * ny + j + 1] = ty;
}
// set diagonal elements
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
result[i * ny + j, i * ny + j] = -2.0 * tx + -2.0 * ty + V[i, j];
// and finally, add periodic boundary conditions where appropriate
if (periodic_in_x)
for (int j = 0; j < ny; j++)
{
result[j, j + (nx - 1) * ny] = tx;
result[j + (nx - 1) * ny, j] = tx;
}
if (periodic_in_y)
for (int i = 0; i < nx; i++)
{
result[i * ny + 1, i * ny] = ty;
result[i * ny, i * ny + 1] = ty;
}
return result;
}
DoubleHermitianMatrix Create_Hamiltonian(ILayer[] layers, Band_Data pot)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(nz);
// set off diagonal elements
for (int i = 0; i < nz - 1; i++)
{
result[i + 1, i] = t; result[i, i + 1] = t;
}
// set diagonal elements
for (int i = 0; i < nz; i++)
{
result[i, i] = -2.0 * t + pot[i];
}
return result;
}
DoubleHermitianMatrix Create_Hamiltonian(ILayer[] layers, Band_Data pot)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(nx * ny);
// set off diagonal elements
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
{
// coupling sites in the transverse direction
if (i != 0)
result[i * ny + j, i * ny + j - ny] = tx;
if (i != nx - 1)
result[i * ny + j, i * ny + j + ny] = tx;
// coupling sites in the growth direction
if (j != 0)
result[i * ny + j, i * ny + j - 1] = ty;
if (j != ny - 1)
result[i * ny + j, i * ny + j + 1] = ty;
}
double[,] potential = new double[nx, ny];
// set diagonal elements
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
result[i * ny + j, i * ny + j] = -2.0 * tx + -2.0 * ty + pot.mat[i, j];
return result;
}
DoubleHermitianMatrix Create_Hamiltonian(double[] V, double t, int N)
{
DoubleHermitianMatrix result = new DoubleHermitianMatrix(N);
// set off diagonal elements
for (int i = 0; i < N; i++)
{
// coupling sites in the growth direction
if (i != 0)
result[i - 1, i] = t;
if (i != N - 1)
result[i + 1, i] = t;
}
// set diagonal elements
for (int i = 0; i < N; i++)
result[i, i] = -2.0 * t + V[i];
return result;
}