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Anisotropic Thermal Conductivity of Suspended Black Phosphorous Probed by Opto-thermomechanical Resonance Spectromicroscopy Arnob Islam, Anno van den Akker, and Philip X.-L. Feng Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b03333 • Publication Date (Web): 29 Oct 2018 Downloaded from http://pubs.acs.org on October 29, 2018

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TOC graphic 545x195mm (120 x 120 DPI)

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Anisotropic Thermal Conductivity of Suspended Black Phosphorous Probed by Opto-thermomechanical Resonance Spectromicroscopy Arnob Islam, Anno van den Akker, Philip X.-L. Feng* Department of Electrical Engineering & Computer Science, Case School of Engineering, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA Abstract Atomic layer semiconducting black phosphorus (P) exfoliated from its bulk crystals offers excellent properties and promises for emerging two-dimensional (2D) electronics, photonics and transducers. It also possesses unique strong in-plane anisotropy among many 2D semiconductors, stemmed from its corrugated crystal structure. As an important thermophysical aspect, probing the anisotropic thermal conductivity of black P is essential for device engineering, especially for energy dissipation and thermal management. Here, we report on measurement and analysis of anisotropic in-plane thermal conductivity of black P crystal, in a mechanically suspended device platform, by exploiting a novel opto-thermomechanical resonance spectromicroscopy (OTMRS) technique. With spatially resolved heating effects and thermomechanical resonance motions of suspended structures, anisotropic in-plane thermal conductivity (κAC and κZZ) is determined for black P crystals of 10 to 100nm thick. This study validates a new non-invasive approach to determining anisotropic thermal conductivity without any requirement of pre-knowledge of crystal orientation or specific configurations of structure and electrodes according to the anisotropy.

Keywords: Black Phosphorus, In-Plane Thermal Anisotropy, Thermal Conductivity, Crystal Orientation, Opto-thermomechanical Resonance Spectromicroscopy (OTMRS).

*

Corresponding Author. Email: [email protected] -1-


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Atomic layers of black phosphorus (P) extracted from its layered bulk, a new two-dimensional (2D), direct-bandgap semiconductor, has stimulated considerable research interest recently, due to its excellent and unique electrical, optical, mechanical and thermal properties,1,2 which can be harnessed to realize next-generation electronic, optoelectronic, photonic devices,3,4,5,6 and various transducers7,8,9,10 for sensory systems. For field effect transistors (FETs), black P offers high fieldeffect hole mobility over 1000 cm2/Vs with excellent current on-off ratio of 106 (ref. 3). Thanks to its relatively narrow bandgap (~0.3 eV) in its multilayer and thin film form, it is possible to achieve broadband photodetection from visible to mid-infrared (~4 µm) regime.11 Further, due to high mechanical strain limit (up to 27-30%) 12 and piezo-resistive property of black P, 13 it is possible to attain ultrasensitive strain gauges with gauge factor of ~185 [ref. 12] and 2D resonant nanoelectromechanical systems (NEMS)7,8,9 for radio-frequency (RF) signal processing and sensing applications. In addition, some distinctive properties, e.g., in-plane anisotropy,2 negative Poisson’s ratio,14 giant quantum Stark effect15 etc., are expected to engender entirely new device functionalities that may not be achievable by using conventional semiconductor materials. The average phonon mean-free path (MFP) in 2D materials (MFP ~ 50300nm) at room temperature (T ~ 300K) is usually larger than the thickness of device structures enabled by these 2D materials, which results in multiple effects, e.g., phonon-boundary scattering at the abrupt interfaces between the 2D materials and their adjacent environment, reduction of thermal conductivity (κ) due to phonon confinement effect.16 These can often cause a significant local temperature rise, i.e., hot spots, in modern electronic and optoelectronic devices and state-of-theart chips featuring very- or ultra-large scale integration (VLSI & ULSI) technologies.17,18 In the emerging 2D electronics, photonics and transducers, and especially new devices based on multilayer stacks of 2D crystals, new intriguing questions and challenges in energy dissipation and -2-


