Even-odd layer-dependent magnetotransport of high-mobility Q-valley electrons in transition metal disulfides

Abstract

In few-layer transition metal dichalcogenides (TMDCs), the conduction bands along the ΓK directions shift downward energetically in the presence of interlayer interactions, forming six Q valleys related by threefold rotational symmetry and time reversal symmetry. In even layers, the extra inversion symmetry requires all states to be Kramers degenerate; whereas in odd layers, the intrinsic inversion asymmetry dictates the Q valleys to be spin-valley coupled. Here we report the transport characterization of prominent Shubnikov-de Hass (SdH) oscillations and the observation of the onset of quantum Hall plateaus for the Q-valley electrons in few-layer TMDCs. Universally in the SdH oscillations, we observe a valley Zeeman effect in all odd-layer TMDC devices and a spin Zeeman effect in all even-layer TMDC devices, which provide a crucial information for understanding the unique properties of multi-valley band structures of few-layer TMDCs.

ARTICLE

Received 17 Nov 2015 | Accepted 15 Aug 2016 | Published 21 Sep 2016

Zefei Wu1,*, Shuigang Xu1,*, Huanhuan Lu1,*, Armin Khamoshi2,*, Gui-Bin Liu3,*, Tianyi Han1, Yingying Wu1, Jiangxiazi Lin1, Gen Long1, Yuheng He1, Yuan Cai1, Yugui Yao3, Fan Zhang2 & Ning Wang1

In few-layer transition metal dichalcogenides (TMDCs), the conduction bands along the GK directions shift downward energetically in the presence of interlayer interactions, forming six

Q valleys related by threefold rotational symmetry and time reversal symmetry. In even layers, the extra inversion symmetry requires all states to be Kramers degenerate; whereas in odd layers, the intrinsic inversion asymmetry dictates the Q valleys to be spin-valley coupled. Here we report the transport characterization of prominent Shubnikov-de Hass (SdH) oscillations and the observation of the onset of quantum Hall plateaus for the Q-valley electrons in few-layer TMDCs. Universally in the SdH oscillations, we observe a valley Zeeman effect in all odd-layer TMDC devices and a spin Zeeman effect in all even-layer TMDC devices, which provide a crucial information for understanding the unique properties of multi-valley band structures of few-layer TMDCs.

DOI: 10.1038/ncomms12955 OPEN

Evenodd layer-dependent magnetotransport of high-mobility Q-valley electrons in transition metal disuldes

1 Department of Physics and the Center for 1D/2D Quantum Materials, the Hong Kong University of Science and Technology, Hong Kong, China.

2 Department of Physics, University of Texas at Dallas, Richardson, Texas 75080, USA. 3 Beijing Key Laboratory of Nanophotonics and Ultrane Optoelectronic Systems, School of Physics, Beijing Institute of Technology, Beijing 100081, China. * These authors contributed equally to this work. Correspondence and requests for materials should be addressed to F.Z. (email: mailto:[email protected]

Web End [email protected] ) or to N.W. (email: mailto:[email protected]

Web End [email protected] ).

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Strong spinorbit couplings in monolayer transition metal dichalcogenides (TMDC)14 entangle the spin and valley degrees of freedom of the valence band states at K valleys5,6.

This gives rise to exciting electronic and excitonic properties, such as optical circular dichroism710, opto-valley Hall effect11,12 and optical valley Zeeman effect1318. Inversion symmetry breaking in monolayer TMDCs plays an important role in opening energy gaps at the energy range of visible lights at K valleys710. Together with the substantial spinorbital couplings, the broken inversion symmetry in monolayer TMDCs further splits the spin-up and spin-down bands at the valence band edge of K valleys, which is absent in any bilayer TMDC10,19. The former feature leads to the optical circular dichroism and valley Hall effect, the latter feature yields the observation of optical valley Zeeman effect. So far, the observed moderate electron mobility in most atomically thin TMDCs has prohibited the exploration of the quantum transport properties. Impurity scattering and non-ohmic contacts are two major obstacles to fabricating high-mobility TMDC-based eld-effect (FE) devices. Several attempts have been made recently in improving the electrical contacts by, for example, using phase engineering20, graphene on MoS2 (ref. 21), Pt electrodes22 or selective etching process23.

