The increasing interest in realizing the full potential of two-dimensional (2D) layered materials for developing electronic components strongly relies on quantitative understanding of their anisotropic electronic properties. Herein, we use conductive atomic force microscopy to study the anisotropic electrical conductance of multilayer MoS2 by measuring the spreading resistance of circular structures of different radii ranging from 150 to 400 nm. The observed inverse scaling of the spreading resistance with contact radius, with an effective resistivity of ρeff  = 2.89 Ω cm, is compatible with a diffusive transport model. A successive etch of the MoS2 nanofilms was used to directly measure the out-of-plane resistivity, i.e., 29.43 ± 7.78 Ω cm. Based on the scaling theory for conduction in anisotropic materials, the model yields an in-plane resistivity of 0.28 ± 0.07 Ω cm and an anisotropy of ∼100 for the ratio between the in-plane and out-of-plane resistivities. The obtained anisotropy indicates that the probed surface area can extend up to 400 times the metal contact area, whereas the penetration depth is limited to roughly 20% of the contact radius. Hence, for contact radius less than 3 nm, the conduction will be limited to the surface. Our investigation offers important insight into the anisotropic transport behavior of MoS2, a pivotal factor enabling the design optimization of miniaturized devices based on 2D materials.

Spreading resistance is an important aspect of nanoscale electronic components since it provides valuable insights into their local electrical properties and is essential for the optimization of their performance.1,2 In electrically isotropic materials, the technique is used to probe a semi-spherical volume within the material with a radius on the order of the contact diameter. Two-dimensional (2D) layered materials, such as graphite and MoS2, comprise highly anisotropic electrical conductivity, particularly between their basal (in-plane) and c-axis (out-of-plane) crystal orientations. For instance, the in-plane (ρab) to out-of-plane (ρc) resistivities ratio γ = ρc/ρab is on the order of 104 for graphite at room temperature.3–6 Correspondingly, the effective planar area for transport below the metal contacts is γ times larger than the penetration depth into the sample, which is in practice only a few graphene layers for contact radii of several hundreds of nanometers.6 Hence, miniaturization and design optimization of electronic devices based on 2D materials require a comprehensive understanding of the anisotropic current flow.7–9 Within the TMDs family, MoS2 stands out as one of the most studied element with its distinctive features such as bandgap tunability,10 valley polarization,11 high carrier mobility,12,13 mechanical strength and flexibility,14 etc., which makes it particularly attractive for a wide range of applications including transistors, photodetectors, sensors, and energy storage devices.15,16 Despite the crucial role as a deterministic factor in device miniaturization, conductance anisotropy in MoS2 remains poorly understood. For instance, attempts to measure the in-plane conductivity of exfoliated MoS2 provides current transport values dependent on gate voltage. Therefore, inherently quantifying anisotropic charge transport provides valuable insight enabling device optimization utilizing 2D materials and heterostructures.

Here, we utilize conductive atomic force microscopy (c-AFM) to measure the spreading resistance of circular mesoscale MoS2 structures with varying radii from 150 to 400 nm [Fig. 1(a)]. We experimentally demonstrate that the spreading resistance follows diffusive transport, with the resistance scales inversely proportional to the contact radius. In addition, the out-of-plane resistivity of ∼ ρc = 29.43 ± 7.78 Ω cm was directly measured by performing successive etching and current vs voltage measurements of the MoS2 circular contacts. This allows us to extract both the in-plane resistivity ρab = 0.28 ± 0.07 Ω cm and the anisotropy ratio γ = ρc/ρab ≈ 100 for MoS2 based on scaling predictions,17–19 which are in good correspondence with prior reports.20,21

FIG. 1.

(a) Schematics of the experimental setup for measuring spreading resistance of metal contacts on exfoliated MoS2 flakes. (b) AFM topography image of metal contacts with radius of 400 nm. (c) Equivalent circuit diagram of the experimental setup. (d) Current volatge characteristics of metal contacts of different radii ranging from 150 to 400 nm. Dashed red square reprensts the voltage range used for resisitacne measurements. Inset represets the current voltage characteristics from −5 to +5 V in log scale.