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thermal management are emerging. In particular, in 2D FETs operating near current saturation condition, collective carrier scattering with phonons, material defects, impurity, etc., can elevate the local temperature in the channel close to melting point of black P and cause detrimental breakdown, known as Joule breakdown.19 Meantime, due to lower thermal conductivity of atomic layers of black P (κavg = 2030 Wm-1K-1) compared to graphene and MoS2, black P may operate as a comparatively inferior heat spreader during device operation, specifically for FETs. Therefore, understanding power dissipation in black P FETs and subsequently innovating thermal management solutions are essential for maintaining reliable and robust performance, in order to fully exploit the attractive and unique properties of this new semiconductor in device applications. Fortunately, although 2D few-layer black P has limited κ values, it has very high electrical conductivity (σ).

Thus, theoretical thermoelectric figure-of-merit (zT=S2(σ/κ)T, where S is

Seebeck coefficient) of black P is predicted to be higher.20 Furthermore, importantly, as an anisotropic semiconductor, black P possesses unique, intrinsic and strong in-plane anisotropy in electrical, optical, mechanical and thermal domains.2 From firstprinciple calculations and several experimental studies, it has been found that black P has higher thermal conductivity (κ) along its zigzag (ZZ) direction than its armchair (AC) direction (κZZ > κAC) (Figure 1a).20 This in-plane anisotropy of κ is related to its anisotropic phonon dispersion along the AC and ZZ directions, derived from its unique corrugated crystal structure (Fig. 1a).21 In addition to high σ/κ ratio, thanks to this thermal anisotropy, coupled with an orthogonal electrical conductivity (σ) anisotropy (σAC > σZZ), zT can be achieved even as high as 3 at 500 K by engineering the thermal and electrical transport,20 which satisfies the criterion for practical thermoelectric applications, thus holding promises for novel thermoelectric and energy applications, such as thermoelectric generators and coolers.20 -3-


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Given the above considerations, toward maintaining functions and performance of black P enabled 2D devices, understanding fundamental thermophysical properties and device characteristics, designing thermal management strategies, and engineering thermal transport, are all critical aspects. To date, however, such studies are still nascent and exploratory; only very limited data and understanding have been obtained in measuring thermophysical properties of black P, its intrinsic anisotropy in thermal properties, and how these would impact device performance. Therefore, there is a great need for thermophysical studies of black P. Techniques established in measuring thermal conduction in conventional nanoscale materials and devices have recently been adopted in exploring black P thermal conductivity.21-24 However, traditional techniques are required to be modified to study anisotropic thermal conductivity of black P. Electrical measurement schemes rely on two or four probe measurements of resistance to infer heating power and temperature distribution, and to derive thermal conductivity,22,23 where lithographically defined electrodes are required along previously calibrated AC and ZZ directions, which demand separate spectroscopic measurements in advance, to resolve AC and ZZ axes before the lithographic patterning and metallization. Optical technique known as opto-thermal microRaman spectroscopy21 exploit effects such as optical absorption induced heating and Raman scattering to measure temperature thus to derive thermal conductivity; the mainstream ones include time-domain thermo-reflectance (TDTR). 24

In these optical methods, while lithographic

electrodes may not be required, it is still necessary to have prior knowledge of the AC and ZZ directions, which in turn demands device structures that can decouple heat transfer along AC and ZZ axes. Further, the above existing schemes are also often limited by samples sitting on substrates, where heat transfer via the substrate complicates the situations and plagues the accuracy in the measurement. Therefore, it is desirable to innovate measurement techniques that are directly

