6. \(g\)-tensor solver

A quantity that may be of interest when considering magnetic effects for a pseudospin confined to a semiconductor nanostructure is the \(g\)-tensor. Here, pseudospin refers to an effective spin-\(\tfrac{1}{2}\) degree of freedom defined by two selected quantum states of the system, \(\left\{|0\rangle, |1\rangle\right\}\), which need not correspond to pure spin states but may instead arise from mixtures of spin, orbital, valley, or band degrees of freedom. Nevertheless, these two states form a qubit subspace that is mathematically equivalent to a spin-\(\tfrac{1}{2}\) system. The \(g\)-tensor characterizes how this effective two-level system responds to an external magnetic field. It is defined through the effective Zeeman Hamiltonian obtained by projecting the full magnetic-field coupling (onsite Zeeman terms and Peierls phases) onto the subspace spanned by \(\left\{|0\rangle, |1\rangle\right\}\). Within this reduced Hilbert space, the magnetic interaction can be written as

(6.8)\[H_Z^{\mathrm{eff}} = \mu_B \mathbf s \cdot \mathbf{g} \cdot \mathbf B,\]

where \(\mathbf{g}\) is the subspace-dependent \(4\times3\) \(g\)-tensor and \(\mathbf s = \frac{1}{2}\left(\sigma_0, \sigma_1, \sigma_2, \sigma_3\right)\) is the pseudospin vector in the \(\left\{|0\rangle, |1\rangle\right\}\) subspace with

(6.9)\[\begin{split}\begin{align} \sigma_0 &= |0\rangle\langle 0| + |1\rangle\langle 1|\,,\\ \sigma_1 &= |0\rangle\langle 1| + |1\rangle\langle 0|\,,\\ \sigma_2 &= -i|0\rangle\langle 1| + i|1\rangle\langle 0|\,,\\ \sigma_3 &= |0\rangle\langle 0| - |1\rangle\langle 1|\,, \end{align}\end{split}\]

which correspond to the identity and the three standard Pauli matrices in the pseudospin basis.

Procedure for computing the \(g\)-tensor

Select a basis for the two-level system of interest.
The two states are chosen by the user to define the pseudospin subspace. These states should be orthonormal and energetically isolated from the rest of the spectrum so that a two-level description is valid. In practice, this typically involves solving the tight-binding (TB) Schrödinger equation using the Schrödinger Solver and choosing two eigenstates of the TB Hamiltonian to define the pseudospin subspace. If the two states are degenerate, any orthonormal linear combination may be used as a basis. The particular choice should be guided by the physical system and the experimental configuration being modeled, since it determines the orientation of the effective pseudospin axes and the representation of the \(g\)-tensor. A common, physically motivated choice is a Kramers doublet, which can be identified by solving the atomistic TB Schrödinger equation (through the relevant Solver) in the presence of a very small magnetic field. This field weakly lifts the Kramers degeneracy and allows the user to select two time-reversed partner states (two states split by the coupling to the magnetic field) of interest. Ideally, the field should be small enough that its effect on the states is negligible beyond lifting the degeneracy. These states can then be taken as the basis states \(|0\rangle\) and \(|1\rangle\) defining the pseudospin subspace.
Compute magnetic-field-dependent matrix elements in the selected basis.
Evaluate the matrix elements of the magnetic-field-dependent terms of the TB Hamiltonian within the selected two-state basis. These matrix elements define the effective Zeeman Hamiltonian, \(H_Z^{\mathrm{eff}}\), acting on the pseudospin. Because \(H_Z^{\mathrm{eff}}\) is linear in \(\mathbf B\) [see Eq. (6.8)], the magnetic-field-dependent terms of the TB Hamiltonian are expanded to first order in \(\mathbf B\). The Zeeman contribution of Eq. (5.27) is already linear in \(\mathbf B\). The Peierls phase of Eq. (5.40) is expanded to first order in \(\mathbf B\), giving

(6.10)\[\exp\left\{ \frac{-ie}{2\hbar} \left(\mathbf{r}_{n} - \mathbf{r}_{n^{\prime}}\right) \cdot \left[ \mathbf A(\mathbf r_{n^{\prime}}) + \mathbf A(\mathbf r_{n}) \right] \right\} \simeq 1 + \frac{ie}{2\hbar} \mathbf{B} \cdot \mathbf{r}_{n} \times \mathbf{r}_{n^{\prime}},\]

where the symmetric gauge for the vector potential is used,

(6.11)\[\mathbf{A}(\mathbf{r}) = \frac{1}{2} \left( \mathbf{B} \times \mathbf{r} \right)\,.\]

The resulting Hamiltonian, linear in \(\mathbf{B}\), is denoted as \(H^{\mathrm{lin}}(\mathbf B)\). Projecting onto the two-level subspace yields