FIG. 1.

(a) Schematics of the experimental setup for measuring spreading resistance of metal contacts on exfoliated MoS2 flakes. (b) AFM topography image of metal contacts with radius of 400 nm. (c) Equivalent circuit diagram of the experimental setup. (d) Current volatge characteristics of metal contacts of different radii ranging from 150 to 400 nm. Dashed red square reprensts the voltage range used for resisitacne measurements. Inset represets the current voltage characteristics from −5 to +5 V in log scale.

Close modal

High-quality MoS2 film (thickness of ∼120 nm, Manchester Nanomaterials) was mechanically exfoliated onto Au/Cr/SiO2/Si wafer (SiO2/Si wafer with 300 nm SiO2 was covered by 5/50 nm thick Cr/Au metal using thermal evaporation). Electron beam lithography was used to fabricate circular metal contacts of different radii (150, 200, 250, 300, and 400 nm) on top of the exfoliated flake using poly-methyl-methacrylate (PMMA) as the resist layer. The metal contacts consisting of 5 and 50 nm Cr and Au, respectively, were deposited using thermal evaporation. Before the evaporation process, a brief oxygen plasma cleaning was performed to eliminate any remaining resist residues from the exposed surface of MoS2. Electrical measurements were carried out using atomic force microscope (AFM, Bruker Dimension V) situated inside a nitrogen filled glove box (H2O and O2 content < 1 ppm). Current vs voltage characteristics were recorded using semiconductor device parameter analyzers (Keysight B1500A), where the voltage was applied by the conductive AFM probe (Rocky Mountain-25pt400B). The precise positioning of the AFM probe on top of the sample (normal force of ∼500 nN) and the following voltage sweep from −5 to +5 V were performed using nanolithography scripts in C++ language [Fig. 1(a)]. Figure 1(b) presents a typical AFM topography image of metal contacts with a radius of 400 nm. An effective equivalent electrical circuit representation is schematically shown in Fig. 1(c), where the top MoS2-Au contact is described by a Schottky element, RSP represents the spreading resistance, and RCL represents the cantilever resistance including both the cantilever itself and tip–Au contact. RCL = 180 ± 3.04 Ω was extracted by measuring the resistance of an Au surface sample (see the supplementary material for I–V curves).

Figure 1(d) presents the current–voltage (I–V) characteristics in the range of 0–5 V obtained for metal contacts of different radii. A set of 10 I–V profiles were measured for each radius (each profile represents a different structure of the same radii). A clear trend of increasing current can be observed with the increase in pillar radius. The complete I–V profiles are shown in log scale as the inset in Fig. 1(d). The obtained I–V characteristics show an asymmetric rectifying behavior, similar to a previous report20 and can be explained by the presence of two back to back MoS2-Au Schottky contacts connected in series. In particular, when a positive voltage bias is applied to the cantilever, the top MoS2-Au contact exhibits forward biasing, whereas the bottom macroscale MoS2-Au contact displays reverse biasing. Nevertheless, due to the significant disparity in size between the top and bottom contact areas the majority of voltage drop occurs across the top MoS2-Au interface, and the bottom contact can effectively be treated as a minor constant resistance. Consequently, the prevailing mode of transportation within the system is primarily influenced by the top nanoscale contact, leading to the observed asymmetric I–V characteristics. Hence, the applied voltage primarily falls across the top Au–MoS2 contact and within the MoS2 film due to spreading resistance. In addition, at higher applied voltage range, i.e., above ∼3 V, the I–V characteristics start to exhibit linear dependence, where the majority of the potential drop occurs within the MoS2.22–24 Therefore, the linear I–V characteristics at the higher voltage range, i.e., between 4 and 5 V, were used to extract the spreading resistance of the MoS2 meso-structures for each of the measured pillar radius [Fig. 2(a)]. The spreading resistance for a circular contact with a radius R and isotropic resistivity ρ can be quantitatively expressed based on a classical diffusive transport model6,20 as follows:
R s p = ρ 4 R .
(1)
FIG. 2.

