Thermometry and thermal management of carbon nanotube circuits Free

Monitoring and controlling the intrinsic temperature and heat flow in a device allow for the study of energy transformations and for the ability to minimize the dissipation on the nanoscale. The energy losses caused by electric current are a fundamental obstacle which obstructs the overall functionality of nano-circuits. This is due to the excessive heating. Electric signals, when controlled by elements of nano-electronic circuits, generate Joule-Lenz heat, thereby raising the intrinsic temperature T on the nanoscale. Since the dimensions of the elements in the nano-circuit are small (relative to cross-section area S), the high current density J/S causes a large local Joule-Lenz heating. Excessive heating due to energy dissipation not only impairs the performance of the nano-electronics elements but can even damage or destroy them. In recent years, because of the wide spread implementation of high-density circuits, local overheating of nano-elements has become an acute issue. Another fundamental aspect of the same problem is that the thermal flow consists of three separate components: electrons, holes, and phonons. Therefore, controlling the heat flow can be achieved by controlling/filtering out the effects from these components. For instance, when designing the nano-circuits, frequently there is a need to separate the flow of charged excitations from the flow of phonons, or shield a certain element thermally while retaining electric contacts. Such a separation helps to carry out the heat from a circuit element away, improving its functionality.

Thus, monitoring of the local intrinsic temperature T along the thermal management is the critical issue in determining the viability of nano-scale applications. Prevention of excessive heating is accomplished using a special geometry for the nano-circuits and implementing advanced semiconducting materials with large thermal conductivity Λ. In this respect, carbon-based low-dimensional materials like carbon nanotubes (CNTs) and graphene along with atomic monolayer dichalcogenide, which are characterized by large Λ = 50–6000 W/m $⋅$ K, are promising candidates. Implementing such nano-devices^1–3 allows for reducing the local heating of the nano-circuit elements, thereby boosting their overall performance.

In this work, we design and test the thermometer for the monitoring of the intrinsic temperature in the carbon nanotube (CNT) thermoelectric circuits.^1–3 We consider a system where a semiconductor-type CNT is a central component (see Fig. 1). Two pairs of metallic (Ti) contacts made to a CNT and side gates form a nano-device which, from electric point of view, is a system of two field-effect transistors (SFTs) connected in series. In addition, a thermometer is placed in the center of the device, which we consider in more detail below.

We also consider a bi-metal multilayer system used for heat flow control in the nano-circuits. The bi-metal multilayer setup is used to filter the heat flow by separating the electron and phonon components. The elements of the CNT nano-circuits form thermoelectric Peltier coolers based on ambipolar field effect transistors (FET).^1–3 In the devices, shown in Fig. 1, the concentration and polarity of the charge carriers are controlled by application of a local electric field, via side-gate voltage. We have created a series of thermoelectric circuits using a variety of geometries and different metals to create gate, source, and drain electrodes. The FETs were fabricated using conventional electron beam lithography techniques to pattern the source, drain, and gate structures on top of the CVD grown CNTs. The CNTs of choice were characterized using AFM scanning techniques and the metallic structures were deposited in an electron beam evaporator. Excess tubes have been removed via an oxygen plasma etch thus preventing the sample leads from shorting together.

Initially, the source and drain contacts were fabricated using a palladium/gold (3–5 nm)/(40 nm) multilayer where the Pd directly contacted the CNT. The Pd based devices exhibited a Pd/CNT interface resistance of 20 kΩ. Subsequently, this interface resistance was reduced to 4–6 kΩ via annealing in a hydrogen-argon atmosphere at 230 °C for 30 min. With the Pd contacts, the CNT nano-circuits exhibited p-type FET behavior. The on state conductance was 0.05 e²/ℏ, whereas the off state conductance is negligible. Regardless of changes in annealing process and/or gate geometry (with some samples having less than 50 nm of separation between the CNT and side gate), we were unable to produce Pd contact devices with ambipolar behavior. This is likely due to palladium's high work function which creates a large barrier for incident electrons.

In order to achieve ambipolar behavior, we have fabricated a series of samples that utilize a titanium layer to contact the CNT. These samples exhibit Ti/CNT interface resistances of 1 MΩ prior to annealing and display hysteric ambipolar behavior. After annealing using the same process as with the Pd contacts the interface resistance reduces to 40 kΩ, much of the hysteresis is eliminated, and the ambipolar behavior is smoothed. The behavior of the conductance versus side-gate voltage of a single CNT device with Ti contacts after annealing is shown in Fig. 2.