Here we present a magnetotransport study of both even- and odd-layer TMDCs with hexagonal boron nitride (BN) encapsulated structures and metal contacts made by the selective etching process. As we will show, the peculiar electronic properties can be extended from monolayer TMDCs to few-layer TMDCs, from K valleys to Q valleys24, and from optical probe to transport detection. This possibility can be explicitly appreciated by comparing the quantum transport measurements in odd-layer and even-layer TMDCs, in which the essential inversion symmetry is broken and respected, respectively. The unique transport properties observed in BN-encapsulated TMDC devices provide useful information for a better understanding of the thickness dependent magnetotransport of high-mobility Q-valley electrons and the spin-valley entangled unconventional quantum Hall (QH) effects in atomically thin TMDCs.

ResultsHigh-mobility TMDC devices. As discussed in our previous work23, to eliminate any impurity effects induced during device fabrication, we employ a polymer-free dry transfer technique25,26 in an inert environment of argon or nitrogen, as schematically demonstrated in Fig. 1ad. Using the encapsulation of few-layer TMDCs in BN sheets and the selective etching process, we can achieve high-quality low-temperature ohmic contacts(0.150.5 kO mm) and ultrahigh FE mobilities (10,500

19,600 cm2 V 1 s 1) in TMDC channels. For example, Fig. 1e,f show the ISDVSD curve of a nine-layer (9L) MoS2 device, where VSD is the voltage source and ISD is the measured current. The linear characteristic of this device is observed at both 300 and 2 K. The contact resistivity at T 2 K is B0.25 kO mm (see details in

Supplementary Fig. 1ac for another 6L WS2 device).

The high quality of our BN-TMDC-BN heterostructures is reected by their four-terminal FE mobilities mF 1Cg dsdVg,

measured at different temperatures, where s is the conductivity and Cg is the gate capacitance (1.11.2 F cm 2, as calculated based on the thickness of SiO2 (300 nm) and the bottom BN layer (1550 nm)). The FE characteristics of 9L MoS2 (Fig. 1g) and 6L WS2 (Supplementary Fig. 1d) are shown, whose FE mobilities at room temperature are B50 cm2 V 1s 1 and B300 cm2 V 1 s 1, respectively. At T 2 K, our TMDC devices show

excellent performance with remarkably improved FE mobilities (mFB10,500 cm2 V 1 s 1 for 9L MoS2 (Fig. 1h) and mFB16,000 cm2 V 1 s 1 for 6L WS2 (Fig. 1i)). The phonon

scattering is suppressed, and the corresponding Hall mobilities (mH

snHe) reach 6,700 and 8,000 cm2 V 1 s 1, where nH is the

carrier density obtained from the Hall measurement and s is the conductivity. The FE mobility in our 6L WS2 device (B16,000 cm2 V 1 s 1) is more than 30 times higher than the previously reported record for WS2 (B486 cm2 V 1 s 1)

(ref. 27). It must be noted that the screening effect may affect the back gating effect and produce inhomogeneous charge density across the layers28. This can give an inaccurate Cg and hence explain the difference between our Hall and FE mobilities.

Quantum oscillations in odd-layer TMDCs. In the representative 9L MoS2 device, the Shubnikov-de Hass (SdH) oscillations in the longitudinal resistance R appear at perpendicular magnetic elds B44 T (Fig. 2a). This property is the hallmark of the high quality and homogeneity of our BN-MoS2-BN devices. Pronounced SdH oscillations are observed at relatively high gate voltages, where mH is sufciently high. Quantitatively, at the low magnetic eld range, the SdH oscillations in the longitudinal resistance R of a single sub-band in two-dimensional electron gas can be described by the LifshitzKosevich formula29:

DR

R 4e p=octq

lsinhl cos

2pEF

oc

where l 2p2kBT/oc. The cyclotron frequency is given by

oc

eB/m*. tq is the quantum scattering time and the Fermi energy is described by EF 2p2n/gvgsm*. kB is the Boltzmanns

constant, T the temperature, the plank constant, e the electron charge, B the magnetic eld, m* the cyclotron mass of carriers, n the charge carrier density, gv the valley degeneracy and gs the spin degeneracy. In a two-dimensional electron gas, the SdH oscillations can display useful information about the quantization of Landau levels (LLs) when plotted versus 1/B. Figure 2b shows the plots of DR (that is, the background has been subtracted from