(a) Spreading resisitance vs the radius R of metal contact represented in log –log scale. Dashed black line represents the 1/R fit of the measured data according to Eq. (1) yeilding an effective resisitivity value of ρeff = 2.89 Ω cm. (b) Total resistance measured before and following two etch steps. Inset shows the schematic of the pillar structure after etching.

FIG. 2.

(a) Spreading resisitance vs the radius R of metal contact represented in log –log scale. Dashed black line represents the 1/R fit of the measured data according to Eq. (1) yeilding an effective resisitivity value of ρeff = 2.89 Ω cm. (b) Total resistance measured before and following two etch steps. Inset shows the schematic of the pillar structure after etching.

Close modal
Whereas for electrically anisotropic materials such as MoS2, it is necessary to account for the variation in resistivity across different directions. To address this, an effective resistivity (ρeff)17–19 can be employed in Eq. (1), defined by
ρ eff = ρ a b × ρ c .
(2)
The dashed line in Fig. 2(a) corresponds to a linear fit according to Eq. (1), yielding an effective resistivity of ρeff  = 2.89 Ω cm, that is in good agreement with previous reported values for exfoliated MoS2 flakes.20 For out-of-plane resistivity measurements, the sample underwent plasma etching to form mesoscopic cylindrical MoS2 pillars with pillar axis aligned along the C-axis.25–27 Reactive ion etching (RIE-Plasma-Therm 790) was employed for successive etching of MoS2 using SF6 gas (200 W, 20 sccm, 20 mT, and etch rate of ∼5 nm/s), resulting in the formation of pillar-like structures where the metal contacts served as the etch mask.
Ohm's law can be employed to determine the electrical resistance of a pillar (Rp) with height h and radius r as follows:
R p = ρ c h π r 2 .
(3)
The equation states that the resistance per unit height (RP/h) is inversely proportional to the pillar area (πr2), whereas the specific pillar resistance (RP × πr2) is linearly proportional to the pillar height. Hence, the increase in total resistance followed by successive etching is purely attributed to the out-of-plane MoS2 resistance. The total resistance following two sequential etch steps was measured by taking the slope at higher applied bias (4–5 V) and plotted against the etch height [Fig. 2(b)]. Employing Eq. (3), the out-of-plane resistivity is determined to be 29.43 Ω cm, with a standard deviation of 7.78 Ω cm. By substituting ρeff and ρc into Eq. (2), we extract ρab = 0.28 ± 0.07 Ω cm. The ratio of in-plane to out-of-plane resistivity was determined to be γ = ρcab ≈ 100, which aligns well with previous reports for MoS2.20,21,28
The potential distribution V(r, z) below a circular metal contact can be described using Eq. (4) for isotropic materials,29 
V r , z = V 0 2 π sin 1 2 R ( r R ) 2 + z 2 + ( r + R ) 2 + z 2 ,
(4)
where V0 is the potential of the metal contact, r and z denote the radial and perpendicular coordinates, respectively. While addressing anisotropic materials, it is necessary to make slight modifications to the equation by introducing the anisotropy ratio in both radial and perpendicular coordinates as follows:18,19,30
r r γ ,
(5)
z z / γ .
(6)

Figure 3 illustrates the potential distribution for V/V0 = 0.3 for several cases of different anisotropy. As the degree of anisotropy increases, the distribution of voltage tends to extend along the planar ab plane instead of permeating into the sample. In the case of graphite, exhibiting a significantly larger anisotropy ratio (as indicated by γ = 3.6 × 104), the probed area is even more extended, where the conduction is primarily confined to the uppermost surface. In particular, the equipotential surface is observed to exhibit a near-collapse behavior, resembling a horizontal line once the conduction becomes limited to the utmost monolayer.

FIG. 3.