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applicable to generic structures, without requiring prior knowledge or separate measurement along the AC and ZZ directions, or the associated special device configurations (e.g., geometric shaping and electrode arrangement). In this work, we describe a new scheme for measuring the in-plane anisotropic thermal conductivity while simultaneously resolving the anisotropic axes, on suspended devices by using the resonant nanoelectromechanical systems (NEMS) platform. In both electrical and thermal techniques, it is desired to attain free-standing, suspended structures to evade the substrate effects and complication. The NEMS approach thus provides an excellent avenue to investigating thermophysical properties of new materials at nanoscale. While thermal conductivity of 1D InAs has been previously measured on nanowire NEMS resonators platform, 25 thermal conductivity measurement using 2D drumhead NEMS has not yet been explored. In this study, we demonstrate that 2D drumhead NEMS resonators can pave the way to precisely probing thermal conductivities of new semiconducting crystals and resolving the in-plane anisotropic κ. By measuring undriven, Brownian-motion thermomechanical resonances of suspended devices, we employ optical detection of intrinsic properties of the crystalline devices, without requiring separate device actuation method (e.g., electrostatic or piezoelectric actuation) and the associated electrode fabrication, thus offering a non-contact and non-invasive approach.

Here, we measure

thermomechanical resonance (fres) of black P drumhead resonators under a scanning laser interferometry to probe the thermal anisotropy of black P. Thanks to the ultrahigh responsivity of fres to variations of optothermally induced surface tension, minute localized opto-thermal effects are saliently manifested in the shift of fres, which provides precise measurement of heat generation, dissipation, thermal conductivity and its anisotropy.

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Figure 1. Effects of thermal anisotropy of black P on laser heating. (a) 3D crystal structure of black P and its inplane thermal anisotropy. (b) An illustration of localized laser heating at the center of a suspended black P, which also demonstrates the difference of temperature profiles due to laser heating between isotropic and anisotropic materials. (c) Two cases of laser heating on suspended black P, where temperature profile dictates higher average thermal strain or opto-thermal stress (th) for case II. (d) Effect of position dependent laser heating on th with a thermal circuit.

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Figure 1 illustrates the basic principles of our new approach and measurement techniques. We first consider the effect of localized laser heating (a 2D Gaussian beam distribution) at the center of a black P flake suspended on a circular microtrench in a SiO2-on-Si substrate. In this study, we neglect heat convection since the device is under vacuum and heat radiation as the device-ambient temperature differential (~50˚C) is not high enough for heat radiation to be a significant energy loss. For an isotropic material, the resulting temperature distribution contours from heating the device at the center would be circular, since heat spreads evenly in all directions. However, with an anisotropic material, we obtain elliptical contours of temperature with the major axis along the crystal axis with higher κ (ZZ in this case) (Figure 1b). This temperature rise (ΔT) induces thermal expansion in the black P device due to its positive linear thermal expansion coefficient (α).26 Thermal expansion induces compressive thermal strain (εth ≈ -αΔT), which is proportional to the temperature distribution (Figure 1c). Opto-thermal surface tension (γth) due to thermal strain is calculated as follows27

 th 

t  EY,x  EY,y  Tavg t 1  x   y  dxdy    ,   A A3 3 1  

Tavg 

1 T  x, y  dxdy , A  A

(1)

(2)

where  x and  y are thermally induced stress components along x and y directions, respectively; average temperature increase due to localized heating on the suspended area is ΔTavg; EY,x and EY,y are anisotropic Young’s moduli along x and y directions, respectively;  is Poisson’s ratio; t

is thickness of black P and A is the area of the circular drumhead resonator. Finally, this surface tension (negative in polarity due to compressive strain) causes the resonance frequency (fres) of the -7-


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device to decrease (as given the amount of temperature increase, 𝐸Y does not change significantly according to Ref. 28), which can be approximated analytically by the following equation29 𝑓𝑟𝑒𝑠 =

𝑘2𝑚𝑛 𝑎 2𝜋

𝐷

√𝜌𝑎4 [

(𝛾0 +𝛾𝑡ℎ )𝑎2 𝐷

+ (𝑘2𝑚𝑛 𝑎)2 ] ,

(3)

