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);
}
/// <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;
}
double Interpolate_Fermi_Function_Derivative(DoubleHermitianEigDecomp eig_decomp_old, DoubleHermitianEigDecomp eig_decomp_new, int wavefunction)
{
double E0 = eig_decomp_old.EigenValue(wavefunction);
double dE_dk = (eig_decomp_new.EigenValue(wavefunction) - eig_decomp_old.EigenValue(wavefunction)) / delta_k;
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func <double, double>)((double k) => - 1.0 * beta * Math.Exp(beta * (E0 + dE_dk * k)) * Math.Pow(Math.Exp(beta * (E0 + dE_dk * k)) + 1, -2.0)));
dos_integrand.Integrator = new GaussKronrodIntegrator();
return(dos_integrand.Integrate(0, delta_k));
}
/// <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);
}
public override DoubleVector Get_EnergyLevels(ILayer[] layers, 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);
// diagonalise matrix for k = 0
int tmp = 0;
DoubleHermitianEigDecomp eig_decomp = Diagonalise_Hamiltonian(Create_Hamiltonian(dft_pot, 0.0), out tmp);
return(eig_decomp.EigenValues);
}
void Output_Wavefunctions(DoubleHermitianEigDecomp eig, int no_wavefunctions)
{
for (int i = 0; i < no_wavefunctions; i++)
{
System.IO.StreamWriter sw = new System.IO.StreamWriter("wav_" + i.ToString("00"));
for (int j = 0; j < nx; j++)
{
sw.WriteLine((eig.EigenVector(i)[j].Real * Math.Sqrt(Get_OneD_DoS(eig.EigenValue(i), no_kb_T))).ToString());
}
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 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);
}
/// <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;
}
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);
DoubleHermitianEigDecomp eig_decomp = Solve_Eigenvector_Problem(dft_pot, ref charge_density);
int max_wavefunction = (from val in eig_decomp.EigenValues
where val < no_kb_T * Physics_Base.kB * temperature
select val).ToArray().Length;
double[] dens_x = new double[nx];
// and generate a density for the y-direction
for (int i = 0; i < nx; i++)
{
for (int k = 0; k < max_wavefunction; k++)
{
dens_x[i] += DoubleComplex.Norm(eig_decomp.EigenVector(k)[i]) * DoubleComplex.Norm(eig_decomp.EigenVector(k)[i]) * Get_OneD_DoS(energies[k], no_kb_T);
}
}
// multiply the z-densities by the y-density
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// 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;
}
charge_density.Spin_Up.mat[i, j] *= dens_x[i];
charge_density.Spin_Down.mat[i, j] *= dens_x[i];
}
}
// and multiply the density by -e to get the charge density (as these are electrons)
charge_density = unit_charge * charge_density;
}
void Write_Out_Wavefunction_Densities(DoubleHermitianEigDecomp eig_decomp)
{
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 k = 0; k < max_wavefunction; k++)
{
System.IO.StreamWriter sw = new System.IO.StreamWriter("dens_wavefunction_" + k.ToString("00"));
DoubleMatrix tmp = new DoubleMatrix(nx, ny, 0.0);
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// 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)
{
continue;
}
// 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
tmp[i, j] = unit_charge * 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);
}
}
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
sw.Write(tmp[i, j].ToString() + '\t');
if (j == ny - 1)
{
sw.WriteLine();
}
}
}
sw.Close();
}
}
public Band_Data Get_KS_KE(ILayer[] layers, 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);
// create temporary density for input into...
DoubleMatrix dens_up = new DoubleMatrix(nx, ny, 0.0);
DoubleMatrix dens_down = new DoubleMatrix(nx, ny, 0.0);
SpinResolved_Data dens = new SpinResolved_Data(new Band_Data(dens_up), new Band_Data(dens_down));
// calculate eigenvectors
DoubleHermitianEigDecomp eig_decomp = Solve_Eigenvector_Problem(dft_pot, ref dens);
int max_wavefunction = (from val in eig_decomp.EigenValues
where val < no_kb_T * Physics_Base.kB * temperature
select val).ToArray().Length;
// generate kinetic energy data
Band_Data ke = new Band_Data(nx, ny, 0.0);
Band_Data dens_tot = dens.Spin_Summed_Data;
double ke_prefactor = -0.5 * Physics_Base.hbar * Physics_Base.hbar / mass;
for (int i = 1; i < nx - 1; i++)
{
for (int j = 1; j < ny - 1; j++)
{
for (int k = 0; k < max_wavefunction; k++)
{
double dy2psi = (dens_tot.mat[i, j + 1] + dens_tot.mat[i, j - 1] - 2.0 * dens_tot.mat[i, j]) * DoubleComplex.Norm(eig_decomp.EigenVector(k)[i]);
double dx2psi = DoubleComplex.Norm(eig_decomp.EigenVector(k)[i + 1] + eig_decomp.EigenVector(k)[i - 1] - 2.0 * eig_decomp.EigenVector(k)[i]) * dens_tot.mat[i, j];
double psi_div2psi = (dens_tot.mat[i, j] * DoubleComplex.Norm(eig_decomp.EigenVector(k)[i])) * (dx2psi + dy2psi);
ke.mat[i, j] += ke_prefactor * psi_div2psi * Get_OneD_DoS(eig_decomp.EigenValue(k), no_kb_T);
}
}
}
return(ke);
}
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);
}
SpinResolved_Data Calculate_Density_Derivative(DoubleHermitianEigDecomp eig_decomp_old, DoubleHermitianEigDecomp eig_decomp_new, int wavefunction)