R) as a function of 1/B at different gate voltages Vg. The equal spacing between SdH valley positions implies the single-band nature at the studied Vg. Thus, extracting further information using the LifshitzKosevich formula is appropriate. In principle, the periodicity of SdH oscillations is 1/BF g/F0n, where

g gs gv is the LL degeneracy and F0 h/e is the ux

quantum. At relatively high elds, the best t of n versus BF/F0

(Fig. 2c) yields g 3.00.1; the linear t of the LL lling factors

versus the SdH valley positions (Fig. 2d) yields a zero Berry phase (the tting results are in the range of 0.030.05 p). As Fig. 2b shows, the lling factors n 36, 42 and 48 are also clearly

observed at relatively low elds with a gate voltage Vg 40 V. The

degeneracy of 6 arises from the degeneracy between the 3 Q and 3 Q valleys because the spin degeneracy within each Q or Q valley is already lifted by the broken inversion symmetry in an odd-layer TMDC. At relatively high magnetic elds, an LL sextet can be lifted into two LL triplets caused by the valley Zeeman effect, which is similar to the K/K valley Zeeman effects observed most recently using optical circular dichroism1316. The Lande factor gL can be roughly estimated using the formula gLmBBc kBTC,

where mB is the Bohr magneton, and BC is the lowest eld and TC is the highest temperature for our observation of the valley Zeeman effect30,31. With a lling factor of 33 at Vg 40 V, the

valley Zeeman splitting disappears at B10 K (see Supplementary Fig. 2a) amounting to gLB3.4, which is comparable to those reported for WSe2 and MoSe2 monolayers. At relatively high gate voltages (60 and 70 V), where the Hall mobility is sufciently high, the Zeeman effect takes place at a small strength of B eld. The LL triplets start to appear where the SdH oscillations emerge.

The cyclotron mass of charge carriers in the 9L MoS2 device is obtained by investigating the temperature dependence of DR

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a b c e

a

b

c d

Etching window

Electrodes

BN XS2

SiO2/Si

1.0

0.0

I SD(mA)

I SD(A)

Bottom BN

XS

40 V 20 V

20 V 0 V20 V40 V

T = 300 K

0 V20 V40 V

Top BN

20

VSD (mV)

d

f

1.020 0

0

50 T = 2 K

0 20

FE mobility Hall mobility

50 20

20,000

VSD (mV)

g

h

i

FE mobility Hall mobility

(mS)

4 1.8 K

10 K20 K30 K40 K50 K60 K80 K 100 K 300 K

10,000

1,000

100

2

(cm2 V 1 S 1 )

(cm2 V 1 S 1 )

2,000

~ 2.25 ~ 2.01

~ 2.17

~ 2.06

0

Vg = 60 V

Vg = 60 V

40 0 40

1 10 100

200 1 10 100

Vg (V)

T (K)

T (K)

Figure 1 | BN-TMDC-BN heterostructure device. (a) The sandwiched TMDC heterostructure. (b) The BN-TMDC-BN heterostructure for selective etching. The etching window is marked by arrows. (c,d) Optical (c) and schematic image (d) of a BN-TMDC-BN FE transistor device with a Hall bar conguration. Scale bar, 10 mm. (e,f) Two-terminal ISD VSD characteristics of a representative MoS2 device at 300 K (e) and 2 K (f). Linear IV behaviour is observed in

both cases. (g) Four-terminal conductance in the WS2 device plotted as a function of the gate voltage at various temperatures. (h,i) FE mobilities and Hall mobilities of MoS2 (h) and WS2 (i) at Vg 60 V at various temperatures.

oscillations (see Supplementary Fig. 2a). For a given EF (Vg 60

V or n 4.32 1012 cm 2) and a given B, the DR peak

amplitudes (see Supplementary Fig. 2b) follow the Ando formula32 DR T

/

4lT

sinhlT e

octq . We obtain m*E0.27.01me, which is smaller than the effective mass (B0.5me) obtained by our density functional theory (DFT) calculations. Depending on the thickness of odd-layer samples, the measured m* can vary from 0.3 to 0.4me partially due to the uncertainty of the measured temperature. The corresponding quantum scattering time is tq

206 fs, which is much shorter than the transport scattering time tt (m*)/(R0e2n) 1,100 fs, thereby demonstrating that

long-range scattering is dominant in our MoS2 sample (see Supplementary Fig. 3).

