Density-Dependent Quantum Hall States and Zeeman Splitting in Monolayer and Bilayer WSe$_2$

TL;DR

This study investigates the quantum Hall states in monolayer and bilayer WSe$_2$, revealing how their sequence depends on hole density and is influenced by electron-electron interactions and Zeeman effects.

Contribution

It demonstrates the density-dependent transition of quantum Hall state sequences and their insensitivity to electric and in-plane magnetic fields in WSe$_2$, highlighting the role of Zeeman-to-cyclotron energy ratio.

Findings

QHSs sequence transitions between even and odd filling factors with density.

QHSs sequence is unaffected by transverse electric field.

Sequence remains insensitive to in-plane magnetic field.

Abstract

We report a study of the quantum Hall states (QHSs) sequence of holes in mono- and bilayer WSe$_2$. The QHSs sequence transitions between predominantly even and predominantly odd filling factors as the hole density is tuned in the range $1.6 - 12\times10^{12}$ cm$^{-2}$. The QHSs sequence is insensitive to the transverse electric field, and tilted magnetic field measurements reveal an insensitivity of the QHSs sequence to the in-plane magnetic field, evincing that the hole spin is locked perpendicular to the WSe$_2$ plane. These observations imply that the QHSs sequence is controlled by the Zeeman-to-cyclotron energy ratio, which remains constant as a function of perpendicular magnetic field at a fixed carrier density, but changes as a function of density due to strong electron-electron interaction.

Full text

In this work, we report on the magnetotransport of holes in mono- and bilayer WSe2 in the quantum Hall regime. An examination of the Shubnikov–de Haas (SdH) oscillations, and the quantum Hall states (QHSs) sequence reveals interesting carrier density-dependent transitions between predominantly even and predominantly odd filling factors (FFs) as the hole density is tuned. Measurements in tilted magnetic fields reveal an insensitivity of the QHSs sequence to the in-plane magnetic field, indicating that the hole spin is locked perpendicular to the WSe2 plane. The QHSs sequence is also found to be insensitive to the applied transverse electric field. These observations can be explained by a Zeeman-to-cyclotron energy ratio which remains constant as a function of perpendicular magnetic field at a fixed carrier density, but changes as a function of density because of strong electron-electron interaction.

Figure 1(a) shows the schematic cross section, and Fig. 1(b) the optical micrograph of an hBN encapsulated WSe2 sample with bottom Pt contacts, and separate local top- and back-gates. The mono- and bilayer WSe2 Hall bar samples were fabricated using a modified van der Waals assembly technique Movva et al. (2015); Kim et al. (2016). The bottom Pt electrodes in combination with a large, negative top-gate bias ($V_{TG}$) ensure Ohmic hole contacts to the WSe2 Movva et al. (2015); Fallahazad et al. (2016). Both $V_{TG}$, and a back-gate bias ($V_{BG}$) were used to tune the WSe2 hole carrier density, $p$. The magnetotransport was probed using low frequency lock-in techniques at a temperature, $T=1.5$ K, and magnetic fields up to $B=35$ T. The $p$ values at which we observe well-defined SdH oscillations are in the range $1.6-12\times 10^{12}$ cm*-2*, as determined from the slope of the Hall resistance, and from the SdH oscillations minima. The weak interlayer coupling in bilayer WSe2 enables the top layer population with holes, while keeping the bottom layer unpopulated Fallahazad et al. (2016). At negative $V_{TG}$, and positive $V_{BG}$, a bilayer effectively acts as a monolayer, albeit with a dissimilar dielectric environment 111While a monolayer has only hBN as the bottom dielectric, a bilayer biased to populate only the top layer has an unpopulated WSe2 bottom layer, which alters the dielectric environment of holes in the top layer.. All the bilayer data presented here were collected under such biasing conditions, and are therefore closely similar to the monolayer data.

