Pulsed Photothermal Heating of One-Dimensional Nanostructures

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Pulsed Photothermal Heating of One-Dimensional Nanostructures Paden B. Roder, Sandeep Manandhar, Arun Devaraj, Daniel E. Perea, E. James Davis, and Peter J. Pauzauskie J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b04592 • Publication Date (Web): 24 Aug 2016 Downloaded from http://pubs.acs.org on September 7, 2016

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The Journal of Physical Chemistry

Pulsed Photothermal Heating of One-Dimensional Nanostructures Paden B. Roder,1 Sandeep Mananadhar,1, 2 Arun Devaraj,2 Daniel E. Perea,2 E. James Davis,3 and Peter J. Pauzauskie1, 4, ∗ 1

Department of Materials Science and Engineering, University of Washington, Seattle USA 98195 2

Environmental Molecular Sciences Laboratory,

Pacific Northwest National Laboratory, Richland, WA 99354 3

Department of Chemical Engineering,

University of Washington, Seattle, WA 98195 4

Fundamental Computational Sciences Directorate,

Pacific Northwest National Laboratory, Richland, WA 99354 (Dated: August 22, 2016)

1

ACS Paragon Plus Environment


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Abstract Pulsed lasers are used in tandem with one-dimensional nanostructures in a wide range of contemporary physical chemistry experiments, including four-dimensional transmission electron microscopy, scanning probe microscopy, and laser-assisted atom-probe tomography (APT). In this work, closed-form solutions for the pulsed photothermal heating of one-dimensional nanomaterials are compared with experimental time-of-flight APT ion spectra from both amorphous and crystalline silicon targets. Analytical results are given for targets with either a uniform cylindrical morphology or an arbitrary degree of conical tapering. Counterintuitively, increasing a conical specimen’s taper-angle is shown to lead to increases in the maximum temperature reached at the tip of the specimen. In particular, the heat source for tapered targets is affected by internal morphology-dependent cavity resonances that increase the maximum tip temperature relative to an untapered cylindrical structure. Experimental time-of-flight ion spectra for both crystallineand amorphous- silicon specimens are observed to agree with pulsed photothermal heating calculations. The results presented here will be of general use for quantifying photothermal heating in a wide range of experiments including tip-enhanced near-field scanning-probe microscopy, time-resolved electron microscopy, and also laser-assisted atom probe tomography. Keywords: Atom probe tomography, silicon nanowire, laser heating, photothermal, pulsed laser.

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I.

INTRODUCTION

Pulsed lasers are used in tandem with one-dimensional nanostructures in a wide range of modern physical chemistry experiments such as optical scanning-probe microscopy (i.e., nano-FTIR1 and tip-enhanced raman spectroscopy2 ), time-resolved transmission electron microscopy3 , and laser-assisted atom-probe tomography (LA-APT)4–6 . Although knowing a specimen’s temperature is central for interpreting experimental results, it is challenging to make direct thermal measurements of nanomaterials following pulsed photothermal heating. LA-APT plays an important role in the characterization of semiconductor nanostructures3,7 . This involves the use of both highvoltage and pulsed lasers to field-evaporate ions from the tip of needle-like specimens8 such as silicon nanowires9 . Use of a local electrode geometry decreases the applied biases needed for field evaporation and increases the analysis yield of non metallic specimens. In semiconductor applications the physical parameters of interest include growth rates, compositions, dopant concentrations and material geometries10 . Recently, it has been shown that the maximum temperature reached following pulsed laser excitation is crucial in interpreting atom probe tomography data11 . Numerous experimental and theoretical studies of the time-dependent tip temperature, temperature distribution, and cooling time for a voltage- or laser-pulsed field ion microscope specimen have been published. Numerical computations include those of Bunton et al.8 , Seidman and Scanlan12 , Liu and Tsong13 , Liu et al.14 , and Perea et al.9 . Most of the analyses use a onedimensional heat conduction model, usually assuming constant thermal properties (thermal conductivity, thermal diffusivity). Using a pump-probe scheme to scan the lattice temperature of a metallic tip after the laser pulse, Vurpillot et al.15 concluded that the one-dimensional model is valid, at least on the nanosecond time scale. Nanowire geometries are idea for atom-probe tomography measurements and modeling given their diameter, shank angle, and length can be controlled through fabrication. However, the earlier analyses of nanowire specimens9 do not take into account adequately the dependence of the thermal conductivity of the specimen on the size of the specimen, or the potential for increased tip heating related to morphology dependent resonances in tapered specimen geometries. Volz and 3



