Creating micro/nanoscale systems with one dimension substantially larger than the others is relatively straightforward, and convenient for studying thermal energy transport. This paper reports the theory for a novel noncontact micro/nanoscale temperature measurement technique named Suspended ThermoReflectance (STR). Here, the heat diffusion equation is applied to a cantilever beam of micro and nanoscale dimensions with Neumann and Dirichlet Boundary Conditions. The Neumann condition takes on constant, decaying, and periodic forms leading to steady-state, transient and harmonic solutions, respectively. Though general solutions are presented for multiple length to width ratios (L/w) in terms of geometry as well as thermal properties, silicon is studied due to its continued technological importance. The analytical solutions are compared to a 3D Finite Element Model to determine at what L/w ratio the analytical solutions accurately represent a 3D microcantilever beam. Thermal conductivity is determined using the steady-state solution. The transient solution yields the overall thermal diffusivity of the system irrespective of any frequency change, while the harmonic solution provides the phase difference along the length of the cantilever beam leading to a thermal diffusivity and the frequency dependent heat capacity. The analytical model can be used to analyze the aforementioned thermal properties for STR measurement at 4% accuracy for the system with length to width ratio of 20 and at 1% accuracy for length to width ratio of 100.

## INTRODUCTION

There is a constant thrust in the semiconductor industry to reduce device sizes to attain increased operational speed, reduced power consumption and to package devices into smaller envelopes. This move to increasingly smaller length scales has led the industry to feature sizes on the order of 7 nm,^{1,2} which leads to high power densities and concomitant, high temperatures. These concerns are further exacerbated by the fact that at lower length scales, e.g. at sub-micron level, the thermal conductivities of materials decrease.^{3} Contact methods for measurement of thermal properties are inviable at this length scale due to the observer effect, which has necessitated the development of non-contact methods for thermal measurements.

Non-contact methods have matured over this time and researchers have been able to measure thermal properties with high thermal and temporal resolution.^{4–6} It is very common to consider 1D heat transfer models for thin films in non-contact methods like Transient Thermoreflectance (TTR), Time Domain Thermoreflectance (TDTR) and Frequency Domain Thermoreflectance (FDTR) because of the simplicity in calculation. Moreover, it is assumed that the heat conduction in the lateral direction is several orders of magnitude larger than the penetration depth into the substrate in order to attribute a 1D model to TTR and TDTR.^{7–10} A 1D model is applied in FDTR assuming heat flows in the planar direction of the thin films.^{11,12} The previously mentioned models all have the assumption of a semi-infinite substrate and require complex heat transfer model to analyze thermal property in the multilayered structure.^{13} Thus, by definition, these systems do not contain the spatial constraints that many semiconductor devices utilize.^{14–16} Therefore, a new method was recently developed termed Suspended Thermoreflectance (STR).^{17,18} In this method a freely suspended μ-cantilever beam is thermally characterized. With STR the effects of confined geometries, e.g. thicknesses on the order of 100nm – 100 μm, can be tested to determine the influence of different heat carriers and their scattering mechanisms.

Heat carriers for conduction (electrons and phonons) have characteristic lengths and time constants, and when a system’s geometry or conditions become comparable to these characteristic values, the material’s thermal properties (thermal conductivity, κ, and heat capacity, C) change.^{19–26} Figure 1 shows the thermal conductivity of bulk silicon with geometry much larger than characteristic length^{27} and silicon thin films with geometries on the order of the characteristic length.^{28–30} It can be seen from the figure that the thermal conductivity reduces by a factor of 10 when transition occurs from bulk to thin film. These are two extreme cases where dimensional ratios are not comparable. The equations in this paper are developed using a continuum-based model that does not take into account quantum, atomic, or even ballistic effects. The heat equation is a diffusive model and thus deviation from this model is an indication that the aforementioned effects may be at play.

The 1D heat transfer model presented in this paper is directly applicable to measure thermal properties using the Suspended Thermoreflectance (STR) technique.^{17,18} Unlike previously used least-square techniques where analytical solutions are fitted to the experimental data by varying an independent variable like thermal conductivity to find its best value,^{8,9,12,31} this model can provide both thermal conductivity and heat capacity if the temperature of two or more points along the length of the cantilever is known. The model in this paper is developed based on a few assumptions that come directly from experimental conditions. Firstly, the beam is under vacuum to a level that makes convection negligible. Secondly, the beam experiences a small increase in temperature, ΔT≤10K, making radiative effects negligible.^{11} Thirdly, when temperature changes are small, temperature-dependent physical properties of the material are assumed to remain constant. Lastly, heat carriers are in local thermal equilibrium, and are explained adequately by diffusive transport model.^{32,33}