Potential distribution in the sample for a circular contact with radius R. r and z denotes the radial cordiante in the ab plane and out-of-plane cordinate along the c axis, respectively. Different anisotrpy ratios are illustrated alongside MoS2 for comparison. Note: X and Y axes are of different size scales to accommodate the large anisotropy.

FIG. 3.

Potential distribution in the sample for a circular contact with radius R. r and z denotes the radial cordiante in the ab plane and out-of-plane cordinate along the c axis, respectively. Different anisotrpy ratios are illustrated alongside MoS2 for comparison. Note: X and Y axes are of different size scales to accommodate the large anisotropy.

Close modal

In contrast to graphite, the conduction in MoS2 is not confined to the topmost layer, even at contact radius as small as tens of nanometers.

To conduct a more comprehensive analysis of the effective probed area and penetration depth for various contact radii, we examined an equipotential surface where the potential is equal to 0.3 times the initial potential (V/V0 = 0.3), where the probed radius (rab) and penetration depth (ζ) can be expressed as follows:
r a b 2 × R × γ ,
(7)
ζ 2 × R / γ .
(8)
Hence, for MoS2, it is anticipated that the probed area will be ∼400 times larger than the contact area. Likewise, the depth of penetration corresponds to approximately 20% of the contact radius. In the present study, the maximum radius examined is 400 nm, which corresponds to a penetration depth of around 80 nm. Conversely, the minimum radius investigated is 150 nm, resulting in an effective voltage penetration of up to ∼30 nm. Considering a thickness of ∼0.65 nm for each MoS2 layer, it is anticipated that the current will effectively probe about 46 layers when a contact radius of 150 nm is employed. Likewise, conduction will be limited to the upper most surface if the contact radius is below ∼4 nm. Interestingly, such threshold contact radius for MoS2 is ∼10 times smaller than that of graphite, owing to the difference in their conduction anisotropy.

In summary, the experimental verification of the scaling relationship between spreading resistance and contact radius, as predicted by the diffusive transport model, has been verified for MoS2 with an effective resistivity value of ρeff = 2.89 Ω cm. The out-of-plane resistivity of the MoS2 nanofilms was directly measured through successive etching, yielding a value of 29.43 ± 7.78 Ω cm.

According to the scaling theory for conduction in anisotropic materials, the model predicts an anisotropy of ∼100 for the ratio between the in-plane and out-of-plane resistivities. Correspondingly, MoS2 shows a lower effectively probed area and larger penetration depth into the sample as compared to that of graphite. Overall, we demonstrate the utilization of C-AFM as an efficient tool to study the electrical anisotropy and current distribution in layered MoS2, that can be extended to various 2D materials. The findings hold great potential for enhancing the efficiency of nanoelectronic devices based on 2D material systems by optimizing the device geometry according to the anisotropic current distribution in miniaturized contacts.

See the supplementary material for MoS2 characterization, calibration of the cantilever resistance, and complete I–V profiles following the two etching steps.

We gratefully acknowledge the Israel Science Foundation (ISF) for financial assistance, under Grant No. 1567/18, and the Micro & Nano Fabrication Unit (MNFU) for the nanofabrication facilities.

The authors have no conflicts to disclose.

Gautham Vijayan: Data curation (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal). Michael Uzhansky: Methodology (supporting). Elad Koren: Conceptualization (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

