Black phosphorus field-effect transistors

Transcription

1 SUPPLEMENTARY INFORMATION DOI: /NNANO Black phosphorus field-effect transistors Likai Li, Yijun Yu, Guo Jun Ye, Qingqin Ge, Xuedong Ou, Hua Wu, Donglai Feng, Xian Hui Chen and Yuanbo Zhang Supplementary Information 1. X-ray diffraction (XRD) data of our bulk black phosphorus crystal (0 4 0) Black Phosphorus b= nm Intensity (arb.unit) (0 2 0) (0 6 0) (degree) Supplementary Figure 1. XRD of bulk black phosphorus crystal. 2. Chemical stability of black phosphorus crystal We find few-layer phosphorene samples chemically unstable in air. Bubble-like features appear on the surface after sample is exposed in air for a few days NATURE NANOTECHNOLOGY 1

2 (Supplementary Fig. 2a). However, samples are very stable if kept in vacuum or in an inert atmosphere. No detectable changes are observed in our sample (Supplementary Fig. 2b) after our week-long measurement in vacuum (~ mbar) or helium vapor, even after annealed at temperatures up to 370 K. Samples do not show obvious degradation if stored in argon atmosphere with O 2 and H 2 O content less than 1 ppm for several weeks. In addition, the samples are able to survive the electron-beam lithography process with no sign of degradation. Although the exact mechanism of sample degradation is not clear, chemical reactions between black phosphorus with O 2 or water vapor is likely the culprit. O 2 and water vapor barriers similar to the ones studied in ref. S1 may be able to protect black phosphorus samples in air. Supplementary Figure 2. a, AFM images of a black phosphorus think flake obtained immediately after cleaving (left) and after exposed to air for 3 days (right). b. Optical images of a device fabricated by EBL process before (left) and after (right) a one-week measurement in helium atmosphere. 2

3 3. Thickness determination of few-layer phosphorene Thickness of few-layer phosphorene can be reliably determined by Atomic Force Microscopy (AFM). As an example, Supplementary Fig. 3a shows an AFM image a 2 nm sample with clearly defined the step-edge. Optical contrast is another good way to determine the sample thickness. The sample is cleaved onto Si wafer covered with 90 nm thermally grown SiO 2. We measure the optical contrast defined by (I s - I 0 )/I 0, where I s and I 0 are the intensity of the reflected light on sample and substrate respectively. At fixed wavelength, the optical contrast exhibits an oscillatory behaviour as the polarization of the incident light is varied (an example is shown in Supplementary Fig. 3b inset), due to the two-fold anisotropy of black phosphorus crystal. We then measure the contrast as a function of the light wavelength at fixed polarization where the contrast is at its minimum. A strong correlation between the optical contrast and the sample thickness determined by AFM is observed for thicknesses down to 2 nm (Supplementary Fig. 3b). At the wavelength ~ 440 nm, such correlation is found to be highly linear (Supplementary Fig. 3c). The sensitive linear dependence of optical contrast on the sample thickness can be employed for easy thickness determination down to few atomic layers. 3

4 Supplementary Figure 3. a, AFM images of a black phosphorus think flake sample. The crosssectional profile along the white dash line, shown as red curve, indicates a 2 nm step-edge. b. 4

5 Sample contrast spectra obtained with incident light linearly polarized along one of the crystal orientations. Inset: contrast measured as a function of the polarization angle of the incident light. Data is taken from a 2 nm sample at 500 nm wavelength. c. Optical contrast at wavelength equal to 440 nm as a function of the sample thickness determined from AFM measurement. It can be well fitted with a linear dependence (dashed line). 4. Energy bands near Z and points measured by ARPES Apart from the energy bands shown in the main text, we have measured the energy bands along Z-T and of the first Bouillon zone on a freshly cleaved bulk black phosphorus sample (Supplementary Fig. 4a). While we see a clear valence band edge at ~ 0.3 ev at the Z point, the band near point is located ~ 1.4 ev away from the Fermi level and does not correspond to a band maximum. We attribute this to the interlayer coupling, which depresses the band energy at point and elevate the energy at Z point. 5

6 Supplementary Figure 4. Photon energy dependent ARPES measurement on bulk black phosphorus. a, Energy dispersion of black phosphorus along Z-T (left) and -T direction (right). b, Photon energy dependent ARPES spectrum. The highlighted 20eV and 27eV data are the crosssection profiles along the dash lines in a. 6