Figure 1(c) shows the longitudinal ($R_{xx}$) and Hall ($R_{xy}$) resistance vs perpendicular magnetic field ($B$) for a bilayer WSe2 sample at the lowest density, $p=1.6\times 10^{12}$ cm*-2*. The $R_{xx}$ data show SdH oscillations starting at $B\cong 5$ T, which translates into a mobility, $\mu\simeq 2000$ cm${}^{2}/$V s. The FFs, $\nu=ph/eB$, at the $R_{xx}$ minima are marked. The $R_{xy}$ data show developed QHSs plateaux at $\nu=2,3$, where $R_{xy}$ is quantized at values of $h/{\nu}e^{2}$; $h$ is the Planck’s constant, and $e$ the electron charge. The QHSs occur at consecutive integer FFs ($\nu=2,3,4,...$) for $B>10$ T, indicating a full lifting of the two-fold Landau level (LL) degeneracy in WSe2 Fallahazad et al. (2016). For $B<10$ T, the QHSs occur at consecutive odd integer FFs ($\nu=7,9,...$). The occurrence of QHSs at odd FFs contrasts previous observations of QHSs at even FFs in mono- and bilayer WSe2 at higher hole densities Fallahazad et al. (2016). In the following, we will use the term “QHSs sequence”, be it even or odd, to refer to the QHSs FFs in the lower range of $B$ values, such that the LL degeneracy is not fully lifted.

To better understand the QHSs sequence, we performed magnetotransport measurements as a function of $p$ in both mono- and bilayer WSe2. Figure 2(a,b) show $R_{xx}$ and $R_{xy}$ vs $B$ measured for the same bilayer sample discussed in Fig. 1 at $p=3.9\times 10^{12}$ cm*-2*, and $p=5.3\times 10^{12}$ cm*-2*, respectively. While the data at $p=3.9\times 10^{12}$ cm*-2* show an odd QHSs sequence, the QHSs sequence is even at $p=5.3\times 10^{12}$ cm*-2*. Figure 2(c) shows $R_{xx}$ vs $\nu$ at various values of $p$ from $6.1\times 10^{12}$ cm*-2* to $2.4\times 10^{12}$ cm*-2*. The data at $p=6.1\times 10^{12}$ cm*-2* show strong $R_{xx}$ minima at even FFs, and weakly developing minima at odd FFs for $\nu<16$, hence a predominantly even QHSs sequence. As $p$ is reduced to $4.4\times 10^{12}$ cm*-2*, the minima at odd FFs become stronger, and equal in strength to the minima at even FFs. The QHSs sequence at this $p$ cannot be unambiguously classified as even or odd. Further reduction of $p$ to $3.9\times 10^{12}$ cm*-2* makes the odd FFs stronger than the even FFs, rendering the QHS sequence as predominantly odd. The odd QHSs sequence is retained down to $p=2.9\times 10^{12}$ cm*-2*. On further reduction of $p$ to $2.4\times 10^{12}$ cm*-2*, the QHSs sequence reverts to even. Figure 2(d) shows a similar data set for monolayer WSe2, where the QHSs sequence transitions from even at $p=9.7\times 10^{12}$ cm*-2* to odd at $p=4.6\times 10^{12}$ cm*-2*, and back to even at $p=3.3\times 10^{12}$ cm*-2*.