1.
S.
Banerjee
,
M.
Sardar
,
N.
Gayathri
,
A. K.
Tyagi
, and
B.
Raj
, “
Conductivity landscape of highly oriented pyrolytic graphite surfaces containing ribbons and edges
,”
Phys. Rev. B
72
(
7
),
075418
(
2005
).
2.
J.
Mody
,
P.
Eyben
,
E.
Augendre
,
O.
Richard
, and
W.
Vandervorst
, “
Toward extending the capabilities of scanning spreading resistance microscopy for fin field-effect-transistor-based structures
,”
J. Vac. Sci. Technol. B
26
(
1
),
351
(
2008
).
3.
G. J.
Morgan
and
C.
Uher
, “
The C-axis electrical resistivity of highly oriented pyrolytic graphite
,”
Philos. Mag. B
44
(
3
),
427
430
(
1981
).
4.
C.
Uher
,
R. L.
Hockey
, and
E.
Ben-Jacob
, “
Pressure dependence of the c-axis resistivity of graphite
,”
Phys. Rev. B
35
(
9
),
4483
4488
(
1987
).
5.
K.
Matsubara
,
K.
Sugihara
, and
T.
Tsuzuku
, “
Electrical resistance in the c direction of graphite
,”
Phys. Rev. B
41
(
2
),
969
974
(
1990
).
6.
E.
Koren
,
A. W.
Knoll
,
E.
Lörtscher
, and
U.
Duerig
, “
Meso-scale measurement of the electrical spreading resistance in highly anisotropic media
,”
Appl. Phys. Lett.
105
(
12
),
123112
(
2014
).
7.
E.
Koren
,
I.
Leven
,
E.
Lörtscher
,
A.
Knoll
,
O.
Hod
, and
U.
Duerig
, “
Coherent commensurate electronic states at the interface between misoriented graphene layers
,”
Nat. Nanotechnol.
11
(
9
),
752
757
(
2016
).
8.
D.
Dutta
,
A.
Oz
,
O.
Hod
, and
E.
Koren
, “
The scaling laws of edge vs. bulk interlayer conduction in mesoscale twisted graphitic interfaces
,”
Nat. Commun.
11
(
1
),
4746
(
2020
).
9.
A.
Oz
,
D.
Dutta
,
A.
Nitzan
,
O.
Hod
, and
E.
Koren
, “
Edge state quantum interference in twisted graphitic interfaces
,”
Adv. Sci.
9
(
14
),
2102261
(
2022
).
10.
B.
Li
,
Q.
Su
,
L.
Yu
,
J.
Zhang
,
G.
Du
,
D.
Wang
,
D.
Han
,
M.
Zhang
,
S.
Ding
, and
B.
Xu
, “
Tuning the band structure of MoS2 via Co9S8@MoS2 core-shell structure to boost catalytic activity for lithium-sulfur batteries
,”
ACS Nano
14
(
12
),
17285
17294
(
2020
).
11.
S.
Katznelson
,
B.
Cohn
,
S.
Sufrin
,
T.
Amit
,
S.
Mukherjee
,
V.
Kleiner
,
P.
Mohapatra
,
A.
Patsha
,
A.
Ismach
,
S.
Refaely-Abramson
,
E.
Hasman
, and
E.
Koren
, “
Bright excitonic multiplexing mediated by dark exciton transition in two-dimensional TMDCs at room temperature
,”
Mater. Horiz.
9
(
3
),
1089
1098
(
2022
).
12.
B.
Radisavljevic
,
A.
Radenovic
,
J.
Brivio
,
V.
Giacometti
, and
A.
Kis
, “
Single-layer MoS2 transistors
,”
Nat. Nanotechnol.
6
(
3
),
147
150
(
2011
).
13.
N. R.
Pradhan
,
D.
Rhodes
,
Q.
Zhang
,
S.
Talapatra
,
M.
Terrones
,
P. M.
Ajayan
, and
L.
Balicas
, “
Intrinsic carrier mobility of multi-layered MoS2 field-effect transistors on SiO2
,”
Appl. Phys. Lett.
102
(
12
),
123105
(
2013
).
14.
S.
Bertolazzi
,
J.
Brivio
, and
A.
Kis
, “
Stretching and breaking of ultrathin MoS2
,”
ACS Nano
5
(
12
),
9703
9709
(
2011
).
15.
O. V.
Yazyev
and
A.
Kis
, “