As expected, the valley Zeeman effect can also be observed in a 3L WS2 sample (Fig. 2e,f). After subtracting the background of the data shown in Fig. 2e, we plot DR as a function of B (Fig. 2f). At high-B elds, the LLs developed from sextets to triplets, with increasing amplitudes of SdH oscillations. Remarkably, the ultrahigh mobility achieved in these TMDC samples even enables us to observe the onset of the QH effect. Figure 2g shows the longitudinal resistance R and Hall resistance Rxy as a function of

B at 2 K in a 3L MoS2 device. Beyond 6 T, Rxy exhibits at least three well-quantized plateaus (n 36, 39 and 42), and they match

very well with the corresponding R valleys. As in other TMDC devices with an odd number of layers, the SdH oscillations in the 3L MoS2 device clearly exhibit an LL degeneracy of 3, implying the valley Zeeman splitting.

Quantum oscillations in even-layer TMDCs. The SdH oscillations in the representative 6L WS2 device emerge when B eld is greater than 2.5 T (Fig. 3a). Although the gate voltages applied (Vg 5070 V) to the 6L WS2 device are similar to those applied

(Vg 40, 60 and 70 V) to the 9L MoS2 device, the period of SdH

oscillations appears twice larger in the 6L WS2 device (Fig. 3b). Given that the experimentally accessible carrier density is low, the Fermi energy crosses only the lowest spin-degenerate sub-band at the Q/Q valleys in our calculated band structure of 6L WS2. The

single sub-band nature is also evidenced by the unique period in

the SdH oscillations (Fig. 3b). The linear t of n versus BF/F0

(Fig. 3c) indicates a LL degeneracy of B11.80.1; the linear t of the LL lling factors versus the SdH valley positions (Fig. 3d) yields a zero Berry phase. At a large eld of 6.5 T, the secondary SdH valleys and doubling of the oscillation frequency are clearly visible because of the spin Zeeman splitting of LL duodectets into LL sextets (see Supplementary Fig. 4). The disappearance of secondary SdH valleys at around 10 K further indicates that the Lande factor is gLB2.2 (Supplementary Table 1). Under similar experimental conditions, the presence of SdH valleys, as a result of the complete lling of a LL duodectet or sextet, has been repeatedly observed; for example, in a 6L MoS2 device (g 12/6 at

low-/high-B elds in Fig. 3e,f), in a 10L WS2 device (g 12 in

Fig. 3g,h) and in a 10L MoS2 (g 6 in Supplementary Fig. 5).

Clearly, in contrast to odd-layer MoS2 devices (for example, 3L and 9L MoS2), even-layer MoS2 devices exhibit doubled LL degeneracies (see for example the data of 6L MoS2 in Fig. 3). We

p

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a b c

9L MoS2 +40 V +60 V +70 V

240

220

0.35 5

4

3

1.0

0.4

0.15 0.20 0.25 0.30

+40 V

+60 V

+70 V

260

33

n (1012 cm2 )

0.0

0.4

0

1

0

2

0.15 0.20 0.25 0.30

Filling factor

30 36 42 48

1 27 33 39 45

48

42

36

30 24

2 30 36 42 48

45

39

2124 27

1.2 1.4 1.6

BF/0 (1012 cm2)

R()

R()

d

200

180

1/B(T1 )

0.3

0.2

0.1

0.0

0 9 18 27 3645

33

27

2 3 4 5

B (T)

6 7 8

1/B (T1)

e

g

3L WS2

3L MoS2

300

250

0.015

0.020

0.025

0.030

R()

9,000

8,500

8,000 4 5 6 7 8

36

39

42

n = 7.5 1012 cm2

8 10 12

R()

Rxy (h/e

2)

2

4 6

B (T)

1/42

f

48

45

2

1

0

1/39

39

45

51

57

R()

66

1/36

36 42

48 54 60

1

2

n = 5.57 1012 cm2

33

9

0.1 0.2 0.3

B (T)

1/B (T1)

