private void Bind_Slice(int slice_no, DoubleComplexMatrix slice_to_bind)
{
DoubleComplexMatrix new_hopping = Generate_Hopping_Hamiltonian();
DoubleComplexMatrix new_hopping_trans = Generate_Hopping_Hamiltonian().Transpose();
DoubleComplexMatrix temp_matrix = new DoubleComplexMatrix(slice_to_bind - Product(Product(new_hopping, onsite_G[slice_no - 1]), new_hopping_trans));
// factorise the matrix
DoubleComplexLUFact factorisation = new DoubleComplexLUFact(temp_matrix);
// the inverse of this matrix is the onsite Green's function for the new slice
onsite_G[slice_no] = factorisation.Inverse();
// the first hop from n to n + 1 uses the onsite Green's function as G_(i,n+1) = G_(i,n) H' G(n+1,n+1) and when i = n, G_(i,n) = G(i,i)
// must do this here before updating the onsite Green's functions
hopping_G[slice_no - 1] = onsite_G[slice_no - 1];
// update the old onsite Green's functions (not including the boundary slice)
for (int i = 0; i < slice_no; i++)
{
onsite_G[i] = onsite_G[i] + Product(Product(Product(Product(hopping_G[i], new_hopping), onsite_G[slice_no]), new_hopping_trans), hopping_G[i].Transpose());
}
// calculate the new hopping Green's functions
for (int i = 0; i < slice_no; i++)
{
hopping_G[i] = Product(Product(hopping_G[i], new_hopping), onsite_G[slice_no]);
}
}
private DoubleComplexMatrix Create_New_Slice(double energy, int slice_no)
{
// work out where this slice is
double x = (slice_no - (nx - 1) / 2) * dx;
// generate the matrices for this slice
DoubleComplexMatrix new_slice = Generate_Slice_Hamiltonian(x);
return((energy * DoubleMatrix.Identity(ny)) - new_slice);
}
DoubleComplexMatrix Generate_Hopping_Hamiltonian()
{
DoubleComplexMatrix result = new DoubleComplexMatrix(ny, ny);
for (int i = 0; i < ny; i++)
{
result[i, i] = alphax;
}
return(result);
}
void Save_BC_Eigvecs(DoubleComplexMatrix eig_mat)
{
StreamWriter sw = new StreamWriter("vecs.dat");
for (int i = 0; i < eig_mat.Rows; i++)
{
for (int j = 0; j < eig_mat.Cols; j++)
{
sw.Write(eig_mat[i, j].Real.ToString() + '\t');
}
sw.WriteLine();
}
sw.Close();
}
void Initiate_Greens_Function(double energy)
{
onsite_G = new DoubleComplexMatrix[nx];
hopping_G = new DoubleComplexMatrix[nx];
for (int i = 1; i < nx - 1; i++)
{
onsite_G[i] = new DoubleComplexMatrix(ny, ny);
hopping_G[i] = new DoubleComplexMatrix(ny, ny);
}
onsite_G[0] = Calculate_Boundary_Conditions(Boundary.left, energy);
onsite_G[nx - 1] = Calculate_Boundary_Conditions(Boundary.right, energy);
LHSboundary = onsite_G[0];
RHSboundary = onsite_G[nx - 1];
}
DoubleComplexMatrix Generate_Slice_Hamiltonian(double x)
{
DoubleComplexMatrix result = new DoubleComplexMatrix(ny, ny);
for (int i = 0; i < ny - 1; i++)
{
double y = (i - (ny - 1) / 2) * dy;
result[i, i] = -2.0 * alphay - 2.0 * alphax + Get_Potential(x, y);
result[i, i + 1] = 1.0 * alphay;
result[i + 1, i] = 1.0 * alphay;
}
// the alphax term is necessary here as we are doing this in 2D
result[ny - 1, ny - 1] = -2.0 * alphax - 2.0 * alphay + Get_Potential(x, (ny - 1) / 2 * dy);
return(result);
}
void Get_Greens_Functions(double energy)
{
Initiate_Greens_Function(energy);
for (int i = 1; i < nx - 1; i++)
{
Add_Slice(energy, i);
}
// factorise and invert the right hand lead Green's function
DoubleComplexLUFact factorisation = new DoubleComplexLUFact(onsite_G[nx - 1]);
DoubleComplexMatrix rhs_inverted_G = factorisation.Inverse();