The heat transfer behavior of the micro/nano-cantilevers examined in this work is theoretically divided into three different regimes. In the first regime, the beam has been heated by a constant heat flux at the free end and is operating at steady-state conditions. Analysis of this state leads to the thermal conductivity of the material. In the second regime, the cantilever is initially at the same temperature as the fixed end and is then instantaneously heated at its tip. As its temperature increases, it exists in a transient state until enough time has passed, after which it is identical to the steady-state regime. Analysis of this transient regime leads to the thermal diffusivity of the system, which, in conjunction with the thermal conductivity from the steady-state results, leads to the heat capacity of the material. In the third regime, the heat flux has a sinusoidal oscillation superimposed at its tip and the system reacts harmonically. The harmonic analysis yields a frequency dependent heat capacity.

The 1D model developed from the heat equation leads to the analytical determination of the time constant of the cantilever and thermal penetration depth. The model is then compared to a 3D Finite Element Model (FEM) in two ways. First, the FEM is setup with exactly the same geometry, boundary, and initial conditions as the 1D analytical model. Results match within 0.6% for different length to width ratios varying from 5 to 100, thus verifying the 1D analytical derivation. Second, in an effort to resemble actual experimental conditions used in STR,^{17,18} heat is applied to the cantilever perpendicular to its length. In STR, heat is applied by focusing a laser onto a cantilever. The smallest spot size to which a laser can be focused is limited by diffraction. Thus, for cantilever thicknesses less than the diffraction limited spot size, (appox. <1 μm) the laser must be focused perpendicular to the cantilever’s length. Therefore, the 3D FEM is used to understand the applicability and limitations of the 1D analytical model to STR. Finally, homogeneity of the temperature in the plane perpendicular to the heat flow in the cantilever is studied using FEM, which leads to the length where temperature becomes homogeneous and hence back side probing can represent the actual temperature of the plane.

## ONE-DIMENSIONAL HEAT TRANSFER MODEL

A schematic of a cantilever beam with heat flux at the tip is shown in Fig. 2. The physically fixed end of the beam is set at the origin, and the beam extends in the positive *x*-direction to its full length (*L*). The governing equation for 1D heat diffusion in this beam is

In this equation, temperature (*u*) is a function of both time (*t*) and position (*x*). The other variables are taken from the material properties of the cantilever: density (*ρ*), heat capacity (*C*), and the coefficient of thermal conductivity (κ). The three material constants are combined to form the coefficient of thermal diffusivity (*α*)

Boundary conditions are chosen to mimic those of the STR experiment.^{17} The fixed end is set to the temperature of the substrate (*T*_{sub}) for all time.

A harmonic flux is modulated on top of constant intensity (I) on the free end. The harmonic flux has both amplitude (*A*) and frequency (*ω*). For the modeled system, the amplitude of the oscillation cannot exceed the constant intensity.

The initial condition specifies that the beam is initially at the substrate temperature.

Nondimensionalization of terms and equations is performed to simplify the analysis and make the solutions more general. Distance, time and temperature are nondimensionalized in the following way.

The dimensionless boundary and initial conditions are then generated through application of Eqns. 7–10 to Eqns. 3, 4 and 6.

The dimensionless intensity ($\xce$), amplitude ($\xc2$), and frequency ($\omega ^$) are defined as follows.

To homogenize the boundary conditions, the temperature function is divided into two functions ($\Theta $ and *S*). Eqn. 12 is used as a boundary condition to solve S, leaving $\Theta $ with homogenous B.C. and initial condition of $\u2212x^\xce$.

Solving S and $\Theta $ with initial and boundary conditions with the help of Fourier transformation yields Eqn. 18 and 19.

where the eigenvalues $kn=n\u221212\pi ,\u2003n=1,2,3\u2026$.

Solutions for steady-state, transient and harmonic heat transfer for the cantilever in Fig. 2 can be obtained from Eqn. (17).

The steady-state condition is achieved by setting the frequency and amplitude to zero and letting time run to infinity.

The operating conditions for the transient regime include constant intensity, however the amplitude of the harmonic intensity should still be set to zero.

The first term represents the steady-state temperature while the second term is an exponential decay. So as time increases, the second term approaches zero and the temperature profile takes a steady-state form.

An important measure of the transient response duration is the time constant (*τ*). For this system, which has infinitely many decaying exponentials, the time constant is based on the slowest rate of decay which depends on the smallest eigenvalue.

When operating in the harmonic regime, sufficient time has passed for the transient response to die out, leaving the sinusoidal oscillations as the sole time-dependent effect.