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Chen16 pointed out that the effective thermal conductivity of small structures is a function of both boundary and internal photon scattering and that the small cross section of nanowires leads to a significant reduction of the thermal conductivity compared to bulk material. They used a molecular dynamics method to explore the thermal conductivity of nanowires. Li et al.17 measured the thermal conductivity of silicon nanowires as a function of diameter and temperature, and Perea et al.9 used those results in an analysis of the tip temperature and cooling time of silicon nanowires. They did not model the time-dependent temperature distribution during the period of the pulse, but they numerically computed the decay of the tip temperature during the cooling period for silicon nanowires of uniform diameter and for tapered nanowires. Yang and Dames18 reported six models for the phonon thermal conductivity of silicon and compared the models with the data of Li et al. The theoretical framework is based on the bulk mean free path spectrum and thermal conductivity accumulation function using kinetic theory. Their simplest model, the gray model, is discussed and applied below. The purpose of this work is to present an analytical approach to predicting peak temperatures as well as heating and cooling rates of one-dimensional structures that are irradiated by a laser pulse. The analysis takes into account the size-dependency of thermal parameters and generalizes system variables such as specimen size, shape, and optical properties which provides a powerful tool for identifying the key parameters and variables that affect heat generation and dissipation in a system of interest. For the application of the theory we focus on silicon specimens with the aim of determining the effects of taper angle, thermal properties, and laser wavelength and polarization on peak specimen temperature and their heating and cooling rate. Comparison with previous numerical calculations as well as experimental data will allow us to elucidate the relative importance of the morphological, optical, and thermal parameters of the system. The remainder of the paper is organized as follows. In Sec. II we present the general theoretical model. The model is then applied to uniform radius wires in Sec. III and needle specimens with tapered radii in Sec. IV, where radius-dependent thermal properties are appropriately treated. Simulations for crystalline silicon and amorphous silicon nanowires that show the dependencies of taper angle and thermal properties on peak specimen temperature and cooling rates are detailed 4


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in Sec. V. The model is then used to simulate and comment on LA-APT experimental data which is presented and discussed in Sec. VI. Finally, Sec. VII summarizes the main conclusions and points out some potential future applications.

II.

THE HEAT CONDUCTION MODEL

FIG. 1. Configuration of a tapered wire irradiated by a focused pulsed laser.

Consider the tapered wire with the geometry shown in Fig. 1. Although the beam axis is often shifted by a small angle from the normal to the sample axis, that angle is small. Here, we shall consider the beam axis to be 7◦ from normal to the specimen axis with the center of the beam intersecting the tip of the specimen. For a Gaussian beam having irradiance I0 at the centerline the z-dependent irradiance is given by 2z 2 I(z) = I0 exp − 2 , w0

(1)

in which w0 is the beam waist, and z is the axial distance from the specimen’s tip. The centerline irradiance is related to the beam power by 1 P0 = πI0 w02 . 2 5


(2)


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For one-dimensional heat conduction, neglecting radiation heat transfer to or from the surfaces, the energy equation for the period during the pulse is given by ∂T 1 ∂ S(z) 2 ∂T ρCp = 2 κr + 2 , ∂t r ∂z ∂z r

(3)

where ρ is the density, Cp is the heat capacity, κ is the thermal conductivity of the wire, and S(z) represents heat-source per unit length within the sample with units of heat-source function (with units of

W . m

The related volumetric

W ) m3

for a Gaussian beam is 2z 2 S(z) 2Qabs I0 exp − 2 . = r2 πr w0

(4)

After the pulse S(z) = 0. For a tapered wire the absorption efficiency factor, Qabs , is a function of z since the radius changes with z. To obtain the local Qabs we shall apply Mie theory for a long rod of constant radius illuminated by a plane wave at an incidence angle of 7◦ to approximate Qabs at each axial position, that is, we model a conical rod as a stack of cylinders with varying radii as did Bodganowicz et al.19,20 . We must also make some assumption about the polarization of the laser beam to compute Qabs , but we first examine the special case of a wire with uniform diameter. A more rigorous approach to the determination of Qabs and the internal heat source is that used by Roder et al.21 who applied the MIT Electromagnetic Equation Propagation (MEEP) package22 to compute the internal source for cylindrical wires. Likewise, Koelling et al.23 used a two-dimensional simulation of optical absorption by a tapered object using a finite-difference time-domain numerical method.

III.

UNIFORM RADIUS NANOWIRES

In the limit of low Biot-number both radial and angular temperature gradients are negligible, and for a wire with r = r0 = constant, the energy equation then reduces to: 2z 2 ∂T ∂ ∂T 2Qabs ρCp = κ + I0 exp − 2 . ∂t ∂z ∂z πr0 w0

(5)

It is important to note that the thermal diffusivity, α = κ/ρCp , is temperature dependent as both κ and Cp have a strong temperature dependence. As we shall show, the choice of which temperature 6


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we select for the thermal properties affects the time scale, which involves the thermal diffusivity, and the maximum temperature, which depends on the thermal conductivity. Consequently, it is convenient to approximate the thermal properties as constant evaluated at the mean temperature of the irradiated wire at the end of the laser pulse. Thus, for constant α we introduce nondimensional variables and parameters to give ∂ 2Θ ∂Θ = + σ(ζ). ∂τ ∂ζ 2

(6)

The nondimensional terms are given by z α T − Tb , τ = 2 t, ζ = , Tb L L 2 2 2Qabs I0 2L ζ L2 σ(ζ) = exp − 2 · πr0 w0 κTb Θ=

(7)

The boundary conditions are Θ(ζ1 , τ ) = 0,

∂Θ (ζ0 , τ ) = 0. ∂ζ

(8)

Using the classical method of a product solution, we assume a solution of the form Θ(ζ, τ ) =

∞ X

Fn (τ )Gn (ζ).