Figure 2 | Quantum oscillations in odd-layer TMDCs. (ad) Quantum oscillations in 9L MoS2. (a) Resistance R as a function of B eld at 40 V (orange

line), 60 V (blue line) and 70 V (black line) gate voltages. The inset shows the sample image. (b) After subtracting the baselines of R B B curves in a,

DR curves plotted as a function of 1/B yields an oscillation period 1/BF, which decreases with increasing gate voltages. The lling factors are labelled for the oscillations valleys. The degeneracy of 6 arises from the degeneracy between the 3 Q and 3 Q valleys; the spin degeneracy within each Q or Q valley is already lifted by the broken inversion symmetry. At relatively high magnetic elds, an LL sextet can be lifted into two LL triplets caused by the valley Zeeman effect. (c) The total carrier density n obtained from the Hall measurements as a function of BF/F0 (black dots) for different gate voltages. The best t (red line) indicates a LL degeneracy of B3.00.1. (d) LL lling factors as a function of 1/B at different gate voltages. The linear t yields a zero berry phase. (e,f) Quantum oscillations in 3L WS2. (e) R plotted as a function of B at the carrier density of 7.5 1012 cm 2 (f) DR curves plotted as a function of

1/B. The LL degeneracy evolves from 6 at low-B elds to 3 at high-B elds. (g) The onset of QH states in 3L MoS2. Magnetoresistance resistance R (blue line) and Hall resistance Rxy (orange line) as a function of B eld at 2 K. The QH states are shown by at least three almost quantized plateaus in Rxy at n 36, 39 and 42.

note that a tilted magnetic eld will be helpful to further exploration of the TMDC QH effect and better determination of the physical parameters.

The cyclotron mass m* in 6L WS2 is also investigated (See Supplementary Fig. 4). On the basis of the DR data plotted as a function of B and kBT at Vg 70 V (n 3.75 1012 cm 2), we

obtain m*E0.200.04me. This indirect experimental value of m* is again smaller than the effective mass B0.5me obtained in our DFT calculations. We noted that the limited amplitude of SdH oscillations and sample temperature uncertainty may cause certain deviation of m*. On the basis of the Ando formula, we further obtain the quantum scattering time tq 586 fs in the 6L

WS2 device, which is smaller than the corresponding transport scattering time tt 1,300 fs (see Supplementary Fig. 6).

Spin-valley coupled Q valleys in few-layer TMDCs. Figure 4a,b show the calculated band structures of 3L MoS2 and 6L WS2 (see

Supplementary Fig. 7 for the band structures of 3L WS2 and 6L MoS2), in which the minima of the conduction bands are not located at the K/K points, but rather at the Q/Q points, that is, between K(K) and G points, with quadratic sub-bands. As illustrated in Fig. 4c, 3 Q and 3 Q valleys exist in the rst Brillouin zone of the few-layer TMDCs. The C3 rotational symmetry

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a b

48 60

20 0.15

5.5 0.40.30.20.1

0.0 0 12Filling factor

24 36 48 60

+50

60

40

20

0 18 24

36 48

+25

500 6L WS2

36

450 +50 V

+60 V +70 V

400

350

R()

R()

36

30

48

60

24

0.20 0.25 0.30 0.35 0.40 0.45 0.50

300

250

200

1,500

1,000

500

1/B (T1)

1/B(T1 )

c d

n (1012 cm2 )

5.0

4.5

4.0

3.5

150 2 3 4 5 6 7 8

0.30 0.35 0.40 0.45

B (T)

BF/0 (1012 cm2)

e

g

6L MoS2 10L WS2

18

R()

800

750

700 2 4 6 8

24

R()

48

36

36

30

n = 4.32 1012 cm2 n = 5.01 1012 cm2

48

24

2

4 6 8

B (T) B (T)

f h

100

0

20

10

0

0.1 0.2 0.3 0.4 0.5

R()

R()

30 36 48 48

36

24

100

0.1 0.2 0.3 0.4 0.5

18

24

10

20

1/B (T1) 1/B (T1)