Bind_Slice(nx - 1, rhs_inverted_G);
//Output_DoS();
for (int i = 0; i < ny; i++)
{
for (int j = 0; j < nx; j++)
{
DensityoS[j, i] = (-1.0 * onsite_G[j][i, i].Imag / (Math.PI * alphax_prime));
}
}
}
DoubleComplexMatrix Generate_Hopping_Hamiltonian()
{
DoubleComplexMatrix result = new DoubleComplexMatrix(ny,ny);
for (int i = 0; i < ny; i++)
result[i, i] = alphax;
return result;
}
private DoubleComplexMatrix Calculate_Boundary_Conditions(Boundary boundary, double energy)
{
DoubleComplexMatrix bc_matrix = new DoubleComplexMatrix(2 * ny, 2 * ny);
// generate Hamiltonians for the boundaries
DoubleComplexMatrix tmp_slice;
if (boundary == Boundary.left)
{
tmp_slice = Generate_Slice_Hamiltonian(-0.5 * (nx - 1) * dx);
}
else if (boundary == Boundary.right)
{
tmp_slice = Generate_Slice_Hamiltonian(0.5 * (nx - 1) * dx);
}
else
{
throw new NotImplementedException();
}
DoubleComplexMatrix tmp_hopping_trans = Generate_Hopping_Hamiltonian().Transpose();
// create temporary matrix for top-left of BC matrix
DoubleComplexMatrix tmp_matrix = Product(tmp_hopping_trans, (new DoubleComplexMatrix(energy * DoubleMatrix.Identity(ny)) - tmp_slice));
// fill with transfer matrix
for (int i = 0; i < ny; i++)
{
for (int j = 0; j < ny; j++)
{
bc_matrix[i, j] = tmp_matrix[i, j];
bc_matrix[i + ny, j] = tmp_hopping_trans[i, j];
bc_matrix[i, j + ny] = -1.0 * tmp_hopping_trans[i, j];
}
}
// solve eigen-problem for transfer matrix
DoubleComplexEigDecomp eig_decomp = new DoubleComplexEigDecomp(bc_matrix);
DoubleComplexEigDecompServer server = new DoubleComplexEigDecompServer();
eig_decomp = server.Factor(bc_matrix);
// fill the eigenvalue matrix, excluding unphysical solutions which blow-up in the wire
DoubleComplexMatrix eig_vals = new DoubleComplexMatrix(ny, ny);
DoubleComplexMatrix eig_vecs = new DoubleComplexMatrix(ny, ny);
int count = 0;
for (int i = 0; i < 2 * ny; i++)
{
if (Add_BC_EigenSolution(i, eig_decomp, boundary))
{
// invert eigenvalues if calculating for the right lead
if (boundary == Boundary.left)
{
eig_vals[count, count] = eig_decomp.EigenValue(i);
}
else if (boundary == Boundary.right)
{
eig_vals[count, count] = 1.0 / eig_decomp.EigenValue(i);
}
else
{
throw new NotImplementedException();
}
// insert corresponding (normalised) eigenvector into the matrix
double norm = 0.0;
for (int j = 0; j < ny; j++)
{
norm += DoubleComplex.Norm(eig_decomp.RightEigenVector(i)[j]) * DoubleComplex.Norm(eig_decomp.RightEigenVector(i)[j]);
}
double inv_norm = 1.0 / Math.Sqrt(norm);
for (int j = 0; j < ny; j++)
{
eig_vecs[j, count] = inv_norm * eig_decomp.RightEigenVector(i)[j];
}
count++;
}
}
// calculate coefficient matrix A and output Green's function
DoubleComplexMatrix coefficient_matrix = new DoubleComplexMatrix(ny, ny);
coefficient_matrix = Product(Product(tmp_hopping_trans.Transpose(), eig_vecs), eig_vals);
DoubleComplexLUFact lu_fact = new DoubleComplexLUFact(coefficient_matrix);
return(new DoubleComplexMatrix(Product(eig_vecs, lu_fact.Inverse())));
/* // allocate for evanescent modes
* if (boundary == Boundary.left && DoubleComplex.Norm(eig_decomp.EigenValue(i)) < 1.0 - propagating_mode_error)
* eig_vals[i, i] = eig_decomp.EigenValue(i);
* else if (boundary == Boundary.right && DoubleComplex.Norm(eig_decomp.EigenValue(i)) > 1.0 + propagating_mode_error)
* eig_vals[i, i] = 1.0 / eig_decomp.EigenValue(i);