This unusual density-dependent QHSs sequence suggests an interesting interplay of the LL Zeeman splitting and the cyclotron energy. The cyclotron energy of the LLs originating in the upper valence band of monolayer WSe2 is $E_{n}=-n\hbar\omega_{c}$; $n$ is the orbital LL index, $\omega_{c}=eB/m^{*}$ is the cyclotron frequency, $m^{*}=0.45m_{0}$ the hole effective mass Fallahazad et al. (2016); $m_{0}$ is the bare electron mass. The LLs with $n>0$ are spin-degenerate, whereas the $n=0$ LL is non-degenerate Li et al. (2013); Rose et al. (2013). Consequently, in the absence of LL Zeeman splitting, an odd QHSs sequence is expected. However, if the LL Zeeman splitting $E_{Z}=g^{*}\mu_{B}B$ is comparable to, or larger than the cyclotron energy $E_{c}=\hbar\omega_{c}$, the QHSs sequence changes accordingly Chu et al. (2014); $\mu_{B}$ is the Bohr magneton. An $E_{Z}/{E_{c}}$ ratio close to an even (odd) integer leads to a QHSs sequence that is predominantly odd (even). Two noteworthy observations can be made based on Fig. 2 data. First, the presence of an even or odd QHSs sequence at a fixed density implies that the $E_{Z}/{E_{c}}$ ratio, and $g^{*}$ do not change with $B$ at low fields. Second, the different QHSs transitions observed in Fig. 2 suggest that the $E_{Z}/{E_{c}}$ ratio, and therefore $g^{*}$ change with density, likely because electron-electron interaction associated with the large $m^{*}$ in this system leads to an enhanced $g^{*}$ as the density is reduced.

Two measurement types have been traditionally used to probe the Zeeman splitting in 2DESs. In a tilted magnetic field, the $B$ component perpendicular to the 2DES plane ($B_{\perp}$) determines the cyclotron energy $E_{c}=\hbar\omega_{c}=\hbar{eB_{\perp}}/m^{*}$, while the Zeeman energy, $E_{Z}=g^{*}\mu_{B}{B}$ depends on the total field Fang and Stiles (1968). At specific angles $\theta$ between the $B$-field and the normal to the 2DES plane, the $E_{Z}/E_{c}$ ratio attains integer values, which leads to a collapse of different QHSs, and allows a quantitative determination of $E_{Z}$. To assess this effect in our samples, Fig. 3(a) shows $R_{xx}$ vs $B_{\perp}$ for a monolayer WSe2 sample, measured at $p=4.6\times 10^{12}$ cm*-2*, and at different values of $\theta$. The $R_{xx}$ at $\theta=0^{\circ}$ shows an odd QHSs sequence, which remains virtually unchanged for all values of $\theta$ up to $77^{\circ}$. A similar behavior was observed even for bilayer WSe2, suggesting indeed that $E_{Z}$ is insensitive to $\theta$, and in turn, to the parallel component of the $B$-field ($B_{||}$) in both mono- and bilayer WSe2. This observation is in stark contrast to the vast majority of 2DESs explored in host semiconductors such as Si Fang and Stiles (1968); Shashkin et al. (2001), GaAs Zhu et al. (2003), AlAs Vakili et al. (2004), black phosphorus Li et al. (2016), and bulk WSe2 Xu et al. (2017).

A second technique used to determine $E_{Z}$ is the magnetoresistance measured as a function of the magnetic field parallel to the 2DES plane. The Zeeman coupling leads to a spin polarization of the 2DES, which reaches unity when $E_{Z}$ is equal to the Fermi energy. Experimentally, $R_{xx}$ vs $B_{||}$ measured at $\theta=90^{\circ}$ shows a positive magnetoresistance, along with a saturation or a marked kink at the $B$-field corresponding to full spin polarization Okamoto et al. (1999); Tutuc et al. (2001); Zhu et al. (2003); Vakili et al. (2004). Figure 3(b) shows $R_{xx}$ vs $B_{||}$ data for the monolayer sample of Fig. 3(a). Surprisingly, yet consistent with Fig. 3(a) data, $R_{xx}$ remains constant over the entire range of $B_{||}$, which implies that $E_{Z}$ depends only on $B_{\perp}$, namely $E_{Z}=g^{*}\mu_{B}B_{\perp}$, via a density-dependent $g^{*}$. The insensitivity of $E_{Z}$ to $B_{||}$ indicates that the hole spin is locked perpendicular to the plane, a direct consequence of the strong spin-orbit coupling, and mirror symmetry in monolayer WSe2 Zhu et al. (2011). Optical experiments on monolayer WSe2 have shown a similar insensitivity of $E_{Z}$ to $B_{||}$ Srivastava et al. (2015). We note that spin-locking along the $z$-direction renders the tilted $B$-field technique ineffective to determine $E_{Z}$.