(9)

n=1

The orthonormal eigenfunctions are given by Gn (ζ) = =

√ √

2 cos(γn ζ) h π i 2 cos (2n − 1) ζ , n = 1, 2, 3, ... 2

(10)

Substituting the assumed solution in the energy equation and applying orthogonality, one obtains dFn (τ ) + γn2 Fn (τ ) dτZ 1 √ = 2 σ(ζ 0 ) cos(γn ζ 0 )dζ 0 .

Ψn =

(11)

0

The integral Ψn is readily obtained using Matlab or any appropriate numerical method, and we note that it involves the absorption coefficient. 7



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If the irradiance, I0 , is constant during the pulse, Eq. (11) can be integrated to yield the solution 2 ∞ 1 − e−γn τ √ X Θ(ζ, τ ) = 2 Ψn cos(γn ζ) (12) γn2 n=1 for τ ≤ ∆τ , in which ∆τ is the dimensionless pulse width defined by ∆τ = α∆t/L2 . The solution for the cooling period becomes Θ∗ (ζ, τ ) =

√

2

∞ X

2

e−γn (τ −∆τ )

1−e

2 ∆τ −γn

γn2

n=1

Ψn cos(γn ζ)

(13)

for τ > ∆τ .

IV.

TAPERED SPECIMENS

The analysis of tapered specimens is more complicated because the thermal conductivity depends on the radius as well as the temperature and the absorption coefficient also depends on the radius, but some approximations can be made to facilitate a solution. To account for the dependence on radius, thermal conductivity data in the range of sizes and temperatures of interest here can be fitted to an equation of the form κ = κ0

r r0

m ,

in which ‘m’ is a constant. Therefore, the energy equation becomes α0 ∂ S(z) ∂T 2+m ∂T = m 2 r + ∂t r0 r ∂z ∂z ρCp

(14)

(15)

in which αo = κo /ρ Cp . The silicon data of Li et al.17 for nanowires show that the thermal conductivity is a weak function of temperature. Consequently, we can consider the constant κo to be independent of temperature. This is consistent with the gray model for nanostructures’ thermal conductivity reported by Yang and Dames18 , which has the form κnano = κbulk · (1 + Kn)−1 8


(16)

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where the Kundsen number is defined by Kn= Λbulk /D in which Λbulk is the mean free path (MFP) of the energy carrier (e.g. phonons, electrons, etc.) in bulk material, and ‘D’ is the nanowire diameter. The MFPs for silicon nanowires can be estimated by applying the data of Li et al.17 . Table I below presents the MFPs calculated using Eq.16 and the data of Li et al.. TABLE I. Mean free paths calculated for silicon nanowires. D, nm

Λnano (70K), nm Λnano (100K), nm Λnano (150K), nm Λnano (200K), nm Λnano (250K), nm

37

4596

2562

783

481

363

56

3375

1910

736

449

344

115

4186

2446

921

557

420

Λaverage (T)=

4053

2306

813

496

376

Using average MFPs at each temperature to calculate κ for various nanowire diameters, we obtain the results shown in Fig.2. Also shown are the average values at each temperature from the data of Li et al.. The gray model of Yang and Dames is in good agreement with Eq.14 with m=1 and κo =10.73 W/m·K. For the geometry shown in Fig. 1 the radius is related to the axial position by r = r0 + z tan(θ). Using Eq. (17) and applying the chain rule for derivatives to eliminate z, we obtain ∂T S(r) α0 tan2 (θ) ∂ 2+m ∂T r + = . m 2 ∂t r0 r ∂r ∂r ρCp

(17)

(18)

In this case we can introduce dimensionless variable and parameters defined by Θ=

r α0 tan2 (θ) T − Tb ,ξ = ,τ = t, Tb r0 r02

(19)

where α0 is an effective thermal diffusivity given by α0 = κ0 /ρCp . Using Eq. (18) the energy equation transforms to ∂Θ 1 ∂ = 2 ∂τ ξ ∂ξ

2+m ∂Θ ξ + σ ∗ (ξ), ∂ξ 9


(20)


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FIG. 2. Calculated and experimental thermal conductivities of Si nanowires. Black circles are experimentally measured thermal conductivities17 at fixed diameters between 70K and 250K. Blue squares show the average thermal conductivity measured between 70K and 250K for specific diameters. The dotted line shows the predictions using the gray model18 based on the mean free paths in TABLE I, which yields κ0 = 10.7 W/m·K and m = 1 in Eq.14.