𝐸Y,avg 𝑡 3

where 𝐷 = 12(1−𝜈2 ) is the flexural rigidity, the density of black P is 𝜌, the Poisson’s ratio of black P is 𝜈, the average Young’s modulus of black P is 𝐸Y,avg , the pre-tension is 𝛾0, and eigenvalue of a particular (m, n) mode is 𝑘2𝑚𝑛 . We now consider the situation when the laser spot is located near the edge of the suspended black P along any arbitrary direction (Figure 1d). When the laser spot moves towards the edge of the device, heat can more readily dissipate through the substrate, which will yield a lower average thermal resistance (Rth1), lesser ΔT and consequently low compressive strain or γth (Figure 1d). Considering the in-plane anisotropic black P crystal, we can think about two critical cases: localized laser heating near the edge of the suspended flake along (I) the low κ (AC) and (II) the high κ (ZZ) directions (Figure 1c). Due to higher κ along ZZ, higher amount of heat will transfer along ZZ, which will lead to two different types of temperature profile for the above mentioned two cases. From the temperature profile obtained by a finite element method (FEM) simulation with κZZ > κAC, intuitively, it can be inferred that ΔTavg is higher for case-II compared to case-I (Figure 1c). As a result, average thermal strain (εth,avg ≈ -αΔTavg) and subsequently, γth at the suspended part should be higher for case-II according to Eq. 1. Therefore, we can say that at a fixed distance (r) away from the center of the suspended flake, γth is higher when the laser induced heating occurs along ZZ compared to that along AC, (γth,ZZ > γth,AC)r≠0. Figure 1d shows the thermal circuit corresponding to the localized laser heating, which includes thermal resistances for in-plane heat conduction and thermal boundary contact between black P and the substrate. In summary, γth -8-


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decreases as the laser spot is moved away from the center at any in-plane direction, while the rate of decrease with respect to r is the highest along AC and the lowest along ZZ (Figure 1d).

Figure 2. Anisotropic effects on heating induced resonance frequency shift and thermomechanical resonance. (a) The block diagram explaining the downshift of fres of black P drumhead resonator due to opto-thermal heating. (b) The schematic of the measurement system. (c) and (e) The optical images of two devices, corresponding thickness obtained from AFM (white lines on the images show the traces for AFM) and determined crystal orientations from polarized reflectance measurement. (d) and (f) The thermomechanical noise spectrum with first three resonance modes for device #1 and device #2, respectively. All measurements are performed at room temperature and in moderate vacuum (10-20 mTorr).

The variation of the γth under localized laser heating, originated from the nature of heat dissipation and thermal anisotropy of black P can be detected from the fres of the out-of-plane

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flexural mode of black P drumhead structure. Here, γth originated from the thermal expansion creates compressive strain in black P, which acts against the initial tensile pre-tension (γ0) and decreases the total surface tension at the resonator. As a result, fres will shift downwards with the increase of γth, and by monitoring the fres, we can sense the variation of γth depending on the position of laser spot. In other words, from the dependence of γth on the location of opto-thermal heating (γth,ZZ > γth,AC)r≠0, fres values will be higher when laser spot is moving in AC direction (fres,AC) than that in ZZ direction (fres,ZZ), (fres,AC > fres,ZZ)r≠0 (Fig. 2a). Since the shift in fres correlates with γth and consequently laser induced heating or thermal conductivity along the two in-plane crystal axes, we can use the resonance data to extract the material’s anisotropic thermal conductivity. In order to demonstrate this idea experimentally, we fabricate two black P drumhead resonators by using dry-transfer method.30 The thickness of the black P flakes used in these two fabricated resonators are tBP 85nm (device #1) and tBP 30nm (device #2), while their diameters are d=9m and d=8m, respectively. The optical images and AFM measurements (by using Agilent 5500) of the two devices are shown in Figure 2c and 2e. The optical system for laser heating and simultaneous detection of thermomechanical fres is shown in Figure 2b. A He-Ne laser (633 nm) with spot size of ~1 µm is used for localized opto-thermal heating. The same laser is employed in the optical interferometry to measure thermomechanical resonance.31 By employing polarized reflectance measurement, we determine the crystal orientation of black P.32 Here device #1 operates in the ‘plate’ regime in terms of elastic behavior, where fres is determined by the geometry and Young’s modulus (EY). On the other hand, device #2 operates in the ‘membrane’ regime, where the fres value is determined largely by the tension in the device in addition to the geometry (see Supporting Information, S2).29 For device #1, we have obtained the