In light of Fig. 2 data which suggest a density-dependent $g^{*}$, one important question is whether the $g^{*}$ variation is determined by the density, or by the applied transverse electric field ($E$), which depends on the applied gate biases and can change concomitantly with the density. The impact of a transverse $E$-field on bandstructure has been experimentally investigated, among others, in 2D electrons in InGaAs/InAlAs Nitta et al. (1997), 2D holes in GaAs Lu et al. (1998); winkler2005anomalous, and has been theoretically considered in TMDs using a Bychkov-Rashba coupling Kormányos et al. (2014). To probe the impact of the $E$-field on the QHSs sequence in WSe2, we performed $R_{xx}$ vs $B$ measurements by varying $E=|C_{TG}V_{TG}-C_{BG}V_{BG}|/2\epsilon_{0}$ at constant $p$; $C_{TG}$ ($C_{BG}$) is the top (back)-gate capacitance, and $\epsilon_{0}$ the vacuum permittivity. Figure 3(c) shows $R_{xx}$ vs $\nu$ measured in bilayer WSe2 at $p=3.5\times 10^{12}$ cm-2*, at different values of $E$. The data show no variation of the QHSs sequence when the $E$-field varies from 0.64 V/nm to 1.15 V/nm. By comparison, the $E$-field changes from 0.92 V/nm to 1.11 V/nm in Fig. 2(c), concomitantly with the density change from $6.1\times 10^{12}$ cm*-2* to $3.9\times 10^{12}$ cm*-2*, a range in which a QHSs sequence transition from even to odd is observed. Based on these observations, we rule out the effect of the $E$-field on $g^{*}$, and in turn, on the QHSs sequence.

In Fig. 4(a), we summarize the QHSs sequence vs $p$ for four monolayer, and four bilayer WSe2 samples. The data points are grouped into an even or odd QHSs sequence over a range of $p$. We attribute the QHSs sequence transitions to a change in the $E_{Z}/E_{c}$ ratio with varying $p$. For instance, $E_{Z}\approx{E_{c}}$ ($E_{Z}\approx{2E_{c}}$) can lead to an even (odd) QHSs sequence [Fig. 4(b) inset]. Generalized further, $|E_{Z}|/E_{c}\in[2k-1/2,2k+1/2]$ yields an odd QHSs sequence, and $|E_{Z}|/E_{c}\in[2k+1/2,2k+3/2]$ yields an even QHSs sequence; $k$ is an integer. Each of the groups of Fig. 4(a) can therefore be assigned an $|E_{Z}|$ within a $[-E_{c}/2,E_{c}/2]$ window. Starting with $|E_{Z}|=E_{Z}^{0}$ at the highest value of $p$ probed, and assuming $|E_{Z}|$ increases with reducing $p$ because of interaction, we can assign an $|E_{Z}|$ increment of $E_{c}$ for every QHSs sequence transition in the direction of decreasing $p$ [Fig. 4(b)]. In the absence of electron-electron interaction, the $g$-factor, referred to as the band $g$-factor ($g_{b}$) is determined by the material bandstructure. Exchange interaction can enhance $g_{b}$ to a value $g^{*}$, which increases with decreasing density, an observation reported for several 2DESs in Si Okamoto et al. (1999); Shashkin et al. (2001), GaAs Zhu et al. (2003), and AlAs Vakili et al. (2004). The interaction strength is gauged by the dimensionless parameter, $r_{s}=1/(\sqrt{\pi{p}}a^{*}_{B})$, the ratio of the Coulomb energy to the kinetic energy; $a^{*}_{B}=a_{B}(\kappa{m_{0}}/m^{*})$, $a_{B}$ is the Bohr radius, and $\kappa$ is the effective dielectric constant of the medium surrounding the 2DES.

