Exploring the phase diagram of InAs/GaSb/InAs trilayer quantum wells

We study topological insulators based on InAs/GaSb/InAs trilayer quantum wells with different InAs thicknesses ranging from 8.0 to 13.5 nm to experimentally ascertain the phase diagram in this material system. Gate-voltage and temperature-dependent transport measurements reveal distinct transport signatures between samples designed from the trivial-insulating, the topological-insulating, up to the semimetallic phase. Two-carrier transport together with pronounced Van Hove singularities are observable only for the samples grown in the topological-insulating phase and if camelback-like dispersions are present in the hybridized valence or conduction band. The different shaping and the amount of Van Hove singularities in our samples deliver information about a turning point inside the topological-insulating phase. Temperature-dependent measurements allow to further confirm the gap energy values for the trivial- and topological-insulating samples. Furthermore, the indirect inverted gaps are found to be more temperature insensitive in comparison to the trivial gap.

Figure 1

(a) Schematic band structure of an inverted TQW in InAs geometry embedded in ${\mathrm{AlAs}}_{0.08}{\mathrm{Sb}}_{0.92}$ barriers. The layer thicknesses for both InAs and GaSb layers are given by $d_1$ and $d_2$, respectively. The hole level $H_1$ is localized in the GaSb layer, while the electron levels $E_1$ and $E_2$ are located in the InAs layers. (b) Calculated phase diagram for the TQW in InAs geometry. The phase diagram displays the three different phases: trivial-insulating, TI, and the semimetallic phase. The TI phase can be further divided into two subphases dependent on the position of the indirect inverted gap, i.e., $H_1\text{−}{E}_1$ (dark gray) and $E_2\text{−}{E}_1$ (gray). The color-coded dots mark the position of the examined samples.

Figure 2

Calculated band dispersions for a constant GaSb thickness $d_2=5.0\mathrm{nm}$ and InAs thicknesses $d_1=8.0$, 13.5, 10.0, and 10.75 nm in (a), (b), (c), and (d), respectively. On the $x$ axis, the Γ point for all dispersion is shown. Additionally, the crossing points for (c) and (d) where $k=k_{\mathrm{cross}}$ are highlighted. (a) In the trivial-insulating phase, $E_2$ and $E_1$ are energetically above $H_1$ and the band ordering is trivial. The band-gap energy has a value of 22.6 meV. (b) In the semimetallic phase, the minimum of the conduction band is energetically below the maximum of the valence band leading to a gapless state. (c) For hyb1, $E_1$ lies energetically below $H_1$. A camelback-like dispersion is observed where the local minimum of the valence band lies below the maximum of the valence band. In the conduction band, a weak camelback-like dispersion is present. The indirect inverted gap has a value of 11.0 meV. (d) For hyb2, $E_2$, and $E_1$ lie below $H_1$ leading to a pronounced camelback for the $E_1$ subband while $E_2$ remains parabolic. The indirect inverted gap has a value of about 6.7 meV.

Figure 3

(a) Extracted band-gap energies $E_{\mathrm{gap}}$ as a function of the InAs quantum-well thickness $d_1$. The GaSb quantum-well thickness is constant ($d_2=5.0\mathrm{nm}$). (b) Upper panel: Energy values of $E_2$ (black) and $E_1$ (blue) measured against the $H_1$ value as a function of $d_1$. Lower panel: Energy differences $\mathrm{\Delta }{H}_1$ (red) and $\mathrm{\Delta }{E}_1$ (blue) between the local energy minima and maxima in the camelback within the corresponding bands as a function of $d_1$.

Figure 4

Gate-voltage traces of the longitudinal resistivity for the trivial, the sm, the hyb1, and hyb2 sample in (a), (b), (c), and (d), respectively. (a) Trivial sample: $E_{\mathrm{F}}$ is in the electron regime for gate voltages above $V_{\mathrm{CNP}}$ and in the hole regime for voltages below $V_{\mathrm{CNP}}$. (b) sm sample: The semimetallic sample shows a gapless resistance trace. The ${\rho }_{xx}$ values are significantly lower compared to all other samples. (c) hyb1 sample: VHS are observable for the CB and VB due to a camelback-like dispersion in both bands and a gap is observed. (d) hyb2 sample: a VHS is only observable in the hole regime due to the camelback-like dispersion only in the VB. The VHS is much more pronounced compared to the hyb1 sample.

Figure 5

Hall resistance of the triv, hyb1, and hyb2 sample in (a) the electron regime and the gap and the (b) hole regime for magnetic fields up to 2 T. The Hall resistance traces of the trivial sample are evolving linearly for both regimes, whereas the traces for both hybridized samples are only linear in the electron regime. In the gap and hole regime, a nonlinearity in $R_{xy}$ can be seen for hyb1 and hyb2.

Figure 6

Charge-carrier densities from the two-carrier fitting for the hyb1 and the hyb2 sample in (a) and (b), respectively. The corresponding longitudinal resistivity ${\rho }_{xx}$ at $B=0\mathrm{T}$ is also shown in blue. The electron densities decrease with decreasing gate voltages for both samples. For both samples a minority electron density is observable in the gap and hole regime due to the hybridized band structure.

Figure 7

(a) $V_{\mathrm{TG}}$ dependency of ${\rho }_{xx}$ at different temperatures (color coded) starting from $T=4.2\mathrm{K}$ for the trivial, the hyb1, and the hyb2 sample in (a), (b), and (c). With increasing temperature, the resistivity peak corresponding to the trivial/inverted gap reduces until it vanishes at higher temperatures. (d) Arrhenius plot of the temperature dependence of the peak resistivity ${\rho }_{xx,max}$ for the three samples triv, hyb1, and hyb2. The inset shows the data in the high-temperature regime. The dashed lines are fits with a single-exponential function to extract the energy gap values.