* // and for propagating modes
* else if (Math.Abs(DoubleComplex.Norm(eig_decomp.EigenValue(i)) - 1.0) < propagating_mode_error)
* eig_vals[i, i]
* else
* throw new InvalidArgumentException("Error - Cannot have eigenvalue of transfer matrix with value " + eig_decomp.EigenValue(i).ToString() + "!");
*
* // fill the eigenvector matrix with only the top half of the eigenvector
* DoubleComplexMatrix eig_vecs = new DoubleComplexMatrix(ny, ny);
* DoubleComplexMatrix inv_eigvec = new DoubleComplexMatrix(ny, ny);
* for (int i = 0; i < ny; i++)
* {
* DoubleComplexVector tmpvec_right = eig_decomp.RightEigenVector(i);
* DoubleComplexVector tmpvec_left = eig_decomp.LeftEigenVector(i);
*
* // normalise the top of half of the eigenvector
* double norm2_right = 0.0;
* double norm2_left = 0.0;
* for (int j = 0; j < ny; j++)
* {
* norm2_right += NMathFunctions.Abs(tmpvec_right[i]) * NMathFunctions.Abs(tmpvec_right[i]);
* norm2_left += NMathFunctions.Abs(tmpvec_left[i]) * NMathFunctions.Abs(tmpvec_left[i]);
* }
* double norm_left = Math.Sqrt(norm2_left);
* double norm_right = Math.Sqrt(norm2_right);
*
* // and insert it into the matrix
* for (int j = 0; j < ny; j++)
* {
* eig_vecs[j, i] = tmpvec_right[j] / norm_right;
* inv_eigvec[i, j] = tmpvec_left[j] / norm_left;
* }
* }
*
* // get the inverse of the eigenvector matrix
* // DoubleComplexMatrix tmp_eigvec = new DoubleComplexMatrix(eig_vecs);
* // DoubleComplexMatrix inv_eigvec = NMathFunctions.PseudoInverse(tmp_eigvec);
* //DoubleComplexLUFact eigvec_fact = new DoubleComplexLUFact(tmp_eigvec);
* //DoubleComplexMatrix inv_eigvec = eigvec_fact.Inverse();
*
* // Calculate the on-site Greens function of the end of the wire and return it
* if (boundary == Boundary.left)
* return Product(Product(Product(eig_vecs, eig_vals), inv_eigvec), tmp_hopping_trans);
* else if (boundary == Boundary.right)
* return Product(Product(Product(eig_vecs, eig_vals), inv_eigvec), tmp_hopping);
* else
* throw new NotImplementedException();
*/
}
DoubleComplexMatrix Generate_Slice_Hamiltonian(double x)
{
DoubleComplexMatrix result = new DoubleComplexMatrix(ny,ny);
for (int i = 0; i < ny - 1; i++)
{
double y = (i - (ny - 1) / 2) * dy;
result[i, i] = -2.0 * alphay - 2.0 * alphax + Get_Potential(x, y);
result[i, i + 1] = 1.0 * alphay;
result[i + 1, i] = 1.0 * alphay;
}
// the alphax term is necessary here as we are doing this in 2D
result[ny - 1, ny - 1] = -2.0 * alphax - 2.0 * alphay + Get_Potential(x, (ny - 1) / 2 * dy);
return result;
}
DoubleComplexMatrix Product(DoubleHermitianMatrix A, DoubleComplexMatrix B)
{
return new DoubleComplexMatrix(NMathFunctions.Product(MatrixFunctions.ToGeneralMatrix(A), B));
}
void Initiate_Greens_Function(double energy)
{
onsite_G = new DoubleComplexMatrix[nx];
hopping_G = new DoubleComplexMatrix[nx];
for (int i = 1; i < nx - 1; i++)
{
onsite_G[i] = new DoubleComplexMatrix(ny,ny);
hopping_G[i] = new DoubleComplexMatrix(ny,ny);
}
onsite_G[0] = Calculate_Boundary_Conditions(Boundary.left, energy);
onsite_G[nx - 1] = Calculate_Boundary_Conditions(Boundary.right, energy);
LHSboundary = onsite_G[0];
RHSboundary = onsite_G[nx - 1];
}
void Add_Slice(double energy, int slice_no)
{
DoubleComplexMatrix new_slice = Create_New_Slice(energy, slice_no);
Bind_Slice(slice_no, new_slice);
}
private DoubleComplexMatrix Calculate_Boundary_Conditions(Boundary boundary, double energy)
