Spin susceptibility in interacting two-dimensional semiconductors and bilayer systems at first order: Kohn anomalies and spin density wave ordering

This work is an analytic theoretical study of a two-dimensional (2D) semiconductor with a Fermi surface that is split by the Zeeman coupling of electron spins to an external magnetic field in the presence of electron-electron interactions. We calculate the spin susceptibility for long-range and finite-range interactions diagrammatically, and we find a resonant peak structure at the Kohn anomaly already in first-order perturbation theory. In contrast to the density-density correlator that is suppressed due to the large electrostatic energy required to stabilize charge density order, the spin susceptibility does not suffer from electrostatic screening effects, thus favoring spin density wave order in 2D semiconductors. Our results impose significant consequences for determining magnetic phases in 2D semiconductors. For example, a strongly enhanced Kohn anomaly may result in helical ordering of magnetic impurities due to the Ruderman-Kittel-Kasuya-Yosida interaction. Furthermore, the spin degree of freedom can equally represent a layer pseudospin in the case of bilayer materials. In this case, the external “magnetic field” is a combination of layer bias and interlayer hopping. The sharp peak of the 2D static spin susceptibility may then be responsible for dipole-density-wave order in bilayer materials at large enough electron-phonon coupling.

Figure 1

First-order diagrams for the irreducible susceptibility, ${\chi }_{ij}^{\ell }\left(z\right)$, $z=(\mathit{r},\tau )$, $\ell \in \{a,b,c\}$; see Eqs. (28), (29), and (30).

Figure 2

Components of the full susceptibilities near $q=2k_F$ [see Eqs. (73, 74, 75, 76)] at $\kappa =0.2k_F$, $r_s=1.0$, and $\delta =0.02k_F$. Irreducible susceptibilities are calculated numerically as a sum of diagrams in Figs. 1 and 6 with the Thomas-Fermi interaction [see Eq. (20)]. Susceptibilities ${\chi }_{\rho }\left(q\right)$, ${\chi }_{33}\left(q\right)$, $-{\chi }_{03}\left(q\right)$ have two peaks at $q=2k_{-}$, $q=2k_{+}$; ${\chi }_{11}\left(q\right)$ has single peak at $q=k_{+}+k_{-}=2k_F$, see Eq. (10). The spin susceptibility ${\chi }_{33}\left(q\right)$ demonstrates the largest peak at $q=2k_{-}$. The charge susceptibility ${\chi }_{\rho }\left(q\right)$ is suppressed by the electrostatic screening. For reference, the noninteracting Lindhard function ${\chi }_{11}^{\left(0\right)}\left(q\right)$, see Eq. (26), is shown by a black dashed curve.

Figure 3

Full charge and spin susceptibilities ${\chi }_{\rho }\left(q\right)$ and ${\chi }_{33}\left(q\right)$ near $q=2k_F$ [see Eqs. (73, 74, 75, 76)] for increasing values of $\kappa$ (curves correspond to $\kappa =0.07k_F$, $0.1k_F$, $0.2k_F$, $0.3k_F$, in decreasing order of opacity). Here we have set $r_s=1.0$ and $\delta =0.02k_F$. Irreducible susceptibilities are calculated numerically with the Thomas-Fermi interaction [see Eq. (20)]. Smaller values of the screening produce sharper peaks in the susceptibilities.

Figure 4

Components of the irreducible susceptibilities given by the sum of diagrams in Figs. 1 and 6 near $q=2k_F$ for the unscreened Coulomb interaction [Eq. (18)] with $r_s=1.0$ and $\delta =0.02k_F$. Note from Eq. (72) that ${\chi }_{\rho }^{\mathrm{irr}}={\chi }_{33}^{\mathrm{irr}}$, which is shown by the blue curve. The noninteracting Lindhard function ${\chi }_{11}^{\left(0\right)}\left(q\right)$, see Eq. (26), is shown by a black dashed curve.

Figure 5

Density response in each layer of a bilayer 2DEG to a local potential difference $\varphi \delta \left(\mathit{r}\right)$. The two layers have decaying Friedel oscillations of wavelength $\pi /k_F$ exactly out of phase with each other.

Figure 6

Free-fermion susceptibility ${\chi }_{ij}^{\left(0\right)}\left(z\right)$, $z=(\mathit{r},\tau )$ [see Eqs. (24) and (25)]. Here, ${\eta }_i$ is either a Pauli matrix for $i\in \{1,2,3\}$ [see Eqs. (6, 7, 8)] or the identity matrix, ${\eta }_0=I$; the same applies to ${\eta }_j$.