{
DoubleMatrix dens_up_deriv = new DoubleMatrix(nx, ny, 0.0);
DoubleMatrix dens_down_deriv = new DoubleMatrix(nx, ny, 0.0);
for (int i = 0; i < nx; i++)
{
for (int j = 0; j < ny; j++)
{
// 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)
{
continue;
}
// calculate the density of this spin configuration this hamiltonian
dens_up_deriv[i, j] += g_1D * DoubleComplex.Norm(eig_decomp_old.EigenVector(wavefunction)[i * ny + j]) * DoubleComplex.Norm(eig_decomp_old.EigenVector(wavefunction)[i * ny + j]) * Interpolate_Fermi_Function_Derivative(eig_decomp_old, eig_decomp_new, wavefunction);
dens_down_deriv[i, j] += g_1D * DoubleComplex.Norm(eig_decomp_old.EigenVector(wavefunction)[i * ny + j + nx * ny]) * DoubleComplex.Norm(eig_decomp_old.EigenVector(wavefunction)[i * ny + j + nx * ny]) * Interpolate_Fermi_Function_Derivative(eig_decomp_old, eig_decomp_new, wavefunction);
}
}
// and multiply the density by -e to get the charge density (as these are electrons)
return(unit_charge * unit_charge * new SpinResolved_Data(new Band_Data(dens_up_deriv), new Band_Data(dens_down_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_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;
}
void Output_Wavefunctions(DoubleHermitianEigDecomp eig, int no_wavefunctions)
{
throw new NotImplementedException();
}
void Output_Wavefunctions(DoubleHermitianEigDecomp eig, int no_wavefunctions)
{
for (int i = 0; i < no_wavefunctions; i++)
{
System.IO.StreamWriter sw = new System.IO.StreamWriter("wav_" + i.ToString("00"));
for (int j = 0; j < nx; j++)
sw.WriteLine((eig.EigenVector(i)[j].Real * Math.Sqrt(Get_OneD_DoS(eig.EigenValue(i), no_kb_T))).ToString());
sw.Close();
}
}
/// <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;
}
double Interpolate_Fermi_Function_Derivative(DoubleHermitianEigDecomp eig_decomp_old, DoubleHermitianEigDecomp eig_decomp_new, int wavefunction)
{
double E0 = eig_decomp_old.EigenValue(wavefunction);
double dE_dk = (eig_decomp_new.EigenValue(wavefunction) - eig_decomp_old.EigenValue(wavefunction)) / delta_k;
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func<double, double>)((double k) => -1.0 * beta * Math.Exp(beta * (E0 + dE_dk * k)) * Math.Pow(Math.Exp(beta * (E0 + dE_dk * k)) + 1, -2.0)));
dos_integrand.Integrator = new GaussKronrodIntegrator();
return dos_integrand.Integrate(0, delta_k);
}
/// <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;
}
SpinResolved_Data Calculate_Density_Derivative(DoubleHermitianEigDecomp eig_decomp_old, DoubleHermitianEigDecomp eig_decomp_new, int wavefunction)
{
DoubleMatrix dens_up_deriv = new DoubleMatrix(nx, ny, 0.0);
DoubleMatrix dens_down_deriv = new DoubleMatrix(nx, ny, 0.0);
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
{
// 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)
continue;
// calculate the density of this spin configuration this hamiltonian
dens_up_deriv[i, j] += g_1D * DoubleComplex.Norm(eig_decomp_old.EigenVector(wavefunction)[i * ny + j]) * DoubleComplex.Norm(eig_decomp_old.EigenVector(wavefunction)[i * ny + j]) * Interpolate_Fermi_Function_Derivative(eig_decomp_old, eig_decomp_new, wavefunction);
dens_down_deriv[i, j] += g_1D * DoubleComplex.Norm(eig_decomp_old.EigenVector(wavefunction)[i * ny + j + nx * ny]) * DoubleComplex.Norm(eig_decomp_old.EigenVector(wavefunction)[i * ny + j + nx * ny]) * Interpolate_Fermi_Function_Derivative(eig_decomp_old, eig_decomp_new, wavefunction);
}
// and multiply the density by -e to get the charge density (as these are electrons)
return unit_charge * unit_charge * new SpinResolved_Data(new Band_Data(dens_up_deriv), new Band_Data(dens_down_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>
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;
}
}
// put the calculated densities into charge_density
charge_density = new SpinResolved_Data(dens_up, dens_down);
return xy_energy;
}
void Output_Wavefunctions(DoubleHermitianEigDecomp eig, int no_wavefunctions)
{
throw new NotImplementedException();
}
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;
}
/// <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);
}
void Write_Out_Wavefunction_Densities(DoubleHermitianEigDecomp eig_decomp)
{
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 k = 0; k < max_wavefunction; k++)
{
System.IO.StreamWriter sw = new System.IO.StreamWriter("dens_wavefunction_" + k.ToString("00"));
DoubleMatrix tmp = new DoubleMatrix(nx, ny, 0.0);
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
{
// 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)
continue;
// 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
tmp[i, j] = unit_charge * 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);
}
for (int i = 0; i < nx; i++)
for (int j = 0; j < ny; j++)
{
sw.Write(tmp[i, j].ToString() + '\t');
if (j == ny - 1)
sw.WriteLine();
}
sw.Close();
}
}
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);
}