{
DoubleComplexMatrix bc_matrix = new DoubleComplexMatrix(2 * ny, 2 * ny);
// generate Hamiltonians for the boundaries
DoubleComplexMatrix tmp_slice;
if (boundary == Boundary.left)
tmp_slice = Generate_Slice_Hamiltonian(-0.5 * (nx - 1) * dx);
else if (boundary == Boundary.right)
tmp_slice = Generate_Slice_Hamiltonian(0.5 * (nx - 1) * dx);
else throw new NotImplementedException();
DoubleComplexMatrix tmp_hopping_trans = Generate_Hopping_Hamiltonian().Transpose();
// create temporary matrix for top-left of BC matrix
DoubleComplexMatrix tmp_matrix = Product(tmp_hopping_trans, (new DoubleComplexMatrix(energy * DoubleMatrix.Identity(ny)) - tmp_slice));
// fill with transfer matrix
for (int i = 0; i < ny; i++)
for (int j = 0; j < ny; j++)
{
bc_matrix[i, j] = tmp_matrix[i, j];
bc_matrix[i + ny, j] = tmp_hopping_trans[i, j];
bc_matrix[i, j + ny] = -1.0 * tmp_hopping_trans[i, j];
}
// solve eigen-problem for transfer matrix
DoubleComplexEigDecomp eig_decomp = new DoubleComplexEigDecomp(bc_matrix);
DoubleComplexEigDecompServer server = new DoubleComplexEigDecompServer();
eig_decomp = server.Factor(bc_matrix);
// fill the eigenvalue matrix, excluding unphysical solutions which blow-up in the wire
DoubleComplexMatrix eig_vals = new DoubleComplexMatrix(ny, ny);
DoubleComplexMatrix eig_vecs = new DoubleComplexMatrix(ny, ny);
int count = 0;
for (int i = 0; i < 2 * ny; i++)
if (Add_BC_EigenSolution(i, eig_decomp, boundary))
{
// invert eigenvalues if calculating for the right lead
if (boundary == Boundary.left)
eig_vals[count, count] = eig_decomp.EigenValue(i);
else if (boundary == Boundary.right)
eig_vals[count, count] = 1.0 / eig_decomp.EigenValue(i);
else throw new NotImplementedException();
// insert corresponding (normalised) eigenvector into the matrix
double norm = 0.0;
for (int j = 0; j < ny; j++)
norm += DoubleComplex.Norm(eig_decomp.RightEigenVector(i)[j]) * DoubleComplex.Norm(eig_decomp.RightEigenVector(i)[j]);
double inv_norm = 1.0 / Math.Sqrt(norm);
for (int j = 0; j < ny; j++)
eig_vecs[j, count] = inv_norm * eig_decomp.RightEigenVector(i)[j];
count++;
}
// calculate coefficient matrix A and output Green's function
DoubleComplexMatrix coefficient_matrix = new DoubleComplexMatrix(ny, ny);
coefficient_matrix = Product(Product(tmp_hopping_trans.Transpose(), eig_vecs), eig_vals);
DoubleComplexLUFact lu_fact = new DoubleComplexLUFact(coefficient_matrix);
return new DoubleComplexMatrix(Product(eig_vecs, lu_fact.Inverse()));
/* // allocate for evanescent modes
if (boundary == Boundary.left && DoubleComplex.Norm(eig_decomp.EigenValue(i)) < 1.0 - propagating_mode_error)
eig_vals[i, i] = eig_decomp.EigenValue(i);
else if (boundary == Boundary.right && DoubleComplex.Norm(eig_decomp.EigenValue(i)) > 1.0 + propagating_mode_error)
eig_vals[i, i] = 1.0 / eig_decomp.EigenValue(i);
// and for propagating modes
else if (Math.Abs(DoubleComplex.Norm(eig_decomp.EigenValue(i)) - 1.0) < propagating_mode_error)
eig_vals[i, i]
else
throw new InvalidArgumentException("Error - Cannot have eigenvalue of transfer matrix with value " + eig_decomp.EigenValue(i).ToString() + "!");
// fill the eigenvector matrix with only the top half of the eigenvector
DoubleComplexMatrix eig_vecs = new DoubleComplexMatrix(ny, ny);
DoubleComplexMatrix inv_eigvec = new DoubleComplexMatrix(ny, ny);