7 5. Band structure calculation We calculated the band structure of bulk black phosphorus using the projector augmented wave method, as implemented in the Vienna ab initio Simulation Package (VASP) code. For the exchange-correlation energy, we used the screened hybrid density functional of the Heyd-Scuseria-Ernzerhof type (HSE06). This functional mixes 1/4 short-range part of the Hartree-Fock exchange with 3/4 short-range part of the Perdew- Burke-Ernzerhof (PBE) exchange [S10], and it retains the PBE correlation and the longrange part of PBE exchange. The range-separation parameter was set at 0.2 Å -1, that defines a characteristic distance (2/ ) at which the short-range exchange interactions become negligible. It turns out that the HSE hybrid functional, without incorporating material-dependent empirical parameters, is capable of calculating accurately the band gap of many semiconductors [S11]. We used a plane-wave basis set of 400 ev, and the K-point meshes with 320 irreducible K-points for the first Brillouin zone sampling of primitive cell, based on the Monkhorst-Pack scheme [S12]. The calculated total energies converged within 1 mev per atom, with respect to a higher plane-wave cutoff energy and denser K meshes. The Gaussian smearing method was chosen with a smearing width of 0.05 ev for a broadening of the calculated energy levels and for a fixing of the Fermi level. 7

8 6. Estimation of few-layer phosphorene band gap from bulk bands Once the band structure of bulk black phosphorus is obtained, one could roughly estimate the band gap of few-layer phosphorene by quantizing the wavevector of the bulk bands in the c direction under appropriate boundary conditions. In other word, the 2D band structure of few-layer phosphorene can be approximated by slicing the 3D bands of the bulk bands in the ab plane. The resulting energy band gap as a function of layer number is shown in Supplementary Fig. 5. The gap decreases dramatically for the first four layers added, and quickly saturates towards the bulk value starting at ~ 4 nm (corresponding to 8 atomic layers). Supplementary Figure 5. Estimated energy band gap of few-layer black phosphorus as a function of sample thickness. The inset shows the calculated energy band dispersion of bulk black phosphorus along, from which the estimation is obtained. 8

9 7. Modeling thickness-dependent carrier mobility Our model of the thickness dependence of the mobility is based on the Thomas- Fermi charge screening which has been successfully employed to describe other layered material systems such as MoS 2 and graphene [S2 5]. Because of screening, the charge concentration per unit volume, n and carrier mobility, are now functions of z, the distance measured from the interface between sample and substrate. Following the resistance network model proposed in ref. S4 6, we consider a thin slab within the sample with thickness z, which has an in-plane conductance of en(z) (z) zw/l (W and L are width and length of the sample respectively). Without affecting the fitting results (to be discussed later), we set the dimensionless parameter W/L = 1 for simplicity. The out-of-plane resistance of the thin slab is model by two resistors, R int, which are proportional to z (shown in Supplementary Fig. 6). We let 2R int = r z, where r is the parameter introduced to characterize out-of-plane resistivity. Next we assume the source-drain current, I ds, is injected from the top surface of the sample: an assumption well justified for both MoS 2 and our few-layer phosphorene devices [S5]. The sample with thickness z + z can be regarded as the sample with thickness z in parallel with a slab (thickness z), connected through two out-of-plane resistors 2R int as shown in Supplementary Fig. 6. The total conductance can be written as: ( ) ( ) ( ) ( ) ( ) (1) which yields a differential equation for (z): ( ) ( ) ( ) (2) 9

10 Once (z) is known, the effective mobility of the sample with total thickness t can be obtained by: ( ) ( ) (3) Supplementary Fig. 6. Resistance network model for multilayer black phosphorus samples. The conductance of a sample with thickness z+ z can be modeled as a thin slab with thickness z in parallel with the rest of the sample, connected through two interlayer resistors. In order to solve Eq. (2) for (z), we need to find out the functional forms of n z and z Due to Thomas-Fermi charge screening, n z decays exponentially as a function of z. ( ) ( ) (4) where is the Thomas-Fermi screening length. Meanwhile, all the free charge on the sample should be induced by the gate, which sets the boundary condition for n z as follows: ( ) (5) where N gate is the gate induced charge per unit area (for our sample on Si wafer with 90 nm SiO 2 layer, a turn-on gate voltage at ~ -35 V corresponds to N gate = cm -2 ). 10

11 The only unknown parameter in Eq. (4) and (5) is now the Thomas-Fermi screening length, which can be calculated using [S7]: ( ) (6) where r = 8.3 is the dielectric constant of black phosphorus in the out-of-plane direction [S8] and E is the Fermi energy. According to the free electron gas model, ( ) (7) Here n = cm -3 is the bulk carrier density obtained from our Hall measurement, and m* = 0.65m 0 is the effective hole mass in black phosphorus in the out-of-plane direction [S9] (m 0 is the bare electron mass). Combining Eq. (6) and (7), we obtain the screening length = 2.9 nm. We note that this free carrier screening length is much smaller than the Debye screening length (estimated to be ~ 10 nm in our samples), and thus dominates the screening process in this model. Armed with the calculated screening length, Eq. (4) and (5) now completely determines n z The gate induced charge also screens the Coulomb potential of the ionized impurities at the sample-substrate interface, which in turn affects the sample mobility. The further away one is from the interface, the smaller influence the ionized impurities have on the carrier scattering (and thus the mobility). Here we assume the same exponential decay that governs n z also describes the decreasing effect of the Coulomb potential from the ionized impurities at the interface. The carrier mobility can then be modeled as[s5]: ( ) ( ) ( ) (8) where 0 is the mobility at the sample/substrate interface, and inf the mobility at infinity. 11