Figure 3 | Quantum oscillations in even-layer TMDC. (ad) Quantum oscillations in 6L WS2. (a) Resistance R as a function of B eld at 50 V (orange

line), 60 V (blue line) and 70 V (black line) gate voltages. The inset shows the sample image. (b) DR plotted as a function of 1/B eld yields an

oscillation period 1/BF. The lling factors are labelled for the oscillation valleys. A twelve-fold LL degeneracy at low elds and six-fold LL degeneracy at high elds is observed, caused by the spin Zeeman splitting within each valley. (c) The total carrier density n obtained from the Hall measurements as a function of BF/F0 (black dots) for different gate voltages. The best t (red line) indicates a LL degeneracy of B11.80.1. (d) LL lling factors as a function of 1/B for different gate voltages. The linear t yields a zero Berry phase (the tting results are in the range of 0.1 to 0.3 p) (e,f) Quantum oscillations in 6L MoS2.

(e) R plotted as a function of B at the carrier density of 4.32 1012 cm 2 (f) DR curves plotted as a function of 1/B. The LL degeneracies derive from 12 at

low-B elds to 6 at high-B elds. (g,h) Quantum oscillations of 10L WS2 show a LL degeneracy of 12. The negative magnetoresistance implies the existence of disorders, which might be the reason for the absence of the sixfold LLs at high-B elds.

dictates the threefold Q-valley degeneracy. For even-layer TMDCs, the Q and Q valleys are further related by both time reversal and spatial inversion symmetries, which require Kramers degeneracy. Consider the low carrier density in our 6L WS2 device, the Fermi energy is B2.9 meV above the valley edge and crosses only one spin-degenerate sub-band at each valley (see the inset of Fig. 4b). Thus, in the SdH oscillations, we observe a twelve-fold LL degeneracy at low elds and sixfold LL degeneracy at high elds caused by the spin Zeeman splitting within each valley, that is, between |Qm4 and |Qk4 states (see Fig. 4d). The valley Zeeman effect is absent because of the inversion symmetry.

In contrast to the even-layer case, the inversion symmetry in the odd-layer devices is intrinsically broken; thus, all the sub-bands at each Q valley are spin non-degenerate. In our 3L MoS2 device (Fig. 4a), for instance, the Fermi energy is B6.0 meV above the valley edge and crosses only the lowest sub-band, for which the spin-up and spin-down sub-bands are lifted by 4.3 meV. Thus, the SdH oscillations exhibit sixfold LL degeneracy, which is reduced to threefold when the Zeeman effect is large, as shown in Fig. 4d. Although both the spin-up and spin-down sub-bands contribute to the quantum transport, the beating pattern cannot be observed since they originate from the

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a

b

2

2

1.5

1

0.5 3L-MoS2 6L-WS2

EF EF

0

1.5

1

0.96

0.95

0.898

0.892

0.5

Energy (eV)

Energy (eV)

0

0.5

1

1.5

2 2

Odd-layer

B = 0 B > 0

0.5

1

1.5

K M

K

M

c

d

Q3

Q1

Q Q Q

Q Q Q Q

K

Q2

Q

Even-layer

B = 0 B > 0

Q1

Q3

Q2

K

Figure 4 | Layer-dependent spin-valley coupled Q valleys in TMDCs. (a) Calculated band structure of 3L MoS2. The bottom of conduction band is located at the Q (Q) valleys. At the edge of each Q (Q) valley, the Fermi level only crosses the lowest non-degenerate sub-band, whose spin-up and spin-down sub-bands are lifted by 4.3 meV. (b) Calculated band structure of 6L WS2. The energy bands are spin-degenerate at the edge of each Q (Q) valley. These spin-valley coupled band edges are further illustrated in (c), where the red and blue colours denote the spin-down and spin-up bands, respectively. Q1, Q2 and Q3 have the same spin, and Q1, Q2 and Q3 are their time reversals. (d) Schematic diagrams for the Bloch bands, showing the valley Zeeman effect in odd-layer devices and the spin Zeeman effect in even-layer devices. For odd-layer samples, the sub-band at Fermi level is non-degenerate at B 0; at

relatively high magnetic eld, the degeneracy between Q and Q valleys is further lifted by the valley Zeeman effect. It follows that an LL sextet can be lifted into two LL triplets caused by the valley Zeeman effect. For even-layer samples, the sub-band at Fermi level is spin-degenerate at B 0; at relatively high

magnetic eld, the degeneracy between up and down spins is lifted by the spin Zeeman effect. It follows that an LL duodectets can be lifted into LL sextets caused by the spin Zeeman effect.

