Magnetotransport in heterostructures of transition metal dichalcogenides and graphene

TL;DR

This study investigates magnetotransport properties in heterostructures of transition metal dichalcogenides and graphene, revealing effects like weak antilocalization and quantum oscillations, with mobility enhancements and boundary scattering effects.

Contribution

It demonstrates fabrication of high-mobility heterostructures and analyzes their magnetotransport behavior, highlighting the transition from diffusive to quasiballistic regimes and associated phenomena.

Findings

Observation of weak antilocalization in certain heterostructures.

Mobility up to 120,000 cm^2/Vs in encapsulated samples.

Detection of complete spin and valley degeneracy lifting in quantum oscillations.

Abstract

We use a van-der-Waals pickup technique to fabricate different heterostructures containing WSe$_2$(WS$_2$) and graphene. The heterostructures were structured by plasma etching, contacted by one-dimensional edge contacts and a topgate was deposited. For graphene/WSe$_2$/SiO$_2$ samples we observe mobilities of $\sim$12 000 cm$^2$/Vs. Magnetic field dependent resistance measurements on these samples show a peak in the conductivity at low magnetic field. This dip is attributed to the weak antilocalization (WAL) effect, stemming from spin-orbit coupling. Samples where graphene is encapsulated between WSe$_2$(WS$_2$) and hBN show a much higher mobility of up to $\sim$120 000 cm$^2$/Vs. However, in these samples no WAL peak can be observed. We attribute this to a transition from the diffusive to the quasiballistic regime. At low magnetic field a resistance peak appears, which we ascribe to a size effect, due to boundary scattering. Shubnikov-de Haas oscillations in fully encapsulated samples show all integer filling factors, due to complete lifting of the spin and valley degeneracy.

Equations

$$\begin{split}\Delta\sigma(B)=-\frac{e^{2}}{2\pi h}\Bigg{[}F\left(\frac{\tau_{B}^{-1}}{\tau_{\phi}^{-1}}\right)-F\left(\frac{\tau_{B}^{-1}}{\tau_{\phi}^{-1}+2\tau_{asy}^{-1}}\right)\\ -2F\left(\frac{\tau_{B}^{-1}}{\tau_{\phi}^{-1}+\tau_{so}^{-1}}\right)\Bigg{]}\end{split}$$

$${\tau }_{so}^{-1}=4{\tau }_e({\Delta }_{DP}/\hslash {)}^2$$

$$R_c(B)=\frac{\hslash k_F}{e{B}_{max}}=\frac{\hslash \sqrt{\pi n}}{e{B}_{max}}.$$

Full text

I Introduction

In recent years, the assembly of van-der-Waals heterostructures containing graphene has gained much attention Geim and Grigorieva (2013). Encapsulating graphene between hBN and employing one-dimensional edge contacts Wang et al. (2013) has proven to be a reliable method to fabricate high mobility devices. With this a number of effects, such as ballistic transport Taychatanapat et al. (2013), viscous electron flow Bandurin et al. (2016) and moiré patterns Dean et al. (2013) have been observed. However, employing other two-dimensional materials for encapsulation allows to further tailor the properties of graphene. One promising objective is to increase the spin-orbit-coupling (SOC) in graphene, as this may offer numerous possibilities, including the generation of a pure spin-current through the spin-Hall effect or the manipulation of spin-currents through an electric field. Bringing graphene into proximity of transition metal dichalcogenides (TMDC) has been predicted theoretically Kaloni et al. (2014); Gmitra et al. (2016) and observed experimentally Wang et al. (2015, 2016); Yang et al. (2016); Avsar et al. (2014) to increase SOC in graphene. Further, transport measurements Kretinin et al. (2014) and recent Raman measurements indicate the suitability of these substrates for high mobility graphene Banszerus et al. (2017). This is in contrast to previously explored methods for increasing SOC in graphene, such as hydrogenation Castro Neto and Guinea (2009); Balakrishnan et al. (2013), fluorination Avsar et al. (2015) or the attachment of heavy atoms Ma et al. (2012); Balakrishnan et al. (2014), as these methods have the disadvantage of increasing the scattering and therefore decreasing the mobility of graphene.