where the dimensionless source function is ( 2 ) 2 2r I Q (ξ) r (ξ − 1) 0 abs 0 0 σ ∗ (ξ) = . exp −2 πα0 Tb ξ tan2 (θ) w0 tan(θ)

(21)

Again we apply a product solution of the form Θ(ξ, τ ) =

∞ X

fn (τ )gn (ξ).

n=1

10


(22)

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The eigenfunctions satisfy the Sturm-Liouville problem d 2+m dgn ξ + λ2n ξ 2 gn = 0, dξ dξ

(23)

with boundary conditions dgn (ξ0 ) = 0, gn (ξ1 ) = 0, dξ

(24)

where λn are the real eigenvalues, ξ0 = 1, and ξ1 = 1+(L/r0 )tan(θ). The eigenfunctions that satisfy this problem are given by 2λn 2−m − m+1 2 2 gn (ξ) = ξ Zp , (25) ξ 2−m p where p = − (m + 1)2 /(m − 2), and Zp (x) takes on one of two forms depending on the value of p. For p equal to 0 or a positive integer, Zp (x) = AJp (x) + BYp (x).

(26)

Zp (x) = AJp (x) + BJ−p (x).

(27)

For all other values of p,

Restricting our analysis to the former case (p=0,1,2,...), Eq. (25) becomes m+1

gn (ξ) = ξ − 2 2λn 2−m 2λn 2−m × Jp ξ 2 + Bn Yp ξ 2 , 2−m 2−m in which the boundary condition at the base yields 2−m 2λn ξ1 2 Jp 2−m Bn = − 2−m . 2λn Yp 2−m ξ1 2

(28)

(29)

Defining the norms of the eigenfunctions as 2

Z

ξ1

kgn k =

ξ 02 gn2 (ξ 0 )dξ 0 ,

ξ0

11


(30)


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the orthonormal eigenfunctions are defined by gn (ξ) . kgn k

(31)

# 2 0 2 (ξ − 1) 2r ξ 0 Qabs (ξ 0 ) exp − 02 un (ξ 0 )dξ 0 , w0 tan2 (θ)

(32)

un (ξ) = Given that Ψn is defined as Ψn = Z

2r02 I0 πα0 Tb tan2 (θ)

ξ1

× ξ0

"

the solution for the heating period (τ ≤ ∆τ ) becomes −λ2n τ ∞ X 1−e Θ(ξ, τ ) = Ψn un (ξ). λ2n n=1 For the cooling period (τ ≥ ∆τ ) the solution is given by −λ2n ∆τ ∞ 1−e X 2 Θ∗ (ξ, τ ) = e−λn (τ −∆τ ) Ψn un (ξ). λ2n n=1 V.

(33)

(34)

SIMULATIONS A.

Crystalline Silicon Nanowires

In the analysis presented above we have considered the effective thermal diffusivity, α0 , to be constant having some mean value in the temperature range encountered. To examine the effect of such an assumption we compared our results with those of Perea et al.9 who used a numerical solution of the energy equation to take into account the temperature and size dependence of the thermal conductivity. They assumed an initial condition of the temperature distribution for the cooling period to be Gaussian, which, as we show below, is questionable for tapered samples. The parameters used by Perea et al. and the various system parameters used in this study are presented in Table II. They used a pulsed laser with a wavelength of 532 nm, a pulse frequency of 100 kHz, a pulse width of 10 ps, and a pulse energy ranging from 20 to 100 pJ (2 ≤ P0 ≤ 10 W). The 12


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representative silicon nanowire specimen had a tip diameter of 30 nm (r0 = 15 nm) and a length, L, of 10 µm. The Gaussian beam was reported to have a spot size of 1 µm at FWHM. For these conditions one obtains w0 = 0.849 µm, and centerline irradiances from 1.765x1012 to 8.826x1012 W/m2 . The taper angle varied from 0◦ to 4◦ .

TABLE II. Parameters and properties used for the calculations. Laser wavelength, nm

532 nm

Laser angle of incidence

7◦

Pulse frequency

100 kHz

Pulse width

10 ps

Centerline irradiances of laser beam 1.765 x 1012 to 8.826 x 1012 W/m2 Nanowire tip diameter

30 nm

Nanowire length

10 µm

Base temperature

70 K

Crystalline silicon refractive index24

4.136 + 0.0102058i

Taper angle

0◦ to 4◦

Amorphous silicon refractive index25

4.4272 + 0.87758i

κo in Eq.14

10.7 W/m·K

Absorption efficiency factor, Qabs

0.1 - 2.4

For temperatures greater than about 70 K the temperature-dependent values of κ0 scatter about κ0 = 10.7 W/m·K. For m = 1 in Eq.14, the eigenfunctions gn (ξ) for the tapered wire given by Eq. (28) become gn (ξ) = ξ where

−1

h p p i J2 2λn ξ + Bn Y2 2λn ξ ,

(35)

√ J2 2λn ξ1 √ . Bn = − Y2 2λn ξ1

(36)

13



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From the boundary condition at the tip, the eigenvalues satisfy the transcendental equation h p p i 0 = 2 J2 2λn ξ0 + Bn Y2 2λn ξ0 p h p p i − λn ξ0 J1 2λn ξ0 + Bn Y1 2λn ξ0 .