The first line of this equation represents steady oscillation of temperature along the cantilever while the summation term contributes to the gradual change of temperature. Coupling frequency with the time constant yields an equation for thermal penetration depth (*L*_{p}), which has been used by researchers to interpret results.^{34,35}

## COMPARISON BETWEEN 1D ANALYTICAL MODEL AND 3D FEM WITH APPLICATION OF HEAT LATERALLY AT TIP

Solutions obtained from the previous section are compared with a 3D Finite Element Model that has exactly the same geometry, boundary, and initial conditions for verification. Silicon beams 50 μm wide, 50 μm thick and with length to width ratios (L/w) of 5-100 are taken into consideration. Choosing the same width and thickness has the advantage of utilizing symmetry and avoiding unnecessary width/thickness and length/thickness ratios. For steady-state and transient heat transfer analyses 0.705 mW heat is applied at $x^=1$, and $x^=0$ is kept at a constant temperature of 300K. For harmonic analysis 0.58 mW of heat is modulated as a sine wave on top of 0.705 mW. In both the 1D and 3D cases, the material properties used were: thermal conductivity^{27} is 147 W/m-K, specific heat^{36} is 679 J/kg-K and density^{37} is 2.328 g/cm^{3}.

Figure 3 shows the result of the comparison between analytical solution and FEM. No difference is found between the two methods for steady-state regime (Figure 3b). In case of transient regime (Figure 3c), differences between analytical solution and FEM varies from 0.003% to 0.2% when L/w varies from 5 to 100. Harmonic mode (Figure 3d) has a constant difference of around 0.6% for all the L/w ratios. The comparison shows accuracy of the analytical solution.

## COMPARISON BETWEEN 1D ANALYTICAL MODEL AND 3D FEM WITH APPLICATION OF HEAT PERPENDICULAR TO TIP

After verifying the 1D analytical model, it’s important to figure out whether it’s applicable to analyze thermal properties in actual STR experiment where heat is applied perpendicular to the tip of the cantilever as shown in Figure 4. All the material properties mentioned in the previous section are also used in this section.

A 3D representation and respective isotherm diagrams are shown in Figure 4. In STR, two lasers are used to thermally characterize the suspended cantilever. One laser, named pump laser, heats the cantilever perpendicularly at the tip from one side while another laser, termed probe laser, scans temperature from the other side of the cantilever as shown in Fig. 4b. The probe laser has the mobility and can be focused at any point along the length of the cantilever. A photodetector registers the change in the probe reflectivity which is corelated to the temperature of those points with the help of thermoreflectance coefficient.^{17,18} Challenges of this technique lies in laser selection since most of the pump and probe power needs to be absorbed and reflected, respectively. Besides, like any other thermoreflectance technique, sample surface needs to be polished in order to avoid unwanted scattering of light. Figure 4a and 4b visually represent the actual system where the fixed end of the cantilever is attached to a substrate. Note that the heat flux (laser heating in the figure) is applied laterally to the cantilever as mentioned before and can be seen from Figure 4a. The associated surface temperature and contour plots exhibit how heat is being transferred along the beam. From these figures it is evident that for Si of this geometry the isotherms quickly, within one thickness of the cantilever, become perpendicular to the flow of heat. Further analysis, regarding this, is presented in next section.

Figure 5 shows the steady state temperature distribution along the length of the cantilever with L/w ranges from 5 to 100. Dimensionless length and temperature are calculated using Equations (7) and (9), respectively. Upon applying the same conditions in FEM model, it is found that difference between two models decreases with the increase of length to width ratio, the difference being 4.8% for L/w of 5 to 0.05% for L/w of 100.

Figure 6 depicts the transient response of the heat at a dimensionless position near the tip of the cantilever. The response of each L/w ratio is plotted to 15τ while the inset shows the plot for (1.5 sec.). It is observed that as the L/w ratio becomes smaller, the temperature of analytical model and FEM deviates from each other. For L/w=5, maximum deviation between two models is 12.7% whereas the maximum deviation is only 0.85% for L/w=100.

In the case of harmonic analysis, heat is applied as a sinusoidal wave form with constant intensity and amplitude. As a result, each point along the cantilever experiences periodic increase and decrease in temperature in time. Figure 7 shows good agreement between two models and the difference is 4.8% at length to width ratio of 5 but goes down to only 1% at L/w of 100. It can be observed from Figure 7 that the cantilever temperature at the same dimensionless point increases with an increase of cantilever length.