same sub-band and have the same effective mass. The Zeeman effect in this case is obviously a valley Zeeman splitting between |Qm4 and |Qk4 states, as a linear combination of the spin, orbital and lattice Zeeman effects. Despite the complex orbital hybridizations and the strong spinorbital couplings, because of time reversal symmetry orbital and spin characters are opposite for the Q and Q valleys, which can be split in the presence of B eld. The lattice Zeeman effect arises from the opposite Berry curvatures of Bloch electrons at two valleys, which is also dictated by time reversal symmetry. In the inversion symmetric even-layer cases, the Berry curvature vanishes and so does the lattice Zeeman effect. Nevertheless, the Q/Q valley Zeeman effect, observed here for the rst time in transport, is analogous to the K/K valley Zeeman effect observed using optical circular dichroism1316.

DiscussionOur realization of high-mobility TMDC devices and observation of their odd-even layer-dependence in the SdH oscillations, as well as the appearance of the onset of QH plateaus, are just the beginning for fully understanding the QH effects in atomically thin TMDCs. The existing theories focus exclusively on the K/K valleys of monolayers. However, the quantum transports at low

density for multilayers are dominated either by the electrons at Q/Q valleys or the holes at the G valley. As there is only one quadratic sub-band across the Fermi level per Q valley in our current study, we are able to use the symmetry arguments and the well-known physics of the quadratic band to understand the observed evenodd layer-dependent behaviours in Zeeman effects. Further theoretical efforts are necessary in order to fully understand the quantum transports in atomically thin TMDCs. On the experimental side, the mobility of WS2 samples often appears higher than those of MoS2 samples; however, the observed SdH oscillations in MoS2 samples appear more pronounced. The observed SdH oscillations in odd-layer WS2 also appear weaker than that of even layers. Higher mobility and clearer knowledge of the stacking orders will be crucial for future systematic observations of QH effects in atomically thin TMDCs.

In summary, we demonstrate high-mobility TMDC FE transistors achieved by encapsulating atomically thin TMDCs between BN sheets. At moderate magnetic elds of 2.54 T and relatively low carrier density B1012 cm 2, the quantum oscillations are dominated by the Q valleys, exhibiting a universal evenodd layer dependence. Above 4 T, we observe spin Zeeman effects in even-layer devices and valley Zeeman effects in odd-layer devices. We also observe the onset of QH plateaus in

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the 3L device. The high-quality atomically thin BN-TMDC-BN-based FE transistors fabricated in this work pave the way for understanding the multi-valley band structures of few-layer TMDCs and for exploring their spin-valley entangled unconventional QH effects33.

Methods

Materials. MoS2 and WS2 crystals are grown by the chemical vapour transport method reported previously34. A mixture of Mo/W and S is rst annealed in a sealed quartz tube at B800 C. Then the polycrystalline powders together with iodine are annealed in a furnace (950 C for the cold zone and 1,050 C for the hot zone) for several weeks to grow large single crystals35,36. The as-grown 2H-type crystals are pre-examined by optical approaches before making FE transistors. The h-BN sources (Polartherm grade PT110) are bought from Momentive and annealed at a high temperature to improve the quality of BN crystals.

Sandwiched heterostructures. To eliminate impurities induced during device fabrication, we employ a polymer-free dry transfer technique in an inert environment of argon or nitrogen, as schematically demonstrated in Supplementary Fig. 8a,b. Atomically thin akes are mechanically exfoliated on 300 nm SiO2/Si substrates by the scotch-tape microcleavage method. A selected

FL XS2 (X Mo/W) is picked up from the SiO2/Si substrate by a thin h-BN ake

(515 nm thick) on PMMA (950 A7, 500 nm) via van der Waals interactions. The h-BN/XS2 ake is then transferred onto a fresh thick h-BN ake at 4050 C, which is exfoliated previously on a different SiO2/Si substrate, to form a BN-XS2-BN heterostructure. The BN-XS2-BN structure and the high-temperature annealing can guarantee the stability of our XS2 devices. In the annealing process (conducted in Ar atmosphere above 300 C), the small bubbles formed at the interfaces between h-BN and XS2 are largely removed and the charge trap density are largely reduced.