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first three fres modes at 8.39 MHz, 15.39 MHz, and 21.45 MHz respectively, with quality (Q) factors in the range of 150200 (laser power, Plas = 1.6 mW). The large anisotropy in Black P’s Young’s modulus is prevalent in the difference in fres between the second and third modes (mode shapes are presented in Supporting Information, S1), whereas, for an isotropic material, these modes should be degenerate pairs and have the identical frequency).8 For device #2, three fres modes are found at 7.45 MHz, 9.92 MHz and 14.39 MHz respectively, with Q-factor ranging from 70 to 280 (Plas = 0.9 mW). As the devices are mounted on a X-Y stage with sub-micrometer precision movement capability, it is possible to perform scanning spectromicroscopy measurements for a particular resonance mode. Therefore, we can obtain the mode shape of each resonance mode (Supporting Information, S1), following the spatial mapping techniques we have demonstrated in detail previously.8 In the present study, while scanning a device with the laser spot (~1μm), we simultaneously exert localized heating and detect shifts in fres.

As a result, we can design a new opto-

thermomechanical resonance spectromicroscopy (OTMRS) measurement to obtain a spatial map of fres values over the suspended device area. We use the fundamental mode for spatial fres mapping, as this mode can be efficiently detected from anywhere on the suspended region because of the mode’s circular and symmetric mode shape along both crystal axes. Figure 3a shows the circular and symmetric mode shape of the fundamental mode of device #1 obtained from scanning spectromicroscopy measurements along with the pre-determined crystal orientation. Spatial fres mapping of the same mode is presented in Figure 3b. This experimentally shows the change of fres depending on the in-plane crystal direction (θ) and r. From the elliptical contours observed at the 2D color plot of mapping, it can be found that fres values along AC (from center to the edge of the suspended part) is higher compared to ZZ, which is in agreement with our earlier prediction, (fres,AC -11-


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> fres,ZZ)r≠0. Obtained fres values from laser positions along the two axes are shown in Figure 3c to illustrate the phenomenon more clearly. Figure 3d shows the increase of fres from center to the edge, while fres values along AC (fres,AC) is clearly higher than that along ZZ (fres,ZZ) for device #1, as we expected. Therefore, the frequency mapping obtained from OTMRS measurement indicates that it is possible to discern crystal orientation without any necessity of performing additional

ZZ (high κ)

measurements beforehand. Similar results are obtained for device #2 (Figure 3e).

1.000

9.140

0.9609

8.625

0.9218

8.110

0.8826

7.595

0.8435

7.080

0.8044

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0.7652

6.050

0.7261

5.535

Device #1

AC (low κ)

S

b

c

0.6

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fres(MHz)

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Device #1

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(V/Hz )

a

8.4

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❺ ❹ ❸ ❷ ❶❷❸❹❺

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0.2 6 7 8 Frequency(MHz)

5


5


5

6.0

0.2



5.5

0.0

0.3

0.6

Figure 3. Opto-thermal resonance spectromicroscopy (OTMRS): (a) The mode shape of the fundamental mode of the device #1 along with the determined crystal orientation. (b) Spatial fres mapping for the fundamental mode of device #1 and yellow dashed ellipse provides the visual aid to discern the effect of thermal anisotropy on fres mapping. (c) The OTMRS measurements along AC and ZZ directions. (d) and (e) The shift of fres while moving the laser spot from center towards the edge along AC and ZZ for device #1 and device #2 respectively.