for (int i = 0; i < ny; i++)
{
DoubleComplexVector tmpvec_right = eig_decomp.RightEigenVector(i);
DoubleComplexVector tmpvec_left = eig_decomp.LeftEigenVector(i);
// normalise the top of half of the eigenvector
double norm2_right = 0.0;
double norm2_left = 0.0;
for (int j = 0; j < ny; j++)
{
norm2_right += NMathFunctions.Abs(tmpvec_right[i]) * NMathFunctions.Abs(tmpvec_right[i]);
norm2_left += NMathFunctions.Abs(tmpvec_left[i]) * NMathFunctions.Abs(tmpvec_left[i]);
}
double norm_left = Math.Sqrt(norm2_left);
double norm_right = Math.Sqrt(norm2_right);
// and insert it into the matrix
for (int j = 0; j < ny; j++)
{
eig_vecs[j, i] = tmpvec_right[j] / norm_right;
inv_eigvec[i, j] = tmpvec_left[j] / norm_left;
}
}
// get the inverse of the eigenvector matrix
// DoubleComplexMatrix tmp_eigvec = new DoubleComplexMatrix(eig_vecs);
// DoubleComplexMatrix inv_eigvec = NMathFunctions.PseudoInverse(tmp_eigvec);
//DoubleComplexLUFact eigvec_fact = new DoubleComplexLUFact(tmp_eigvec);
//DoubleComplexMatrix inv_eigvec = eigvec_fact.Inverse();
// Calculate the on-site Greens function of the end of the wire and return it
if (boundary == Boundary.left)
return Product(Product(Product(eig_vecs, eig_vals), inv_eigvec), tmp_hopping_trans);
else if (boundary == Boundary.right)
return Product(Product(Product(eig_vecs, eig_vals), inv_eigvec), tmp_hopping);
else
throw new NotImplementedException();
*/
}
DoubleComplexMatrix Product(DoubleHermitianMatrix A, DoubleComplexMatrix B)
{
return(new DoubleComplexMatrix(NMathFunctions.Product(MatrixFunctions.ToGeneralMatrix(A), B)));
}
void Save_BC_Eigvecs(DoubleComplexMatrix eig_mat)
{
StreamWriter sw = new StreamWriter("vecs.dat");
for (int i = 0; i < eig_mat.Rows; i++)
{
for (int j = 0; j < eig_mat.Cols; j++)
sw.Write(eig_mat[i, j].Real.ToString() + '\t');
sw.WriteLine();
}
sw.Close();
}
DoubleComplexMatrix Product(DoubleComplexMatrix A, DoubleComplexMatrix B)
{
return NMathFunctions.Product(A, B);
}
DoubleComplexMatrix Product(DoubleComplexMatrix A, DoubleComplexMatrix B)
{
return(NMathFunctions.Product(A, B));
}
private void Bind_Slice(int slice_no, DoubleComplexMatrix slice_to_bind)
{
DoubleComplexMatrix new_hopping = Generate_Hopping_Hamiltonian();
DoubleComplexMatrix new_hopping_trans = Generate_Hopping_Hamiltonian().Transpose();
DoubleComplexMatrix temp_matrix = new DoubleComplexMatrix(slice_to_bind - Product(Product(new_hopping, onsite_G[slice_no - 1]), new_hopping_trans));
// factorise the matrix
DoubleComplexLUFact factorisation = new DoubleComplexLUFact(temp_matrix);
// the inverse of this matrix is the onsite Green's function for the new slice
onsite_G[slice_no] = factorisation.Inverse();
// the first hop from n to n + 1 uses the onsite Green's function as G_(i,n+1) = G_(i,n) H' G(n+1,n+1) and when i = n, G_(i,n) = G(i,i)
// must do this here before updating the onsite Green's functions
hopping_G[slice_no - 1] = onsite_G[slice_no - 1];
// update the old onsite Green's functions (not including the boundary slice)
for (int i = 0; i < slice_no; i++)
onsite_G[i] = onsite_G[i] + Product(Product(Product(Product(hopping_G[i], new_hopping), onsite_G[slice_no]), new_hopping_trans), hopping_G[i].Transpose());
// calculate the new hopping Green's functions
for (int i = 0; i < slice_no; i++)
hopping_G[i] = Product(Product(hopping_G[i], new_hopping), onsite_G[slice_no]);
}