Here, we report on a comparison of magnetotransport in graphene/TMDC heterostructures in a broad mobility range, realized by different material combinations in the van-der-Waals stacked layer sequence. We integrate one-dimensional contacts into the TMDC/graphene processing scheme, achieving a high yield of functional devices and include top gates using a TMDC layer as a gate dielectric. In diffusive samples, we observe weak antilocalization and study proximity-induced spin-orbit interaction at different out-of-plane electric fields, while in high mobility samples, a ballistic size effect and the quantum Hall effect are observed.

II Sample Fabrication

Heterostructures were fabricated by using a dry pickup process Wang et al. (2013). Three different types of devices were fabricated. For device type 1 (see Fig. 1 (a)) monolayer graphene was picked up by exfoliated multilayer WSe2 and placed onto a standard p*++*-doped Si/SiO2 chip.

For device type 2 monolayer graphene was encapsulated between hBN and WS2, while for device type 3 (see Fig. 1 (b)) bilayer graphene was encapsulated between hBN and WSe2. After assembly all three devices were annealed for 1 hour at 320 *∘*C in vacuum and 1 hour at 320 *∘*C in forming gas. Annealing removes contaminations between the layers, as well as the remaining PPC on top of the WSe2 (WS2) flake. Then electron-beam lithography (EBL) and reactive ion etching (RIE) with CHF3/O2 were employed to define a Hall-bar structure. The graphene was then contacted by 5 nm Cr/ 80 nm Au side contacts. These edge contacts showed high reliability as 70 of 74 contacts were functional. As a last step 10 nm Al2O3 were deposited by atomic layer deposition (ALD), followed by a Au topgate. The Al2O3 layer is necessary to prevent any leakage between topgate and graphene at the sides of the stack.

III Experimental Results

III.1 Diffusive Regime

For measurements in the diffusive regime monolayer graphene/WSe2 is placed onto SiO2 in device type 1. We therefore observe a mobility of only $\mu=12\,000$ cm2/Vs at $T=1.65$ K. Figure 2 depicts the magnetoconductivity of this sample at different temperatures. In order to suppress universal conductance fluctuations an average over 15 curves at slightly different backgate voltages with a mean charge carrier concentration of $n=1.0\cdot 10^{12}$/cm2 was taken. The curves were obtained in a four-point lock-in measurement with an AC-current of $I_{AC}=10$ nA for the curves at $T=1.65$ K and $T=4.2$ K, $I_{AC}=50$ nA at T=10 K and $I_{AC}=100$ nA at T=100 K at a frequency of f=13 Hz.

The occurrence of a sharp peak in the magnetoconductivity can be explained by weak-antilocalization, stemming from spin-orbit coupling. For the case that the intervalley scattering rate exceeds the decoherence rate, the low magnetic field dependence of the conductivity correction, due to WAL can be described as McCann and Fal’ko (2012):

[TABLE]

where $F(x)=\ln(x)+\Psi(1/2+1/x)$, with $\Psi(x)$ being the digamma function, $\tau_{B}^{-1}=4DeB/\hbar$, $\tau_{\phi}$ the phase coherence time, $\tau_{so}$ the spin-orbit scattering time and $\tau_{asy}$ a scattering time that takes into account only spin-orbit coupling that is asymmetric in $z\rightarrow-z$ direction. Here $\tau_{so}$ combines symmetric and antisymmetric spin-orbit scattering: $\tau_{so}^{-1}=\tau_{sym}^{-1}+\tau_{asy}^{-1}$ McCann and Fal’ko (2012).

Fitting the curve in Fig. 2 at $T=1.65$ K (red curves in Fig. 2) gives $\tau_{\phi}=25.7$ ps, $\tau_{so}=0.57$ ps and $\tau_{asy}=1.71$ ps. These are comparable to the values that were reported for graphene placed on WSe2 Wang et al. (2016) and WS2 Wang et al. (2016, 2015); Yang et al. (2016). $\tau_{so}$, which is an upper bound for the spin-relaxation time is therefore much shorter than the values typically found in pristine graphene (100 ps-1 ns) Tombros et al. (2007); Han et al. (2010); Volmer et al. (2013). The occurrence of WAL with such small $\tau_{so}$ is therefore a clear indication of strong SOC in this device. With increasing temperature the feature in Fig. 2 decreases as the phase coherence time $\tau_{\phi}$ decreases and the peak disappears at $T=20$ K.

