/// <summary>
/// checks to see whether the solution number eigenvalue and eigenvector is unphysical and therefore shouldn't be added
/// to the boundary condition matrix
/// </summary>
bool Add_BC_EigenSolution(int solution_no, DoubleComplexEigDecomp eig_decomp, Boundary boundary)
{
double eigenvalue_norm = DoubleComplex.Norm(eig_decomp.EigenValue(solution_no));
// check for propagating solutions
if (Math.Abs(eigenvalue_norm - 1.0) < propagating_mode_error)
{
if (eig_decomp.EigenValue(solution_no).Imag > 0 && boundary == Boundary.left)
{
return(true);
}
else if (eig_decomp.EigenValue(solution_no).Imag < 0 && boundary == Boundary.right)
{
return(true);
}
else if (eig_decomp.EigenValue(solution_no).Imag == 0)
{
throw new NotImplementedException();
}
}
// for evanescent solutions, check for unphysical solutions, ie. |alpha| > 1 for left lead
else if (eigenvalue_norm > 1.0 && boundary == Boundary.left)
{
return(true);
}
// and |alpha| < 1 for right lead
else if (eigenvalue_norm < 1.0 && boundary == Boundary.right)
{
return(true);
}
// if these properties are not satisfied, do not add the eigenvalue and eigenvector to the BC (default behaviour)
return(false);
}
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();
*/
}
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();
*/
}
/// <summary>
/// checks to see whether the solution number eigenvalue and eigenvector is unphysical and therefore shouldn't be added
/// to the boundary condition matrix
/// </summary>
bool Add_BC_EigenSolution(int solution_no, DoubleComplexEigDecomp eig_decomp, Boundary boundary)
{
double eigenvalue_norm = DoubleComplex.Norm(eig_decomp.EigenValue(solution_no));
// check for propagating solutions
if (Math.Abs(eigenvalue_norm - 1.0) < propagating_mode_error)
{
if (eig_decomp.EigenValue(solution_no).Imag > 0 && boundary == Boundary.left)
return true;
else if (eig_decomp.EigenValue(solution_no).Imag < 0 && boundary == Boundary.right)
return true;
else if (eig_decomp.EigenValue(solution_no).Imag == 0)
throw new NotImplementedException();
}
// for evanescent solutions, check for unphysical solutions, ie. |alpha| > 1 for left lead
else if (eigenvalue_norm > 1.0 && boundary == Boundary.left)
return true;
// and |alpha| < 1 for right lead
else if (eigenvalue_norm < 1.0 && boundary == Boundary.right)
return true;
// if these properties are not satisfied, do not add the eigenvalue and eigenvector to the BC (default behaviour)
return false;
}