The $|E_{Z}|$ vs $p$ of Fig. 4(b) can therefore be converted to a $|g^{*}|$ vs $r_{s}$ dependence. We first address the value of $E_{Z}^{0}=g^{*}_{0}\mu_{B}B$. The even QHSs sequence at the highest $p$ probed implies that $E_{Z}^{0}=(2k+1)E_{c}$, or equivalently, $g^{*}_{0}=4.44(2k+1)$; $k$ is an integer 222We assume a density independent $m^{*}=0.45m_{0}$ Fallahazad et al. (2016). Recent magneto-reflectance measurements that resolve the LL spectrum report a $g_{b}=8.5$ for holes in monolayer WSe2 Wang et al. (2016). To account for the uncertainty in $E_{Z}^{0}$, we consider two scenarios of $g^{*}_{0}$ corresponding to $k=1$ ($E_{Z}^{0}=3E_{c}$), and $k=2$ ($E_{Z}^{0}=5E_{c}$). We rule out the case $k=0$ based on the reported $g_{b}$ value Wang et al. (2016). The $E_{c}$-step increments of $|E_{Z}|$ between groups are equivalent to a $|g^{*}|$ increment of $\Delta{g^{*}}=2m_{0}/m^{*}=4.44$ Note (2). Within this framework, Fig. 4(c) and Fig. 4(d) show $|g^{*}|$ vs $r_{s}$ for the mono- and bilayer samples, respectively. Because of the difference in dielectric environment, slightly different $\kappa$ values were used to convert $p$ into $r_{s}$ for mono- and bilayer WSe2 333For a 2DES with an asymmetric dielectric environment, $\kappa(t,b)=[(\epsilon^{||}_{t}\epsilon^{\perp}_{t})^{1/2}+(\epsilon^{||}_{b}\epsilon^{\perp}_{b})^{1/2}]/2$; $\epsilon^{||/\perp}_{t(b)}$ are the top (bottom) medium dielectric constants. For monolayer WSe2, $\kappa=\kappa(hBN,hBN)$. For bilayer WSe2 Note (1), $\kappa=[\kappa(hBN,hBN)+\kappa(hBN,WSe_{2})]/2$. We use $\epsilon_{hBN}^{||}=3.0$, $\epsilon_{hBN}^{\perp}=6.9$ Geick et al. (1966), $\epsilon_{WSe_{2}}^{||}=7.2$ Kim et al. (2015), and $\epsilon_{WSe_{2}}^{\perp}=14$ Li et al. (2014b)..

For comparison, in Fig. 4(c,d) we include the $g_{b}$ value multiplied by the interaction enhanced spin susceptibility obtained from quantum Monte Carlo (QMC) calculations Attaccalite et al. (2002). The QMC calculations along with the $g_{b}$ of Ref. Wang et al. (2016) match well with $|g^{*}|$ determined using $E_{Z}^{0}=5E_{c}$ for both mono- and bilayer WSe2. Noteworthy, the relatively large $m^{*}=0.45m_{0}$ leads to moderately large $r_{s}$ values, and potentially strong interaction effects even at high carrier densities Tanatar and Ceperley (1989).

In summary, we present a density-dependent QHSs sequence of holes in mono- and bilayer WSe2, which transitions between even and odd filling factors as the hole density is tuned. The QHSs sequence is insensitive to the in-plane $B$-field, indicating that the hole spin is locked perpendicular to the WSe2 plane, and is also insensitive to the transverse $E$-field. The QHSs sequence transitions stem from an interplay between the cyclotron and Zeeman splittings via an enhanced $g^{*}$ due to strong electron-electron interaction.