As the harmonic thermal wave propagates through the beam, a phase shift occurs because of time delay in the transport of thermal energy. The phase shift is calculated from the lag time between dimensionless position 1 and corresponding other positions along the beam by dividing the time delay with the period of a single wave and multiplying by the total phase of that wave. Figure 8 shows a larger phase shifts for longer beams than the shorter beams as expected. For the beam with L/w = 10, phase shift is almost 0 which indicates that heat propagates through the beam instantaneously. If the phase shift and temperature of two different points along the beam is known, heat capacity of a beam can be calculated using Equation (21).

There are differences between one dimensional analytical model and FEM as mentioned in the figures above. Figure 9 illustrates the differences for all three heat conduction modes. The maximum temperature difference is calculated for the same dimensionless position at any given time by taking 3D model as the base. In case of steady state, the difference between the models reduces from 4.8% to 0.05% as the length to width ratio increases from 5 to 100. This difference for transient heat transfer is 12.7% for L/w of 5 and as the L/w ratio increases, the difference becomes 3.7%, 1.6% and 0.85% at L/w of 20, 50 and 100 respectively. The largest difference for harmonic heat transfer is same 4.8% as steady state at length to width ratio of 5, but reduces to only 1% at L/w of 100.

## BACKSIDE PROBING VALIDATION FOR STR MEASUREMENT

Since STR measurement reports only one temperature at a point on the surface of the cantilever using backside probing, cf. Fig. 4b, it is important that the temperatures along a given y-z plane are equivalent, i.e. isotherms are perpendicular to the heat flux – 1D heat flow. The FEM is modified to heat the tip of the cantilever as in Fig. 4a and then 13 points in a y-z plane are compared as is shown in the inset of Figure 10. Figure 10 shows how far from the cantilever tip, the difference among these points becomes negligible i.e. temperature profile becomes 1D. Temperature differences are shown as 1% and 0.5% standard deviation from the mean. It can be noticed that when thickness is larger than the width, the temperature profile does not become 1D until distance from the tip equivalent to several widths. But as the thickness becomes smaller than the width, both the differences become zero within the distance from the cantilever tip equivalent to 50% of the width.

## CONCLUSION

An analytical model with three different regimes of 1D heat transfer; steady state, transient and harmonic; is presented which will be used to analyze thermal properties in an emerging temperature measurement technique named Suspended ThermoReflectance (STR). The analytical solution is compared with a 3D Finite Element Model for Si as it is the most important material in the semiconductor industry. The temperature difference between the models is higher in all three regimes for the lower length to width ratios. This indicates that a micro/nano-cantilever beam, with length comparable to width, acts like a three-dimensional object and 1D model cannot predict its properties properly. However, as the length to width ratio increases, specifically for L/w>20, the difference becomes smaller than 4% and at L/w of 100 three-dimensional beam acts like one-dimensional object (less than 1% difference). This finding is highly important in micro-nano electronics in case of analyzing thermal properties of these devices. Instead of using complex 3D models and finite element simulation, the closed form temperature equations of this paper can be used to analyze thermal properties like thermal conductivity, heat capacity, diffusivity etc. The harmonic solution is particularly important to find heat capacity of MEMS devices like high speed computer processors because it provides the phase shift between two points on the surface for the propagating heat wave. This phase shift is directly related to the heat capacity of the material under study. This method can be used to analyze how heat capacity of computer processor and chips changes with the change in speed from MHz to GHz.

## ACKNOWLEDGMENTS

Co-author Singh would like to thank National Science Foundation (award number 1662879) for financial support.

## NOMENCLATURE

- u
temperature, K

- T
_{sub} environment temperature, K

- C
specific heat capacity, J/(kg.k)

- k
thermal conductivity, W/(m.k)

- α
thermal diffusivity, m

^{2}/s- L
length of the sample, μm

*x*distance along horizontal direction, m

- $x^$
dimensionless distance along horizontal direction

- t
time, s

- $t^$
dimensionless time

- I
intensity of laser, W/m

^{2}- A
intensity of modulated frequency, W/m

^{2}*ω*angular frequency, rad/s

*θ*dimensionless temperature distribution

- $\xce$
dimensionless laser intensity

- $\xc2$
dimensionless modulation intensity

- $\omega ^$
dimensionless angular frequency

*k*_{n}eigenvalue constant

- $\theta ssx^$
dimensionless steady-state temperature

*θ*_{transient}dimensionless transient temperature

*θ*_{harmonic}dimensionless harmonic temperature

- n
integers

*τ*time constant

*L*_{p}thermal penetration depth

## REFERENCES

*Doctoral dissertation*,

*Doctoral dissertation*,