Thickness-independent transport in thin (001)-oriented cadmium arsenide films

The three-dimensional Dirac semimetal is a parent phase for a variety of topological phases that can be generated by tuning parameters in material growth or device operation. Notably, it has recently been found that cadmium arsenide, which is ordinarily a three-dimensional Dirac semimetal, can nevertheless realize a three-dimensional topological insulator in (001)-oriented films about 50-nm thick. In this work, we study the quantum Hall effect in thin (001)-oriented cadmium arsenide films, their thickness ranging from 12 to 24 nm. When the carrier density is kept approximately constant across the different films, quantum transport reveals an identical underlying picture. The result is shown to be consistent with the transport's origin in the surface states of a three-dimensional topological insulator, but problematic for a perspective in which the quantum Hall effect originates from the confined subbands of the bulk band structure. These thin-film results complement previous studies of the quantum Hall effect in 50-nm-thick films.

Figure 1

Model Hamiltonian spectra. (a)–(d) Zero-field spectra for small $k$ and $v_F=8\times {10}^5\mathrm{m}/\mathrm{s}$. The parameter values are chosen for clarity. In panel (a), the cones are doubly degenerate everywhere because ${\mathrm{\Delta }}_i={\mathrm{\Delta }}_h=0$. (b) The effect of finite ${\mathrm{\Delta }}_i$ is to displace the two cones in energy. The upper and lower cones are each associated to one of the eigenvalues of the ${\tau }_z$ operator. (c) By contrast, a finite ${\mathrm{\Delta }}_h$ opens a gap in the spectrum, minimal at $k=0$, which affects both surfaces in equal measure; the two bands are doubly degenerate everywhere. (d) When both ${\mathrm{\Delta }}_h$ and ${\mathrm{\Delta }}_i$ are appreciable, a gap opens and the two bands are in general nondegenerate. Whether away from $k=0$ or $E=0$, however, the spectrum is qualitatively and quantitatively similar to that in panel (b). (e) and (f) Landau level spectra as a function of ${\mathrm{\Delta }}_h$ and ${\mathrm{\Delta }}_i$. Throughout $g^{*}$ is set at +25. (The positive sign means that the zeroth Landau level disperses lower in energy with increasing field.) The changing parameter is colored according to the scale at the far right. (e) Effect of changing ${\mathrm{\Delta }}_h$ with finite ${\mathrm{\Delta }}_i$. The Landau levels are shifted in energy, which is more noticeable for Landau levels with low indices. (f) Effect of changing ${\mathrm{\Delta }}_i$ with finite ${\mathrm{\Delta }}_h$. Small changes to ${\mathrm{\Delta }}_i$ cause large changes to the spectrum because the two fans [more visible in panel (e)] are pushed to higher and lower energies, respectively.

Figure 2

The quantum Hall effect in sample D ($n_{2D}=6.95\times {10}^{11}\mathrm{c}{\mathrm{m}}^{-2}$). (a) The longitudinal magnetoresistance, $R_{xx}$, acquired at 2 K, is plotted against magnetic field. Minima are indicated with arrows, matching those in the lower panel. (b) The von Klitzing constant ($R_K=h/e^2$) is divided by $R_{xy}$, the Hall resistance, to show that the plateaus match integer filling factors $\nu$, indicated by the dashed lines and labels. Arrows correspond to the minima in $R_{xx}$, shown in panel (a). Though a plateau does not form with $\nu =5$, a corresponding minimum in $R_{xx}$ is nevertheless visible (dashed arrows).

Figure 3

A comparison of the quantum Hall effect, measured at 2 K, in samples A, B, C, and D at various values of carrier density. (a) The quantum Hall effect in sample D. Each trace is recorded under a different top gate bias, corresponding to a different carrier density. For legibility, the traces in the longitudinal magnetoresistance, $R_{xx}$, are offset from each other by sequential multiples of 250 Ω. The true values are all comparable to the lowest resistance trace, which is not offset. (b)–(d) A comparison of the quantum Hall effect in samples A, B, C, and D at carrier densities of (b) about $6.5\times {10}^{11}\mathrm{c}{\mathrm{m}}^{-2}$ (traces for samples A and B were acquired prior to gate deposition), (c) about $7\times {10}^{11}\mathrm{c}{\mathrm{m}}^{-2}$ (traces for samples A and B were acquired prior to gate deposition), and (d) about $1.5\times {10}^{12}\mathrm{c}{\mathrm{m}}^{-2}$ (the trace for sample D was acquired prior to gate deposition). No offsets have been added in panels (b)–(d).