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(37)

Table III in the appendix lists the first few eigenvalues and norms for nanowire tapers of 1◦ , 2◦ , and 4◦ . Furthermore, we note that the integral in Eq. (32) involves the radius-dependent absorption coefficient. Since the radius varies substantially with a tapered wire we need to consider how Qabs changes over the length of the wire, which is presented in Fig. 3a for a beam that is polarized parallel with the wire axis (TM) and with a wavelength of 532 nm. The taper angle has a large effect on Qabs , and morphology-dependent resonances are seen in the absorption profiles for angles greater than 1◦ (for diameters greater than 60 nm). Since the resonances are superposed on the irradiance of the Gaussian beam, the heating is highly non-uniform; a result also demonstrated in the simulations of Koelling et al.23 and which appear in the scanning electron micrographs of Kelly et al.26 . This effect can be understood by considering the corresponding dimensionless source functions defined by Eq. (21), presented in Fig. 3b. To help qualitatively compare the heating source profiles, the dimensionless source functions in Fig. 3b have been normalized to the wire tip value. For the wire with uniform diameter the source function has the expected Gaussian profile, but significant distortion of that profile arises for tapered specimens. In the computations of the source function we assumed the laser power to be 5.8W so that the tip temperature at the end of the pulse matched the value of 250K assumed by Perea et al.9 for the wire with uniform diameter. The axial temperature distribution for the wire with r0 = 15 nm has the Gaussian profile assumed by Perea et al., but that is not the case for the tapered wires as shown in Fig. 4. Consequently, the initial condition of a Gaussian temperature distribution used in their computations for the tapered wires is not appropriate. Furthermore, if the same laser power is used for uniform and tapered wires, the tip temperature at the end of the laser pulse varies with the taper angle. In some cases, increasing the taper angle leads to a counterintuitive increase in the maximum tip temperature due to morphology dependent diameter resonances near the tip of the specimen as shown in Fig. 4b. 14


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FIG. 3. a, Absorption efficiency Qabs of crystalline silicon nanowires with taper angles of θ = 0◦ (black), 1◦ (blue), 2◦ (green), and 4◦ (red) in vacuum irradiated with a 532nm wavelength TM-polarized laser at 7◦ incidence. b, Dimensionless source functions (Eq. (21), normalized to the wire tip value) showing Qabs resonances superposed with the irradiance of the Gaussian beam.

The tip temperature calculated for uniform diameter and tapered silicon wires during the heating and cooling periods are presented in Fig. 4. For the tapered wires the tip temperature falls to the base temperature of 70K within 100 ns, but that for the uniform diameter wire takes an order of magnitude longer to decay to the base temperature. In all cases, however, the base temperature is reached well before the next pulse occurs for a pulse frequency of 100 kHz (10 µs period). The results calculated of our analysis are compared with the numerical results of Perea et al.9 15



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FIG. 4. a, Results for the axial temperature distributions taken at the end of the 5.8W laser pulse for silicon nanowires with taper angles of θ = 0◦ (black), 1◦ (blue), 2◦ (green), and 4◦ (red). b, Corresponding temperatures at the nanowire tip for the heating and cooling periods for the various wire shapes.

for the identical system in Fig. 5. We predict larger rates of cooling even for the uniform diameter wire, which can be explained by how the appropriate choice of temperature-dependent thermal parameters affects the cooling time scale defined by Eq. (19). In the numerical computations of Perea and co-workers the temperature dependence of nanowire thermal conductivity was neglected due to the fact it showed little variation in the temperature range of interest in their study (70K to 250K). The faster cooling rates observed in the present analysis suggest that their prior choice of thermal diffusivity was lower than that in the simulations discussed here. 16


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FIG. 5. Comparison between our results (solid lines) and the results of Perea et al.9 (dashed lines) of the temperatures at the silicon nanowire tip for the cooling periods for taper angles of θ = 0◦ (black) and 4◦ (red).

One cannot expect the temperatures for tapered wires from Perea et al. to agree with our results because of the effects of morphology-dependent resonances (MDRs) on the initial conditions used in the computations, but the decay times show similar trends. It is clear from Fig. 4a that the initial temperatures used for the cooling period are not Gaussian as assumed by Perea et al.

B.