12 Having determined the functional forms of n z and z, we numerically extract (z), and in turn eff, from Eq.(2) and (3), with three free parameters: r, 0, and inf. Before we move on to fit eff with our experimental data, we notice that there is an inactive layer at the sample/substrate interface. The inactive layer has a very small carrier mobility which is reflected by the sharp drop of the mobility at ~ 3 nm. We therefore set the thickness of the inactive layer to be d = 3 nm, and a constant mobility 0 = 5.5 cm 2 /Vs (the smallest mobility we have measured on a 3nm sample) is assumed for the inactive layer. Our fitting (see below) is found not sensitive to the exact values of d and 0. Finally we fit eff with our experimental data on the thickness-dependent mobility with only two fitting parameters, inf and r. The best fit is obtained with inf = 4500 cm 2 /Vs and r = /cm (Fig. 3c inset, main text). 8. Modeling thickness-dependent on-off ratio In the on state, conductance can be regarded as a combination of two parts: 1) contribution from free carriers induced by gate en gate eff, and 2) intrinsic conductivity 0 from initial impurity doping and thermally excited carriers: (9) where N gate = cm -2 is the gate induced carrier concentration in on state (corresponding to V g = -35 V). In the off state, the initially p-doped bottoms layers of the sample are depleted by the gate, and we assume a residual conductivity dep for the depletion layer. The off state conductance can then be written as: 12

13 { ( ) (10) where t dep = 8.5 nm is the thickness of the depletion layer obtained from the experiment data. We manually fit our experimental data using Eq. (9) and (10) by adjusting the two free parameters 0 and dep. The best fitting are obtained with 0 = 12.4 S/cm and dep = 1.66 ms/cm (Fig. 3c inset, main article). We note that our model predicts a sharp decrease in the on-off ration for samples thinner than 4 nm, due to the diminishing on state carrier mobility in those samples. This predicted sharp decrease in on-off ratio is not experimentally observed. Further study is needed to elucidate the detailed charge transport mechanism at the sample/substrate interface. 9. Carrier density determination Two methods are utilized to extract the carrier density in few-layer phosphorene samples: the carrier density can be obtained from Hall coefficient R using the expression n H 1/ er, where e is the charge of an electron; or it can be calculated from applied gate voltage V g assuming a parallel-plate capacitor model, ng ( Vg Vth) 0 r / d, where Vth is the threshold gate voltage, 0 is the vacuum permittivity, d is the thickness of the SiO 2 gate dielectric and 3. 9 r is its dielectric constant. As shown in the inset of Supplementary Fig. 7a, the two methods yield the same carrier density value for a typical multi-terminal sample. Two opposing electrical contacts in the middle are used to measure the Hall coefficient R. The measured nh at different gate voltages (black squares, 13

14 black solid line is the line fit) agree with the calculated n G (red dashed line) very well, as shown in Supplementary Fig. 7a. However, if the contacts used to measure the Hall coefficient are placed too close to the large Source/Drain contacts, the measured nh may deviate significantly from n G. One such device is shown in Supplementary Fig. 7b inset. nh measured in this device is shown as black squares in Supplementary Fig. 7b, which is ~2.6 times larger than n G (red dashed line). Such a large discrepancy is found due to sample geometry. Specifically, the large source/drain contact near the Hall voltage probes creates a low-resistance path, which significantly reduces the Hall voltage measured by external circuits. In other words, the Hall voltage probes are shorted by the source/drain contact and produce an artificially low Hall coefficient, which in turn yields a significantly over-estimated carrier density. On the other hand, n G does not surfer from this problem, and for this reason is always used to determine the carrier density in our samples. 14

15 Supplementary Figure 7. Carrier density determination in few-layer phosphorene FETs. a, Measured carrier density n H as a function of gate voltage (black squares) for the device pictured in the inset. Two opposing contacts in the middle are used to measure the Hall coefficient. The black solid line is a linear fit of the data. They agree well with the calculated dashed line. b, Measured carrier density 15 n G shown as the red n H as a function of gate voltage (black squares) for another device shown in the inset. This device has large source/drain contacts too close to the Hall voltage probes. Black solid line is a line fit of the data, which significantly deviates from the calculated n G (red dashed line).

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