Figure 4

Frequency analysis of quantum oscillations. (a)–(d) Oscillations with background subtracted for samples A through D, respectively. (e)–(h) Fourier transform of the background-subtracted data from samples A through D, respectively. (i)–(l) Fits (lines) of the background-subtracted oscillations (markers) according to $\mathrm{\Delta }{R}_{xx}=Aexp\left(-B_0/B\right)cos\left(2\pi F/B+\varphi \right)$ and $\mathrm{\Delta }{R}_{xx}=A_{1}exp\left(-B_{0,1}/B\right)cos\left(2\pi F_1/B+{\varphi }_1\right)+A_{2}exp\left(-B_{0,2}/B\right)cos\left(2\pi F_2/B+{\varphi }_2\right)$, where $A,A_1,A_2,B_0,B_{0,1},B_{0,2},F,F_1,F_2,\varphi ,{\varphi }_1$, and ${\varphi }_2$ are fit parameters. The extracted oscillation frequencies $F$, $F_1$, and $F_2$, are compared to those derived from the Fourier transform. Note that the oscillation data are interpolated on even intervals in 1/B. Panels (i)–(l) are labeled by the difference $\mathrm{\Delta }F=F_{\mathrm{fit}}\text{–}{F}_{FT}$. Where two cosines are used to fit the data, two values are reported, the lower first. The fit did not converge for the middle sample B trace, and no difference is reported. The agreement for the high-frequency peak is generally very good, less than 2 T, that is, $\mathrm{\Delta }F$ and $\mathrm{\Delta }{F}_2$ are generally small.

Figure 5

Thickness dependence in the subband picture. (The k·p model and coefficients are those from Ref. [27]). In-plane spectra ($k_{\left|\right|}=k_x=k_y$) are plotted for a film thickness L equal to (a) 6 nm, (b) 12 nm, (c) 15 nm, (d) 18 nm, (e) 24 nm, and (f) 30 nm. (g) Evolution of the gap $E_G$ as a function of L. The thicknesses corresponding to panels (a)–(f) are marked with labeled, colored vertical lines.

Figure 6

Background-subtracted oscillations in conductivity (a)–(d) and corresponding fan diagrams (e)–(n). Panels (a)–(d) show quantum oscillations, plotted against magnetic field, for samples A–D, respectively. For samples B, C, and D, traces are offset by 40 and 80 μS (zero is denoted by a horizontal dashed line). In all cases the higher density traces are on the top; the scheme matches the labeling of the fan diagrams in the bottom half of the figure. Peaks are identified in the conductivity. The magnetic field values for the peak centers are used to make the fan diagrams in the bottom half of the figure, in panels (e) through (n). These fan diagrams are identified by the sample (a letter) and the carrier density (in units of ${10}^{11}\mathrm{c}{\mathrm{m}}^{\text{–}2}$). Note that not all peaks (open circles) are fitted (solid lines). The $x$ intercept is identified as $n_0$ in each panel, along with an error (one standard deviation). Finally, note that the abscissa of the fan diagrams (e)–(n) is in all cases simply an unadjusted integer index of the conductivity peaks, counting up from 1 for the peak at the highest field.

Figure 7

Fan diagrams extracted from the magnetoresistance data shown in Fig. 3. Each minimum in $R_{xx}\left(B\right)$ is indexed by the concurrent value of $\nu =R_K/R_{xy}$, rounded to an integer, which is plotted against the value of the inverse of the magnetic field ($1/B$) where the minimum occurs. Linear fits are shown; $y$ intercepts are consistent with zero except for the high-density traces in samples C and D.