Hall effect measurements using low ac magnetic fields and lock-in technique on field effect transistors with molybdenum disulfide channels

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Figure captions

Fig. 1. Schematics of setup for measurement of frequency response of the lock-in amplifier (a) and magnet

(b). Dependence of Vmeas on Vin and that of B0 on I0 are measured, where Vin, B0, and I0, are the oscillation

amplitudes of the input signal, magnetic field, and magnet current, respectively. Vmeas is the amplitude

recorded by the lock-in amplifier. The magnetic field is measured using a Hall sensor, of which the Hall

voltage is read out using the differential amplifier and oscilloscope. In the actual Hall effect measurements

on MoS2 devices, the Hall sensor is replaced with a MoS2 flake, and the very small output signal of the

differential amplifier is fed into the lock-in amplifier.

Fig. 2. Frequency characteristics of the ac Hall effect measurement system used in this work. The ratios

Vmeas/Vin (black squares) and B0/I0 (blue triangles) are shown as a function of frequency, where Vmeas Vin, B0,

and I0 are the voltage amplitude measured by the lock-in amplifier, the amplitudes of input-voltage

oscillations, magnetic field measured by a Hall sensor, and the applied ac magnet current, respectively.

Here, B0/I0 is shown as a normalized value so that the value is unity as the frequency tends to zero.

Fig. 3. The ratio Vmeas/I0 as a function of frequency, which is the overall frequency characteristics of the ac

Hall effect measurement system. When the amplitude of the magnet current is fixed, the measured Hall

voltage is proportional to this value.

Fig. 4. Optical micrograph of the MoS2 FET device. Scale bar is 50 m. The dc bias current flowed

between the drain and source terminals (D and S). The Hall voltage was measured using the Hall voltage

probes (V1 and V3). The four-terminal resistance was obtained using the voltage between the voltage probes

(V1 and V2). The channel length between the drain and source terminals and the channel width were L =

26m and W = 65 m, respectively.

Fig. 5. The four-terminal sheet conductance Gs of the MoS2 FET device as a function of Vg. The field effect

mobility is derived from the slope of the Gs–Vg curve in the linear regime as shown in the text.

Fig. 6. Time traces of drain-to-source voltage Vds and amplitude Vm* of the oscillation component of the

transverse voltage that are in phase with magnetic field oscillations, acquired by the lock-in amplifier. The

unit of time T was 160 s. Vds was switched between 2.5 V and –2.5 V, with a period of 4T. For each

value of Vds, Vm* was recorded for the time interval of T after waiting time of T, so that the device

reaches an equilibrium state.

Fig. 7. Transverse resistance Rxy as a function of B (amplitude of ac magnetic fields) at Vg = 50 V. From the

slope of the fitted line, the sheet electron density was estimated to be 3.7 × 1012 cm-2. This value is different

from that shown in Fig. 8 (2.7 × 1012 cm-2), because the data of Fig. 7 were measured six months after the

measurement for Fig. 8.

Fig. 8. The sheet electron density n and the Hall mobility H as a function of Vg acquired in the ac Hall

effect measurements. H is obtained from n and the four-terminal sheet resistance. The frequency of the ac

magnetic fields was 0.9 Hz in this measurement.

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.01

0.1

10

0.0

100

Vmeas / I0 (normalized)

1.0

B0 / I0 (normalized)

Vmeas / Vin

Fig. 1

0.5

0.4

0.3

0.2

0.1

0.0

10

Frequency (Hz)

Frequency (Hz)

Fig. 2

Fig. 3

Fig. 4

10

Vds (V)

Time / T

Vds = 2.5 V

Vds = -2.5 V

-100

-40 -20

20

40

60

Vg (V)

Time / T

Fig. 6

16

14

-2

n (10 cm )

10

12

12

0.01

0.02

0.03

B (T)

-40

-20

20

Vg (V)

Fig. 7

Fig. 8

11

dn = Ci

dVg e

40

60

Fig. 5

0.00

H (cm /Vs)

100

Rxy ()

200

Vm (V)

Gs (S)

-1

-2

-3

...