Amorphous Silicon

We also simulated heating and cooling of amorphous silicon nanowires. There is much greater uncertainty about the thermal properties of amorphous Si (a-Si and a-Si:H) than there is for crystalline Si as indicated by the review by Cahill et al.25 and the data of Zink et al.27 . In the absence of data on nanowires of a-Si, and particularly on the size dependence, we have taken m = 0 and κ = κ0 = 1 W/m·K in Eq. (14) for the heating and cooling calculations for amorphous nanowires. We used a complex refractive index of n = 4.4272 + 0.87758i24 and heat capacity data reported by Zink et al.27 , using the mean value of α0 as discussed above. In this case, the eigenfunctions gn (ξ) 17



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for the tapered wire become i 1 h gn (ξ) = √ J 1 (λn ξ) + Bn J− 1 (λn ξ) , 2 ξ 2

(38)

where Bn = −

J 1 (λn ξ1 ) 2

J− 1 (λn ξ1 )

.

(39)

2

Since J1/2 (z) = (2/πz)1/2 sin(z) and J−1/2 (z) = (2/πz)1/2 cos(z), the boundary condition at the wire tip can be written in terms of trigonometric functions to yield the following transcendental equation that the eigenvalues must satisfy

0 = sin (λn ξ0 ) + Bn cos (λn ξ0 ) − λn ξ0 [cos (λn ξ0 ) − Bn sin (λn ξ0 )] .

(40)

Table IV in the appendix lists the first few eigenvalues and norms for nanowire tapers of 1◦ , 2◦ , and 4◦ . The low thermal conductivity of amorphous silicon (κ0 ≈ 1 W/m·K) and large Qabs (Fig. 6a) lead to much higher tip temperatures at the end of a pulse compared with geometrically similar crystalline nanowires. For example, for a 30 nm diameter nanowire with the length and base temperature considered above for a crystalline nanowire the laser power is only 0.23 W to reach a tip temperature of 250K at the end of a pulse compared with 5.8 W for a crystalline sample. The dimensionless source functions for uniform and tapered amorphous silicon nanowires show comparable trends to those discussed for the crystalline case and are presented in Fig. 6, resulting in nanowire temperature distributions shown in Fig. 7a. The time required for the tip temperature to decay to the base temperature of 70K (Fig. 7b) in this case is more than an order of magnitude greater than for the crystalline material and is close to the start of the next laser pulse, designated by the arrow labeled fp in Fig. 7b. 18


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FIG. 6. a, Absorption efficiency Qabs of amorphous silicon nanowires with taper angles of θ = 0◦ (black), 1◦ (blue), 2◦ (green), and 4◦ (red) in vacuum irradiated with a 532nm wavelength TM-polarized laser at 7◦ incidence. b, Corresponding dimensionless source functions (Eq. (21), normalized to the wire tip value) showing Qabs resonances superposed with the irradiance of the Gaussian beam. VI.

EXPERIMENTS

To explore the effects of thermal properties on the evaporation of ions using laser-assisted atom probe tomography (LA-APT) we performed experiments on single tapered specimens of crystalline and amorphous silicon. The CAMECA LEAP 4000XHR LA-APT system was used for these experiments. A TM-polarized, focused / pulsed UV laser (355nm wavelength) with a spot size of 1-5 microns, a 100 (amorphous silicon) or 200 (crystalline silicon) kHz pulse repetition 19



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FIG. 7. a, Results for the axial temperature distributions taken at the end of the 0.23W laser pulse for amorphous silicon nanowires with taper angles of θ = 0◦ (black), 1◦ (blue), 2◦ (green), and 4◦ (red). b, Corresponding temperatures at the nanowire tip for the heating and cooling periods for θ = 0◦ (solid black) and 4◦ (solid red) taper angles. For comparison, results for the corresponding taper angles of crystalline silicon nanowires for θ = 0◦ (dashed black) and 4◦ (dashed red) show cooling rates orders of magnitudes larger than amorphous silicon nanowires (solid lines), which approach cooling times of the pulse frequency interval fp .

rate, and 20pJ (2W) laser pulse was used in addition to a standing DC voltage to result in field evaporation of the specimen. The analysis was conducted in ultra-high vacuum (UHV) system (1.4x10−11 Torr) with a cryogenically cooled specimen maintained at Tb = 44.2K, and the applied 20


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DC voltage was controlled to maintain a uniform evaporation rate of 0.005 atoms/pulse throughout the length of the experiment. For the silicon nanowire specimens the tip radius, r0 , was measured to be 12nm, and the shank angles were θ=9.6◦ for the crystalline sample and θ=12.2◦ for the amorphous silicon specimen. For the heat transfer calculations the pulse width was taken to be 1 ps, and the lengths to be 10 µm. The Gaussian beam was considered to have a spot size of 1 µm at FWHM, resulting in a beam waist of w0 = 0.849 µm. The absorption efficiency Qabs was calculated using the 355 nm wavelength at an incidence angle of 7◦ with complex refractive indices for the silicon and amorphous silicon specimens given by n = 5.657 + 3.0186i and n = 3.7414 + 2.7813i, respectively24 . During an APT experiment, the specimen is held at cryogenic temperatures (typically 20-60 K) while applying a standing voltage. This standing voltage creates a (thermal) barrier for field evaporation, the height of which scales with the field strength. It is not accurate then to consider a critical temperature for ion emission as the necessary thermal energy to overcome the barrier for field evaporation depends on the applied field. Moreover, the field evaporation rate constant is known to follow an Arrhenius-type relation with an exponential temperature dependence. This means that field evaporation takes place only near the peak of the temperature pulse imparted from the laser. In this context, both the peak temperature and an effective operating temperature are of interest, the former being central to this study. Ion mass-to-charge state spectra (for