(6.12)\[H_Z^{\mathrm{eff}} = \sum_{\sigma \sigma^{\prime}\in \{0, 1\}} |\sigma^{\prime}\rangle\langle \sigma^{\prime}| H^{\mathrm{lin}}(\mathbf B) |\sigma\rangle\langle \sigma| = \mu_B \mathbf s \cdot \mathbf{g} \cdot \mathbf B\,,\]

a \(2\times2\) Hermitian operator acting in the pseudospin space.
Extract the \(\mathbf{\mathit{g}}\)-tensor.
Any \(2\times2\) operator can be decomposed into the identity and Pauli basis. Using the trace orthogonality relations,

(6.13)\[\mu_B \sum_{j}g_{ij}B_j = 2 \mathrm{Tr}\left[H_Z^{\mathrm{eff}} s_i\right]\,.\]

Because the components of the magnetic field vector, \(B_j\), are independent, this relation allows the extraction of the \(i^{\mathrm{th}}\) row of the \(\mathbf{g}\) by identifying the coefficients of the terms proportional to each \(B_j\). Repeating this for \(i = x, y, z\) yields all components of \(\mathbf{g}\).

This procedure is implemented in the Solver class of the qtcad.atoms.g_tensor module.

Effective \(g\)-factor and Zeeman splittings

While the \(g\)-tensor parameterizes the effective Zeeman Hamiltonian in a given two-level subspace, it can also be used to compute quantities that are directly measurable in experiments. For example, the \(g\)-tensor can be used to compute the splitting \(\Delta\) between the two levels of the two-level system in a given magnetic field \(\mathbf B\). Concretely, this splitting is given by the difference in eigenenegies of the effective Zeeman Hamiltonian \(H_Z^{\mathrm{eff}}\) of Eq. (6.8). Starting from this Hamiltonian, we write

\[H_Z^{\mathrm{eff}} = \mu_B B \mathbf s \cdot \left( \mathbf{g} \cdot \hat{\mathbf{B}} \right) = \mu_B B g^{\star} \mathbf s \cdot \hat{\mathbf{n}} = \mu_B g^{\star} \mathbf{s} \cdot \mathbf{B}_{\mathrm{eff}}\,,\]

where \(\hat{\mathbf{B}}\) is a unit vector defining the direction of the magnetic field, \(B = \left|\mathbf{B}\right|\) is its magnitude, \(\hat{\mathbf{n}} = \mathbf{g}\cdot\hat{\mathbf{B}} / \left|\mathbf{g}\cdot\hat{\mathbf{B}}\right|\) is a unit vector defining the direction of the effective field \(\mathbf{B}_{\mathrm{eff}} = B\hat{\mathbf{n}}\), and \(g^{\star} = \left|\mathbf{g}\cdot\hat{\mathbf{B}}\right|\) is the effective \(g\)-factor for the given direction of magnetic field \(\hat{\mathbf{B}}\) (we have suppressed this dependence in the equations to simplify the notation).

Note

Under the definition used here, \(g^{\star} = |\mathbf g \cdot \hat{\mathbf B}|\), the effective \(g\)-factor is always non-negative. The sign information is instead contained in the direction of the effective field \(\mathbf B_{\mathrm{eff}}\), or equivalently in the unit vector \(\hat{\mathbf n}\). In some parts of the literature, a signed effective \(g\)-factor is reported. In that convention, the sign is absorbed into \(g^{\star}\) rather than into the direction of \(\mathbf B_{\mathrm{eff}}\). For example, if \(\mathbf B_{\mathrm{eff}}\) is antiparallel to \(\hat{\mathbf B}\) (so that the effective field points opposite to the applied field), one may write \(H_Z^{\mathrm{eff}} = \mu_B g^{\star}_{\text{(signed)}} \, \mathbf s \cdot \mathbf B\), with \(g^{\star}_{\text{(signed)}} < 0\). Both conventions describe the same physics: the energy splitting depends only on \(|g^{\star}|\), while the sign determines which state moves up in energy and the direction of pseudospin precession.

In principle, the energy splitting described above can be computed directly from a full TB calculation by looking at the eigenvalues of the full TB Hamiltonian. However, these calculations, performed over the entire atomic structure, are often computationally expensive since they require a full calculation for each considered magnetic field. In contrast, using the method outlined above, the \(g\)-tensor can be computed from a single TB calculation. Once the \(g\)-tensor is computed, the Zeeman splitting for any magnetic field can be computed by diagonalizing the associated \(2 \times 2\) effective Zeeman Hamiltonian of Eq. (6.8).

Zeeman splittings can be computed from the \(g\)-tensor using the get_Zeeman_splitting method of the Solver class. Additionally, the eigenenergies and eigenstates of the effective Zeeman Hamiltonian can be computed using the diagonalize_effective_Zeeman method of the Solver class.