SUPPLEMENTARY INFORMATION - Nature...5 . Sample contrast spectra obtained with incident light...

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NNANO.2014.35

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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

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

10 20 30 40 50 60 70

(0 6 0)

(0 4 0)

Inte

nsity

(ar

b.un

it)

2 (degree)

Black Phosphorus

(0 2 0)

b= 1.04831 nm

Black phosphorus field-effect transistors

© 2014 Macmillan Publishers Limited. All rights reserved.

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(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 (~1×10-5

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 O2 and H2O 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 O2 or water vapor is likely the culprit. O2 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.


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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 SiO2. We measure

the optical contrast defined by (Is - I0)/I0, where Is and I0 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.


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Supplementary Figure 3. a, AFM images of a black phosphorus think flake sample. The cross-

sectional profile along the white dash line, shown as red curve, indicates a 2 nm step-edge. b.


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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.


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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 cross-

section profiles along the dash lines in a.


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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 long-

range 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 10 8 8 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.


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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.


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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 MoS2 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, Rint, which are

proportional to z (shown in Supplementary Fig. 6). We let 2Rint = rz, where r is the

parameter introduced to characterize out-of-plane resistivity.

Next we assume the source-drain current, Ids, is injected from the top surface of

the sample: an assumption well justified for both MoS2 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 2Rint as shown in Supplementary Fig. 6. The total conductance can be written as:

( ) ( ) ( ) ( )

( )

(1)

which yields a differential equation for (z):

( ) ( ) ( ) (2)


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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 nz

and z Due to Thomas-Fermi charge screening, nz 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 nz as

follows:

∫ ( )

(5)

where Ngate is the gate induced charge per unit area (for our sample on Si wafer with 90

nm SiO2 layer, a turn-on gate voltage at ~ -35 V corresponds to Ngate = 8.6×1012

cm-2

).


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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 = 1.18×1017

cm-3

is the bulk carrier density obtained from our Hall measurement,

and m* = 0.65m0 is the effective hole mass in black phosphorus in the out-of-plane

direction [S9] (m0 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 nz

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 nzalso 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.


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Having determined the functional forms of nz 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 cm2/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

cm2/Vs and r = 6.51×10

8 /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 eNgate×eff, and 2) intrinsic conductivity 0

from initial impurity doping and thermally excited carriers:

(9)

where Ngate = 8.6×1012

cm-2

is the gate induced carrier concentration in “on” state

(corresponding to Vg = -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:


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{ ( )

(10)

where tdep = 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

eRnH /1 , where e is the charge of an electron; or it can be calculated from applied gate

voltage Vg assuming a parallel-plate capacitor model, dVVn rthgG /)( 0 , where thV is

the threshold gate voltage, 0 is the vacuum permittivity, d is the thickness of the SiO2

gate dielectric and 9.3r 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 Hn at different gate voltages (black squares,


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black solid line is the line fit) agree with the calculated Gn (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 Hn may deviate significantly from Gn .

One such device is shown in Supplementary Fig. 7b inset. Hn measured in this device is

shown as black squares in Supplementary Fig. 7b, which is ~2.6 times larger than Gn

(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, Gn does not surfer from this problem, and for this reason is

always used to determine the carrier density in our samples.


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Supplementary Figure 7. Carrier density determination in few-layer phosphorene FETs. a,

Measured carrier density Hn 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 Gn shown as the red

dashed line. b, Measured carrier density Hn 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 Gn (red dashed line).


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