28

Si2+ -

30

Si2+ ) and calculated tip temperatures for the

crystalline and amorphous samples are shown in Fig. 8. As the ion mass-to-charge spectra is generated by a time-of-flight detection technique, the mass-to-charge counts generated by the detector depends on time elapsed between the laser pulse and the detection event9 . Therefore, the longer the nanowire persists at elevated temperatures after the initial laser pulse, the more spread in the ion mass-to-charge spectra is to be expected. As shown in Fig. 8, the increased spread in the ion mass-to-charge state spectrum of the amorphous silicon specimen can be explained by a slower cooling rate and thereby prolonged time at higher temperatures. Although the FWHM in Fig. 8b appears to be comparable between amorphous and crystalline specimens, the data are plotted versus the logarithm of time, and the amorphous specimen takes two orders of magnitude longer to 21



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FIG. 8. a, Time-of-flight mass-to-charge state spectrum of the 28 Si2+ - 30 Si2+ ions from the silicon nanowire (black) and amorphous silicon nanowire (red). b, Corresponding predicted temperatures at the nanowire tip of the heating and cooling periods for the experimental TM polarization (solid lines) and example TE polarization (dashed lines), showing the strong peak temperature dependence on polarization. The increased width of the amorphous silicon nanowire’s Si2+ peaks are attributed to the decreased cooling rate of the nanowire.

return to the base temperature. The maximum temperature that is predicted to be reached by the crystalline silicon specimen under TM polarization in 8b is 1618 K (1345◦ C) which is below the melting point of crystalline silicon (1414◦ C). This prediction agrees with the presence of poles in detector event histograms 22


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for the crystalline sample (SI Figure 1b), suggesting that the crystalline silicon specimen is not amorphized after repeated laser pulses28–30 . Similarly, for the amorphous silicon specimen the absence of poles in APT detector event histograms indicates that the amorphous sample is not crystallized after repeated laser pulses (SI Figure 1a). Additionally, the calculated temperature evolution of the crystalline silicon specimen under TE polarization (Fig.8b) predicts that the temperature at the tip of the specimen actually continues to increase following the end of the laser pulse. This counter-intuitive prediction is due to the diffusion of heat toward the tip of the specimen from a nearby MDR, and can be considered another non-trivial impact of MDRs that can be predicted using the model derived here.

VII.

DISCUSSION OF RESULTS

The use of Eq.(14) for the dependence of the thermal conductivity on size appears to be adequate for the data of Li et al.17 for crystalline silicon nanowires with radii greater than 25 nm, but it is not accurate for the smallest wires studied. It overpredicts κ for r < 25 nm. Furthermore, we note in Sec. V B that there is uncertainty about the thermal properties of amorphous Si reported by Cahill et al.25 and the data of Zink et al.27 . The results of these investigators are for thin films of a-Si deposited on substrates such as MgO or Si25 . There is considerable scatter in the thermal conductivity data of various investigators for film thicknesses ranging from 0.33 µm to 50 µm near room temperature. Earlier data of Cahill et al.31 show a plateau in the thermal conductivity versus temperature for T > 50 K with κ ∼ = 1.8 W/m·K whereas the temperature-dependent thermal conductivity reported by Pompe and Hegenbarth32 show κ varying from about 0.4 to 1.8 W/m·K in the temperature range 10 K to 120 K. The thin film κ data for amorphous Si presented in Figs. 5 and 6 of Cahill et al.25 and Fig. 1 of Zink et al.27 fall in the range 0.5 to 2 W/m·K. In this case, it is a reasonable assumption that the thermal conductivity has a relatively weak dependence on the radius. Another important consideration for our analysis is the treatment of thermal parameters as independent of temperature. As noted in Sec. III, the thermal diffusivity is a temperature dependent 23



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parameter as both κ and Cp typically have strong temperature dependencies. However, obtaining an analytical solution to Eq. (3) necessitates that the thermal parameters be independent of temperature. To account for this, the analysis for the temperature evolution of the laser pulsed system was iterated so that the constant temperature thermal properties are updated to better approximate the mean nanowire temperature at the end of the laser pulse in the irradiated region. This technique helps to mitigate the assumption of constant temperature thermal parameters, but the approximation becomes less appropriate as large thermal gradients are generated in the irradiated region. The peak temperature of tapered wires is affected by MDRs as well as the polarization of the pulsed laser since Qabs depends on the polarization. Figure 8b illustrates the sensitivity of the incident polarization on the resulting tip temperature of the specimen. Figure 9 shows the sensitivity of Qabs to the polarization, and it varies by multiple orders of magnitude over the length of a wire. Therefore, the choice of the laser polarization plays a key role in modulating the specimen temperature during experiments.

