Magnetic properties of HgTe quantum wells

Using analytical formulas as well as a finite-difference scheme, we investigate the magnetic field dependence of the energy spectra and magnetic edge states of HgTe/CdTe-based quantum wells in the presence of perpendicular magnetic fields and hard walls for the band-structure parameters corresponding to the normal and inverted regimes. Whereas one can not find counterpropagating, spin-polarized states in the normal regime, below the crossover point between the uppermost (electronlike) valence and lowest (holelike) conduction Landau levels, one can still observe such states at finite magnetic fields in the inverted regime, although these states are no longer protected by time-reversal symmetry. Furthermore, the bulk magnetization and susceptibility in HgTe quantum wells are studied, in particular their dependence on the magnetic field, chemical potential, and carrier densities. We find that for fixed chemical potentials as well as for fixed carrier densities, the magnetization and magnetic susceptibility in both the normal and the inverted regimes exhibit de Haas–van Alphen oscillations, the amplitude of which decreases with increasing temperature. Moreover, if the band structure is inverted, the ground-state magnetization (and consequently also the ground-state susceptibility) is discontinuous at the crossover point between the uppermost valence and lowest conduction Landau levels. At finite temperatures and/or doping, this discontinuity is canceled by the contribution from the electrons and holes and the total magnetization and susceptibility are continuous. In the normal regime, this discontinuity of the ground-state magnetization does not arise and the magnetization is continuous for zero as well as finite temperatures.

Figure 1

Calculated energy spectra of (a) a semi-infinite system and (b) a finite strip of width $w=200$ nm for $B=10$ T, $\mathcal{A}=364.5$ meV nm, $\mathcal{B}=-686.0$ meV nm${}^2$, $\mathcal{C}=0$, $\mathcal{D}=-512.0$ meV nm${}^2$, $\mathcal{M}=-10.0$ meV, and $g_{\mathrm{e}}=g_{\mathrm{h}}=0$. Here, the energy spectra are plotted versus $y_k=l_B^{2}k$. The solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively, which have been calculated using the analytical methods from Sec. 3a [case (ii) for panel (a) and case (iii) for panel (b)]. Results obtained by the finite-difference method from Sec. 3b are represented by circles (spin up) and diamonds (spin down) in panel (b).

Figure 2

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=5.5$ nm, $w=200$ nm, and $B=0$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 3

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=5.5$ nm, $w=200$ nm, and $B=0.1$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 4

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=5.5$ nm, $w=200$ nm, and $B=1$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 5

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=5.5$ nm, $w=200$ nm, and $B=10$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 6

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=7.0$ nm, $w=200$ nm, and $B=0$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 7

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=7.0$ nm, $w=200$ nm, and $B=0.1$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$-direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 8

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=7.0$ nm, $w=200$ nm, and $B=1$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 9

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=7.0$ nm, $w=200$ nm, and $B=10$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 10

Magnetic field dependence of the states at $k=0$ in a finite strip of width $w=200$ nm compared to the bulk Landau levels given by Eqs. (25, 26, 27, 28). The thinner solid and dashed lines represent bulk Landau levels for $s=\uparrow$ and $\downarrow$, respectively. The levels of the finite-strip geometry are displayed by thick lines. All levels displayed here have been calculated for band parameters corresponding to $d=7.0$ nm.

Figure 11

Magnetic field dependence of the states at $k=0$ in finite strips with the widths (a) $w=25$ nm, (b) $w=50$ nm, (c) $w=75$ nm, and (d) $w=100$ nm compared to the bulk Landau levels given by Eqs. (25, 26, 27, 28). The thinner solid and dashed lines represent bulk Landau levels for $s=\uparrow$ and $\downarrow$, respectively. The levels of the finite-strip geometry are displayed by thick lines. All levels displayed here have been calculated for band parameters corresponding to $d=7.0$ nm.

Figure 12

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=6.3$ nm, $w=200$ nm, and $B=0$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 13

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=6.3$ nm, $w=200$ nm, and $B=0.1$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 14

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=6.3$ nm, $w=200$ nm, and $B=1$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 15

(a) Calculated energy spectrum and (b), (c) probability densities $\rho (x,y)={\left|\Psi (x,y)\right|}^2$ of selected states for $d=6.3$ nm, $w=200$ nm, and $B=10$ T, where solid and dashed lines represent $s=\uparrow$ and $\downarrow$ states, respectively. Here, the states shown in Figs. (b) and (c) are marked in the energy spectrum, Fig. (a), by dots. The velocity with which the states propagate along the $x$ direction is given by $v_k=[\partial E\left(k\right)/\partial k]/\hslash$.

Figure 16

The nonvacuum magnetization $M\left(T,\mu ,B\right)$ (corresponding to a quantum-well thickness of $d=7.0$ nm) plotted versus $1/B$ for a fixed chemical potential $\mu =20$ meV and different temperatures ($T=1,10,100$ K).

Figure 17

The nonvacuum susceptibility $\chi \left(T,\mu ,B\right)$ (corresponding to a quantum-well thickness of $d=7.0$ nm) plotted versus $1/B$ for a fixed chemical potential $\mu =20$ meV and different temperatures ($T=1,10,100$ K).

Figure 18

Magnetic field dependence of the chemical potential $\mu \left(T,n_d,B\right)$ (corresponding to a quantum-well thickness of $d=7.0$ nm) for $n_d={10}^{16}1/{\text{m}}^2$ and different temperatures ($T=0,10,100K$).

Figure 19

Magnetic field dependence of the chemical potential $\mu \left(T,n_d,B\right)$ (corresponding to a quantum-well thickness of $d=7.0$ nm) for $T=10$ and 100 K and different densities ($n_d={10}^{14},{10}^{15},{10}^{16}1/{\text{m}}^2$).

Figure 20

Magnetic field dependence of the contribution $M_{\mathrm{dis}}\left(B\right)+M\left(T,\mu ,B\right)$ (corresponding to a quantum-well thickness of $d=7.0$ nm) for $T=10$ K and different densities ($n_d={10}^{14},{10}^{15},{10}^{16}1/{\text{m}}^2$).

Figure 21

The nonvacuum magnetization $M\left(T,\mu ,B\right)$ (corresponding to a quantum-well thickness of $d=5.5$ nm) plotted versus $1/B$ for a fixed chemical potential, $\mu =20$ meV, and different temperatures ($T=1,10,100$ K).

Figure 22

The nonvacuum susceptibility $\chi \left(T,\mu ,B\right)$ (corresponding to a quantum-well thickness of $d=5.5$ nm) plotted versus $1/B$ for a fixed chemical potential, $\mu =20$ meV, and different temperatures ($T=1,10,100$ K).

Figure 23

Magnetic field dependence of the magnetization $M_{\mathrm{dis}}\left(B\right)$ (corresponding to a quantum-well thickness of $d=7.0$ nm).

Figure 24

Magnetic field dependence of the nonvacuum magnetization $M\left(T,\mu ,B\right)$ (corresponding to a quantum-well thickness of $d=7.0$ nm) for $T=10$ K and different densities ($n_d={10}^{14},{10}^{15},{10}^{16}1/{\text{m}}^2$).

Figure 25

Magnetic field dependence of the vacuum magnetization ${\tilde{M}}_0\left(B\right)$ corresponding to a quantum-well thickness of $d=7.0$ nm and $\alpha =10$.

Figure 26

Magnetic field dependence of the vacuum magnetization ${\tilde{M}}_0\left(B\right)$ corresponding to a quantum-well thickness of $d=5.5$ nm and $\alpha =10$.