MoS2 and semiconductors in the flatland
,”
Mater. Today
18
(
1
),
20
30
(
2015
).
16.
S.
Manzeli
,
D.
Ovchinnikov
,
D.
Pasquier
,
O. V.
Yazyev
, and
A.
Kis
, “
2D transition metal dichalcogenides
,”
Nat. Rev. Mater.
2
,
17033
(
2017
).
17.
G.
Wexler
, “
The size effect and the non-local Boltzmann transport equation in orifice and disk geometry
,”
Proc. Phys. Soc.
89
(
4
),
927
941
(
1966
).
18.
M. M.
Yovanovich
, “
On the temperature distribution and constriction resistance in layered media
,”
J. Compos. Mater.
4
(
4
),
567
570
(
1970
).
19.
Y. S.
Muzychka
,
M. M.
Yovanovich
, and
J. R.
Culham
, “
Thermal spreading resistance in compound and orthotropic systems
,”
J. Thermophys. Heat Transfer
18
(
1
),
45
51
(
2004
).
20.
F.
Giannazzo
,
G.
Fisichella
,
A.
Piazza
,
S.
Agnello
, and
F.
Roccaforte
, “
Nanoscale inhomogeneity of the Schottky barrier and resistivity in MoS2 multilayers
,”
Phys. Rev. B
92
(
8
),
081307
(
2015
).
21.
A. M.
Hermann
,
R.
Somoano
,
V.
Hadek
, and
A.
Rembaum
, “
Electrical resistivity of intercalated molybdenum disulfide
,”
Solid State Commun.
13
(
8
),
1065
1068
(
1973
).
22.
Z.
Zhang
,
K.
Yao
,
Y.
Liu
,
C.
Jin
,
X.
Liang
,
Q.
Chen
, and
L. M.
Peng
, “
Quantitative analysis of current-voltage characteristics of semiconducting nanowires: Decoupling of contact effects
,”
Adv. Funct. Mater.
17
(
14
),
2478
2489
(
2007
).
23.
Y.
Liu
,
Z. Y.
Zhang
,
Y. F.
Hu
,
C. H.
Jin
, and
L. M.
Peng
, “
Quantitative fitting of nonlinear current-voltage curves and parameter retrieval of semiconducting nanowire, nanotube and nanoribbon devices
,”
J. Nanosci. Nanotechnol.
8
(
1
),
252
258
(
2008
).
24.
Z. Y.
Zhang
,
C. H.
Jin
,
X. L.
Liang
,
Q.
Chen
, and
L. M.
Peng
, “
Current-voltage characteristics and parameter retrieval of semiconducting nanowires
,”
Appl. Phys. Lett.
88
(
7
),
073102
(
2006
).
25.
E.
Koren
,
A. W.
Knoll
,
E.
Lörtscher
, and
U.
Duerig
, “
Direct experimental observation of stacking fault scattering in highly oriented pyrolytic graphite meso-structures
,”
Nat. Commun.
5
,
5837
(
2014
).
26.
E.
Koren
,
C.
Rawlings
,
A. W.
Knoll
, and
U.
Duerig
, “
Adhesion and friction in mesoscopic graphite contacts
,”
Science
348
,
679
684
(
2015
).
27.
R.
Bessler
,
U.
Duerig
, and
E.
Koren
, “
The dielectric constant of a bilayer graphene interface
,”
Nanoscale Adv.
1
(
5
),
1702
1706
(
2019
).
28.
A.
Pisoni
,
J.
Jacimovic
,
O. S.
Barišic
,
A.
Walter
,
B.
Náfrádi
,
P.
Bugnon
,
A.
Magrez
,
H.
Berger
,
Z.
Revay
, and
L.
Forrö
, “
The role of transport agents in MoS2 single crystals
,”
J. Phys. Chem. C
119
(
8
),
3918
3922
(
2015
).
29.
A. D.
Bejan
and
A.
Kraus
,
Heat Transfer Handbook
, 1st ed. (
John Wiley & Sons
,
2003
).
30.
E.
Slot
,
H. S. J.
Van Der Zant
, and
R. E.
Thorne
, “
Electric-field distribution near current contacts of anisotropic materials
,”
Phys. Rev. B
65
(
3
),
033403
(
2001
).

Supplementary Material