FIG. 9. Absorption efficiency Qabs diameter dependence of a silicon nanowire (black) and amorphous silicon nanowire (red) irradiated with a 355nm wavelength 7◦ from normal with TM (solid lines) and TE (dashed lines) polarizations.

24


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VIII.

CONCLUSION

An analytical model of pulsed laser heating of uniform and tapered nanowires has been presented, which takes into account size-dependent thermal parameters. The interplay between the specimen size, taper angle, thermal properties, and laser wavelength and polarization has been investigated by taking silicon nanowires as a representative example. We observe that the taper angle, the size-dependence of the thermal conductivity, and the choice of temperature that constant thermal parameters are evaluated at become increasingly important when determining the rate of heat generation and dissipation. Furthermore, we find that internal temperature distributions and peak specimen temperatures are highly sensitive to the incident laser polarization and subsequent MDRs. By comparing our simulation results with experimental data, we see that these effects on peak temperatures and cooling rates correspond to broadening in the produced ion mass-to-charge spectra. The results can thereby be used to predict the critical time it takes a pulsed-laser-heated specimen to cool to its base temperature prior to the next pulse, thereby informing ideal APT experimental parameters so that minimal fields for ion evaporation and maximum pulse rates can be attained. Concerning alternative applications, another interesting extension would be to comment on experimental parameters needed to induce phase changes in laser pulsed nanowires4 using ultrafast pumps5,6 . Likewise, this analysis could be used to explore heat generation and thermal effects of tip-enhanced spectroscopy for techniques such as nanoFTIR1 and tip-enhanced raman spectroscopy (TERS)2 , and time-resolved transmission electron microscopy3 .

ACKNOWLEDGMENTS

This work was supported by the Air Force Office of Scientific Research Young Investigator Award, Contract #FA95501210400 as well as by start-up funding from the University of Washington. PBR acknowledges NSF-GRFP support (#DGE-1256082) and PJP acknowledges support from an NSF-CAREER award (#1555007). The APT experiment was conducted in the William 25



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R. Wiley Environmental Molecular Sciences Laboratory (EMSL), a national scientific user facility sponsored by DOEs Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory (PNNL). The PNNL is a multiprogram national laboratory operated for Department of Energy by Battelle under Contract No. DE-AC05-76RL01830. AD would like to acknowledge EMSL directors discretionary funding for supporting the experimental APT work. PJP and SM acknowledge support from the Materials Synthesis and Simulation across Scales (MS3) Initiative and the PNNL’s Laboratory Directed Research and Development program.

IX.

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∗

Corresponding author: [email protected]

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Absorption in Conical Silicon Particles. Opt. Express 2013, 21, 3891–3896 20

Bogdanowicz, J.; Gilbert, M.; Koelling, S.; Vandervorst, W., Impact of the Apex of an Elongated Dielectric Tip Upon its Light Absorption Properties. Appl. Surf. Sci. 2014, 302, 223–225

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X.

TOC FIGURE

29



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Page 30 of 31

Appendix: Simulation Details

To help inform the reader, a partial list of the first 5 eigenvalues and norms have been included in tables III and IV. Videos of time-dependent cooling of simulations performed in this study can be found online in supporting information. TABLE III. A partial list of eigenvalues and norms used for silicon nanowire simulations presented in Sec. V A. n 1 θ=1◦

2

3

4

5

λn 0.5820 1.0203 1.4793 1.9592 2.4529 ||gn || 1.5458 1.1512 1.0130 1.3339 525.3549

θ=2◦

λn 0.3769 0.6558 0.9358 1.2217 1.5137 ||gn || 2.4108 1.8313 1.4969 1.2838 1.1883

θ=4◦

λn 0.2497 0.4329 0.6141 0.7958 0.9790 ||gn || 3.6423 2.8014 2.3457 2.0387 1.8119

30


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TABLE IV. A partial list of eigenvalues and norms used for amorphous silicon nanowire simulations presented in Sec. V B. n 1 θ=1◦

λn

2

3

4

5

0.2490 0.5001 0.7544 1.0119 1.2721

||gn || 4.0100 2.8154 2.2891 2.0020 1.8583 θ=2◦

λn

0.1294 0.2590 0.3889 0.5192 0.6500

||gn || 7.7252 5.4556 4.4466 3.8444 3.4365 θ=4◦

λn

0.0660 0.1320 0.1980 0.2640 0.3301

||gn || 15.1563 10.7152 8.7464 7.5718 6.7695

31


Pulsed Photothermal Heating of One-Dimensional Nanostructures

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