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Solving Heat Transfer Equation

Laser Heating (633 nm)

Thermal Strain Calculation

Temperature Distribution, T (K)

Thermal Strain Distribution, εth

Calculation of fres by FEM

Resonance ` (fres ) Frequency Shift

a 7.5

Measurement (AC) Measurement (ZZ) Simulation (AC) Simulation (ZZ)

κ AC = 13 Wm -1K-1 κ ZZ = 49 Wm -1K-1

9.2

7.0 κ = 9 1 Wm -1K-1 AC κ ZZ = 37 8 Wm -1K-1

fres (MHz)

fres(MHz)

κ AC = 13 1 Wm -1K-1 8.5 κ ZZ = 49 1 Wm -1K-1

6.5

9.0

7

Device #1

8.0 6.0

Device #1 7.5

κ AC = 9 Wm -1K-1 κ ZZ = 37 Wm -1K-1

8

fres (MHz)

Measurement (AC) Measurement (ZZ) Simulation (AC) Simulation (ZZ)

9.0

fres (MHz)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.0

0.3

0.6

0.9

Device #2 5.5

b

0.0

0.3

Measurement Simulation

8.8

0.6

0.0

c

d

Device #2 Measurement Simulation

6

0.5 1.0 1.5 Laser Power, Plas (mW)

0.0

e

0.5 1.0 Laser Power, PLas (mW)

Figure 4. Modeling and simulation of the measurement: (a) The multi-stage block diagram of the simulation. (b) and (c) Fitting of the simulation with the measurement results of position dependent heating induced fres shift for device #1 and device #2, respectively, from which κAC and κZZ values are determined. (e) and (f) Comparison of the simulated and measured fres downshift with the increasing Plas by using the determined anisotropic κ values.

In order to extract anisotropic thermal conductivity of black P from the measurement results, we model the opto-thermal effects on resonance through a multi-stage physical process shown in Figure 4a. First, we treat the laser source on the device as a heat source with Gaussian-distribution and simulate the temperature distribution (T) within the suspended black P by solving following Eq. (4),

𝜅AC

𝜕2 𝑇 𝜕𝑥 2

+ 𝜅ZZ

𝜕2 𝑇 𝜕𝑦 2

+ 𝜅OP

𝜕2 𝑇 𝜕𝑧 2

=

𝑃𝑙𝑎𝑠 𝐴𝐵𝑃 𝑡𝜋𝜎𝑥 𝜎𝑦

−(𝑥−𝑥0 )2 (𝑦−𝑦0 )2 − ] 2𝜎2 2𝜎2 𝑙𝑥 𝑙𝑦

[

𝑒

,

(4)

where the absorbance of black P is 𝐴𝐵𝑃 , out-of-plane thermal conductivity is 𝜅OP , the standard deviations of the laser beam distribution are 𝜎𝑙𝑥 and 𝜎𝑙𝑦 and the laser spot location is (𝑥0 , 𝑦0 ). The forcing term in Eq. (4) assumes that the heat originated from absorption is evenly distributed in the out of plane axis (z-axis) in the black P since the device thickness is small compared to the laser wavelength. By employing finite element method (FEM) simulation (via COMSOL), we solve the Eq. (4) and obtain the T distribution. The calculated temperature distribution is then -13-