The dual gated device allows us to examine the WAL peak with an applied transverse electric field, while leaving the charge carrier density unchanged.

Figure 3 shows the magnetoconductivity at three different top- and backgate voltage combinations. Applying the electric field strongly decreases $\tau_{so}$ from $\tau_{so}=1.5$ ps to $\tau_{so}=0.91$ ps in one direction of the electric field and $\tau_{so}=1.25$ ps in the other direction. The SOC strength is expected to increase with an electric field, due to the Rashba effect Bychkov and Rashba (1984). However, $\tau_{so}$ depends on the total out-of-plane electric field acting on the carriers, which is composed of the externally applied field, as well as an internal field, due to the WSe2-graphene interface. The weak asymmetry in the external electric field therefore points to a small contribution of an internal field. This is in contrast to the findings of Yang et al. in graphene/WS2 samples, who reported a linear dependence of the spin orbit scattering rate $\tau_{asy}$ with the applied electric field, while they assume the symmetric part of the scattering rate to be zero Yang et al. (2016).

Spin relaxation is expected to be dominated by the Dyakonov-Perel mechanism. The SOC strength $\Delta_{DP}$ can be estimated by Huertas-Hernando et al. (2009):

[TABLE]

This results in a SOC strength of $\Delta_{DP}=0.7-1.0$ meV, which agrees well with theoretical predictions Gmitra et al. (2016). For the case of Elliot-Yafet dominated spin relaxation the SOC strength can be estimated by Huertas-Hernando et al. (2009): $\tau_{so}^{-1}=\tau_{e}^{-1}\Delta_{EY}^{2}/E_{F}^{2}$. This results in an unrealistically large SOC strength of $\Delta_{EY}=35-65$ meV. Further, we observe a decrease of $\tau_{so}$ with increasing charge carrier concentration, which indicates that spin relaxation is dominated by the Dyakonov-Perel mechanism.

III.2 Ballistic Regime

In order to increase the mobility of graphene we have encapsulated graphene between WSe2 (WS2) and hBN (see Fig. 1 (b)). Figure 4 shows Shubnikov-de Haas oscillations (black curve) and quantum-Hall effect (blue curve) of device 2, containing monolayer graphene between hBN and WS2.

This device showed mobilities of $\mu=50\,000$ cm2/Vs on the hole side and $\mu=120\,000$ cm2/Vs on the electron side. In Fig. 4, a lifting of the spin and valley degeneracies can be observed, which results in integer filling factors in addition to the expected values of $4n+2$ for monolayer graphene. This behavior is typical for high mobility graphene Young et al. (2012). The resistance peak at low magnetic field, followed by a negative magnetoresistance behavior will be discussed in the following sections.

In order to directly compare the substrates WSe2 and SiO2 in device 3, a bilayer graphene/hBN stack was placed in such a way that part of the stack lies on a WSe2-flake and part of it lies directly on the SiO2 substrate (see Fig. 1 (c)). Figure 5 shows topgate-sweeps of the four-point resistance of these two areas at $T=1.7$ K.

From this we extract a mobility of $\mu=3200$ cm2/Vs on the hole side and $\mu=5300$ cm2/Vs on the electron side for the graphene on SiO2. For the graphene on WSe2 we extract $\mu=57\,000$ cm2/Vs and $\mu=92\,000$ cm2/Vs for hole and electron sides. The overall high mobilities resulting from encapsulation confirm the suitability of WS2 and WSe2 as substrates for high mobility graphene.

Figure 6 (c) shows the magnetoresistance of the graphene on the SiO2 substrate.

Here we observe a peak in the resistance around $B=0$ T, which we ascribe to weak localization. Fitting this peak with the formula for weak localization in bilayer graphene Gorbachev et al. (2007) reveals a phase coherence length of $L_{\phi}\sim 490$ nm and an intervalley scattering length of $L_{i}\sim 420$ nm.