Thickness determination. The thicknesses of XS2 akes are characterized by atomic force microscopy (AFM, Veeco-Innva) as illustrated in Supplementary Fig. 9. The measured sample thicknesses dm and their numbers of layers are listed in Supplementary Table 2 (ref. 37,38). Technically, the roughness of SiO2 substrate surface is B1 nm, so there is a large variance when measuring the thickness of TMDCs lying on SiO2/Si substrates. In this study, we use a different method to determine the sample thickness (see Supplementary Fig. 9). We leave a small part of WS2 uncovered (without the top BN) when making the sandwiched structure.

The bottom BN layer provides a smooth background in the AFM signal. The variance of sample thickness is smaller than 0.1 nm and the accuracy of the calculated number of layers is thus ensured.

Selective etching process. To fabricate the metal electrodes, a hard mask is patterned on the BN-XS2-BN heterostructure by the standard electron-beam (e-beam) lithography technique using PMMA 950 A5 (see SupplementaryFig. 8c,d). Since the etching rate of XS2 by our reactive ion etching(CHF3:O2 4:40 s.c.c.m.) is lower than that of h-BN, the exposed top BN layer is

then etched and XS2 is partially exposed. The electrodes are then patterned by e-beam lithography followed by O2 plasma etching (to remove PMMA residues) and standard e-beam evaporation (Ti/Au). Figure 1ad show the schematic and the optical images of a typical BN-XS2-BN device with Hall-bar congurations. After the metal electrode deposition, the contact resistance is further reduced by a post annealing treatment at 300 C in ambient pressure for B12 h.

Transport measurement. The ISDVSD curves are measured by Keithley 6430. Other transport measurements are carried out using the standard lock-in technique (SR 830 with SR550 as the preamplier and DS 360 as the function generator) in a cryogenic system. The cryogenic system provides stable temperatures ranging from1.8 to 300 K and magnetic elds up to 9 T. For comparison, the MoS2 device with graphene electrodes are fabricated and the device shows a complicated feature (see Supplementary Fig. 10). However, our MoS2 FE transistors connected by standard metal electrodes produce clean data which can be explained by DFT calculations fairly well. Since we use the same quality of MoS2 channel material to fabricate these two kinds of devices, we conclude that MoS2-graphene-mixed devices always display complicated oscillations of magnetotransport which cannot be understood yet.

Data availability. The data that support the ndings of this study are available from the authors upon request.

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Acknowledgements

We thank Wing Ki Wong for her assistance in the sample preparation process. Financial support from the Research Grants Council of Hong Kong (Project Nos 16302215, HKU9/CRF/13G, 604112 and N_HKUST613/12) and technical support from the Raith HKUST Nanotechnology Laboratory at MCPF are hereby acknowledged. F.Z. and A.K. are supported by UT Dallas Research Enhancement Funds and UT Dallas Undergraduate Research Scholar Awards. F.Z. is grateful to the Kavli Institute for Theoretical Physics for their hospitality during the nalization of this work, which is supported in part by the National Science Foundation under Grant No. PHY11-25915. G.-B.L is supported by the NSFC of China with Grant No. 11304014 and the 973 Program of China with Grant No. 2013CB934500. Y.Y. is supported by the MOST Project of China (Nos 2014CB920903 and 2016YFA0300603), the National Natural Science Foundation of China (Grant Nos 11574029 and 11225418).

Author contributions

N.W. and Z.W. conceived the projects. Z.W., H.L. and S.X. conducted the experiments, including the crystal growth, sample fabrication and the data collection. Z.W., S.X. and

F.Z. analysed the data. N.W. directed the project. N.W. and F.Z. were the principal investigators. F.Z. and A.K. analysed and interpreted the SdH oscillations. F.Z. andG.-B.L. computed and interpreted the band structures. Z.W., F.Z. and N.W. wrote the manuscript. The other authors provided technical assistance in the sample preparation, data collection/analyses and experimental setup.

Additional information

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How to cite this article: Wu, Z. et al. Even-odd layer-dependent magnetotransport of high mobility Q-valley electrons in transition metal disuldes. Nat. Commun. 7, 12955 doi: 10.1038/ncomms12955 (2016).

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