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used to calculate thermally induced strain, consequently surface tension, which is used to calculate the resonant frequencies of the resonator. As we want to extract the anisotropic thermal conductivity by fitting the multi-physics FEM simulation results with our obtained measurement results, we need to know the anisotropic EY (for resonators in plate regime), 𝛾0 (for resonators in membrane regime), 𝐴𝐵𝑃 and α in advance. Anisotropic EY can be determined by fitting the measured multimode fres values (in low laser power to avoid heating effects) of a drumhead resonator in the plate regime with a FEM simulation, which is described in detail in Supporting Information (S1).8 Using this method, we have obtained EY,AC =22 GPa and EY,ZZ = 72 GPa from device #1. Similarly, for tension dominated device (device #2), we can obtain 𝛾0 = 0.448 N/m by fitting measured fres values with simulation. The value of 𝐴𝐵𝑃 is calculated using Fresnel’s law by using published refractive indices for black P,33 presented in the supporting information of S3. For device #1 and #2, 𝐴𝐵𝑃 values are obtained as 31.5% and 8.2%, respectively. There are several contradictory reports on the values of α.28,34,35 We use thermomechanical resonance shift due to thermal stage heating/cooling to extract the value of α (see Supporting Information, S4).36 We obtain α = 5.5 × 10-6 K-1 by using this method. From our measurement results presented in the Supporting Information of S4, essentially, we obtain average α, if there is indeed any anisotropy. The important thing to consider in this case is that we use the fundamental mode to probe the thermal anisotropy, which is circularly symmetric. Therefore, the effect of anisotropic α should be minimal. The use of average α should be a good approximation. Moreover, we use anisotropic α values reported in Ref. 28 in the simulation, which always results in smaller (fres,AC ˗ fres,ZZ)r≠0 compared to the experimental results, if we use αAC> αZZ. Therefore, using these anisotropic α values, we obtain quite unrealistic values of thermal conductivity from fitting, which suggests that the anisotropy in α may be quite small or negligible and requires further -14-


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experimental investigation and validation. We use 𝜅OP = 4 Wm-1K-1 (ref. 24) for the simulation. We model the thermal resistance to the substrate (RTh2) by considering interfacial thermal resistance at the boundary between black P and underneath SiO2. We use interfacial thermal boundary conductance value of GB ~7 MW/m2K for the simulation (see detail discussions in the Supporting Information, S5).16 This leaves anisotropic κ values of black P as the only unknown parameters for the model. We sweep the anisotropic κ values and generate the fres vs r ¢plots (for the 1st mode) along both AC and ZZ directions by simulations, until these simulated plots best match the measured data shown previously in Figure 3e and 3f. The agreement between our simulations and measured data is shown in Figure 4b and 4c for both devices. Through these fitting methods, we obtain κAC =13 ± 1 Wm-1K-1, κZZ =49 ± 1 Wm-1K-1 for device #1 and κAC =9 ± 1 Wm-1K-1, κZZ =37 ± 8 Wm-1K-1 for device #2. The error bars in the simulations have not been generated from the uncertainties in obtaining accurate resonance frequency, as frequency measurements alone provide excellent and higher resolution. Nonetheless, there are also a few reasons which can cause measured resonance frequencies to deviate from the expected values, given the device geometry and opto-thermal heating. The most probable reason of the deviation or error in resonance frequency is the drift of stage during scanning the laser probe on the sample. Furthermore, non-idealities in device structure (e.g., thickness variation, wrinkles, anisotropic initial tension built in thin membrane resonators during dry-transfer process, etc.) can also introduce deviation from the resonance frequency we should observe from an ideal drumhead resonator. As a result, this deviation or error in measured resonance frequency actually propagates to the uncertainties in determined anisotropic thermal conductivity values of black P. We fit the measured data points by multiple fres versus r curves with a range of anisotropic thermal conductivity values. The error bars in the simulated resonance frequency values are generated

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from the standard deviation of anisotropic thermal conductivity values used in the fitting process. One way to reduce the error is to improve the instrumentation to reduce the drift. Another possible way for reducing the error is to improve the fabrication process. We also measure the downshift of fres with the increase of laser power (Plas) (r = 0). Using these extracted κAC and κZZ values, we have also generated the fres vs Plas plots for both devices from simulations, and we find that they match quite well with the measured plots (Figure 4d and 4e). We also present the simulated temperature and thermal strain magnitude profile for device #1 in Supporting Information (S6) by using the extracted anisotropic thermal conductivity values. 10 µm

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Figure 5. OTMRS measurement of three additional devices, with smallest thickness down to 10nm. (a)-(c) Optical images of additional three black P circular drumhead resonators (devices #3 and #5 diameters d=8m; devices #4 diameter d=9m). (d)-(f) OTMRS measurement results from these three devices, along with simulation results.