For the part of the bilayer graphene on WSe2 we observe a dip in the resistance around $B=0$ T in Fig. 6 (a). At first glance this feature might be interpreted as WAL. However, this dip is much too large ($\Delta\sigma=20$ e2/h) and too broad to be fitted with equation 1. Further, the temperature dependence is much weaker and the dip is still visible at $T=60$ K, in contrast to the WAL feature in Fig. 2. Figure 6 (b) shows the magnetoresistance of two bilayer graphene samples with different width. While the black curve shows the magnetoresistance of the sample from Fig.  5 and 6 (a), with a width of $W=4\,\mu$m, the red curve shows the magnetoresistance of a sample with width $W=1.5\,\mu$m. The mobility of this sample was $\mu=90\,000$ cm${}^{2}/$Vs on the hole side and $\mu=100\,000$ cm${}^{2}/$Vs on the electron side. This behavior, i.e. the resistance peak at finite $B$, we ascribe to a ballistic effect, stemming from diffusive boundary scattering Beenakker and van Houten (1991); Thornton et al. (1989); Masubuchi et al. (2012). A schematic description of this effect is shown in the inset of Fig. 6 (a). At low magnetic fields the scattering between boundaries and therefore, the overall resistance, is initially increased (solid lines in the inset of Fig. 6 (a)). When the cyclotron diameter becomes smaller than the sample width, the scattering between boundaries is suppressed and therefore the resistance decreases (dashed lines in the inset of Fig. 6 (a)). From the curves in Fig. 6  (b), the cyclotron radius $R_{c}$ at the magnetic field, where the resistance reaches the maximum can be calculated as:

[TABLE]

The calculated cyclotron radii are $R_{c}=2.17\,\mu$m for the sample with width $W=4\,\mu$m and $R_{c}=0.60\,\mu$m for the sample with width $W=1\,\mu$m. This shows that $R_{c}$ scales with the sample width $W$. For semiconductor 2DEGs, a relation $W=0.55R_{c}$ was found Thornton et al. (1989), whereas for hBN encapsulated graphene a different prefactor was observed Masubuchi et al. (2012). The resistance peak at low magnetic field in Fig. 4 is also attributed to this effect.

No WAL behavior could be observed for graphene encapsulated between hBN and WSe2 (WS2). We attribute this to a transition from the diffusive to the quasiballistic regime. Since, equation 1 was developed in the diffusive regime, it is only valid for the case of: $\tau_{\phi}>\tau_{asy}>\tau_{so}>\tau_{e}$. Due to the higher mobility for decives of type 2 and 3, we find $\tau_{e}$ to be in the range of $\tau_{e}\approx 1\,$ps. Therefore the relation $\tau_{so}>\tau_{e}$ may not be valid here. We expect WAL to be suppressed, due to reduced backscattering and the WAL peak to be narrower, resulting from the higher mobility in these samples (a similar behavior has been observed in GaAs heterostructures Grbić et al. (2008)). Therefore the absence of WAL in these samples is not indicative of a lower SOC strength.

IV Conclusion

In conclusion we investigated charge transport in several graphene/WSe2 (WS2) heterostructures. We successfully employed the established fabrication techniques for hBN/graphene/hBN stacks to heterostructures containing WSe2 (WS2) and graphene. Placing a graphene/WSe2 stack on SiO2 resulted in a mobility of $\mu=12\,000$ cm2/Vs. In this sample we observed a peak in the magnetoconductivity, which we attributed to the WAL effect, stemming from SOC. Applying an electric field increased the SOC strength in this sample. Encapsulating graphene between WSe2 (WS2) and hBN increased the mobility to up to $\mu=120\,000$ cm2/Vs. No WAL behavior could be observed in these samples. We attribute this to a transition from the diffusive to the quasiballistic regime. This is further confirmed by the occurrence of a quasiballistic size effect, due to diffusive boundary scattering. These results confirm the suitability of WSe2 (WS2) as a substrate for high quality graphene with strongly increased SOC.

Acknowledgements.

Financial support by the Deutsche Forschungsgemeinschaft (DFG) within the programs GRK 1570 and SFB 689 is gratefully acknowledged. The authors would like to thank J. Fabian and T. Korn for fruitful discussions.