In addition to the OTMRS results shown in Figure 4a-b, to further extend the thickness range of this study, we have performed the same measurements on additional three devices (device #3: thickness tBP 10nm, diameter d=8m, device #4: tBP 20nm, d=9m; and device #5: tBP 70nm, -16-


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d=8m) (Figure 5a-c). Results of OTMRS measurements and determined anisotropic κAC and κZZ values of corresponding three devices are shown in Figure 5d-f. We obtain κAC =9 ± 3.5 Wm-1K1

, κZZ =37 ± 5 Wm-1K-1 for device #3, κAC =10 ± 2 Wm-1K-1, κZZ =35 ± 3 Wm-1K-1 for device #4,

and κAC =20 ± 2 Wm-1K-1, κZZ =45 ± 2 Wm-1K-1 for device #5. By using our method, we are able to determine anisotropic thermal conductivity of suspended black P devices as thin as ~10 nm, which is among the thinnest black P flakes used for thermal conductivity measurement.

100

10 Luo et al. [21] Lee et al. [22] Jang et al. [24] Smith et al. [23] This Work

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Figure 6. Comparison and benchmarking of the determined anisotropic κ values from five devices, along with the κ values reported in literature by using other measurement techniques.

To date, in order to determine anisotropic κ values of black P using electrical methods, it requires complicated fabrication procedure and thick black P flakes, as we need to fabricate black P nanoribbons oriented along two anisotropic crystal axes with metal electrodes. Moreover, during these multi-step fabrication processes, the intrinsic thermal properties of black P might be altered, as it is very prone to oxidization under ambient conditions.37 In addition to electrical methods, TDTR method can only measure anisotropic κ values for very thick black P flakes. As a new alternative, the OTMRS method we present here requires very simple suspended device structure without any complicated fabrication and patterning steps. We have demonstrated our method -17-


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successfully in both thinner (tension dominated resonator) and thicker (plate resonator) suspended black P resonators. This method is ideally applicable for a wide range of thickness starting from atomically thin layers to thin films of black P, unlike TDTR, four-probe measurements, which are limited to bulk or thin films of black P23,24 (see Supporting Information, S7 for the comparison table). Figure 6 exhibits the comparison and benchmarking of the determined anisotropic κ values (black P flakes ranging from ~10 nm to 90 nm) from our measurements with the those obtained in other studies in literature. In summary, we have demonstrated a new method of quantifying in-plane thermal conductivity and discerning anisotropy of 2D materials by using the suspended nanomechanical resonator platform. The OTMRS technique validated in this work is a non-invasive, non-contact, and an alloptical method for determining in-plane anisotropic thermal conductivity of black P. Furthermore, this method does not require to resolve crystal orientation beforehand and applicable for thinner black P flakes (~10 nm). The method does not need specific device structure to decouple heat transfer along AC and ZZ. As a result, in comparison to other existing methods of determining anisotropic thermal conductivity, our approach is directly applicable to pristine suspended structures without any electrodes, and requires much simpler device structures with much less fabrication complexities. We believe that this method can be effectively applied for investigating thermal conductivity of other isotropic or anisotropic 2D materials. Furthermore, techniques demonstrated in this study will also contribute to developing electronic and thermoelectric applications by utilizing anisotropic thermal conductivity of black P.

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Supporting Information Available The Supporting Information is available free of charge on the ACS Publications website. Mode Shapes of First Three Modes of Device #1; Frequency Scaling of Resonance Frequency with Thickness and Diameter; Theoretical Calculation of Absorption due to Laser Heating; Determination of Thermal Expansion Coefficient (α) of Black Phosphorus (P); Discussions on Thermal Boundary Contact Resistance; Simulated Temperature and Thermal Strain Profile due to Laser Heating; Comparison of OTMRS with Other Methods.

Acknowledgments: We thank Jaesung Lee and Hao Jia for helpful technical discussions. We acknowledge the support the National Science Foundation CAREER Award (ECCS #1454570). Part of the device fabrication was performed at the Cornell Nanoscale Science and Technology Facility (CNF), a member of the National Nanotechnology Infrastructure Network (NNIN), supported by the National Science Foundation (ECCS #0335765).

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