In addition to the effort made to fabricate the CNT devices, there are still challenges that need to be addressed in the measurement of these devices. In order to ascertain if these devices work and to further quantify the device's characteristics, a measurement of the local temperature must be made. This requires the thermometer to be fabricated onto the active portion of the device, restricting possible thermometry options. Additionally, the proposed device is to be tested at a large range of temperatures, thus the thermometer must operate at the same temperature range. Finally, the temperature measurement must be non-invasive. The thermometer must be designed in such a way as to not be a heat leak into or out of the system. Therefore, the measurement must not use a lot of power (current) to prevent changing the local temperature due to heating.

To overcome these challenges, it was determined that a tunnel junction should be used as a thermometer. First, in-situ fabrication of a tunnel junction based thermometers onto the CNT devices as currently designed is relatively easy. Second, since the interfaces between the thermometers and devices are not conducting they will prevent heat leakage. Third, if designed correctly, they can be operated with a very small current that minimizes any interaction with the CNT device. Finally, tunnel junctions have been shown to have a T² dependence with conductance, providing the possible sensitivity needed to see any temperature changes provided by the CNT device.³ 

Before such a device could be implemented as a usable thermometer, the concept had to be turned into a standardized measurement device. The test thermometers were fabricated using e-beam lithography on a silicon substrate with a 300 nm thermally grown oxide layer. A gold meander (a long gold wire) is first placed onto the sample set between two thermal reservoirs. It will then be possible to model the meander's temperature profile for a given dc current. The known temperature profile will then allow the thermometer to be calibrated.

To test the double tunnel junction thermometers, the differential conductance of the thermometer (through the double tunnel junction only) was measured (using standard lock in techniques) from room temperature to liquid nitrogen temperatures. The results in Fig. 3 are consistent with those found by Koppinen.³ 

The thermometer is fabricated using a double aluminum-aluminum oxide tunnel junction. These junctions are fabricated on top of the gold meander by using angled e-beam evaporation. First, the central aluminum island is evaporated onto the meander (when the sample is perpendicular to the evaporation source). Without breaking vacuum, the aluminum is then oxidized. Aluminum oxidizes on the surface only, creating a small insulating layer on top of the island. Without breaking vacuum, a second layer of aluminum is then evaporated at an angle (55°) on top of the aluminum island, creating the double tunnel junction. This geometry allows the middle island to thermalize with the gold meander (and later the CNT device), while the oxide layer provides the tunnel barrier, which thermally insulates the island from the thermometer leads. The last layer of aluminum then provides the tunnel contacts, such that small currents can pass through the island without passing it through the gold meander (altering the potential calibration).

To summarize the work thus far, we have fabricated a number of nano-circuits. Our devices consist of two ambipolar CNT FET connected in series (see image of the CNT FET with Ti source and drain electrodes in Fig. 1). The plot of conductance G(V_G) of our ambipolar CNT FET versus side gate voltage V_G is shown in Fig. 2. In the course of the work, the thermometer (see Fig. 4) which measures local changes in temperature was designed and fabricated.

Along with the problem of direct temperature monitoring, we also studied the issue of controlling the heat current. We consider a bi-metal multilayered structure depicted in Fig. 5(a) representing the electric/heat conductance valve. The suggested system is intended to separate different components of the heat current, consisting of the flow of charged excitations and the flow of phonons. Such a multilayered system might be implemented, e.g., to insulate individual circuit elements thermally, while preserving the electric contacts. We performed analytical and numeric modelling of propagating the heat through multilayered metallic stacks. The random layer thickness change is introduced to eliminate resonance transmission of phonons. Thus, the phonon flow across the multilayer is greatly reduced due to the low transmission probability $ζ≪1$⁠. However, the contribution of phonons to the heat transfer might exceed the contribution due to the electrons and holes. The multilayer actually acts like a filter for the two components of microscopic heat transport.

The heat flow filtering mechanism is understood within the context of a simple analytical multilayer model. The multilayer represents a sequence of bilayers shown in Fig. 5(a), where metallic layers A and B are characterized by different Fermi velocities (⁠$vFA, B$⁠) and sound velocities (⁠$sA, B$⁠). The phonon heat conductance through the multilayer $Λph$ is obtained from the Landauer formula.^6–9 One writes

where $TH, C$ and $NωH, C$ are the temperatures and the Bose-Einstein distribution functions in the “hot” and “`cold” ends. The phonon transmission probability $ζω$ is obtained by the mode matching method.⁹ If the temperature gradient across the junction is small, $TH−TC≪TH+TC$⁠, and the junction is “ideally transparent for phonons,” $ζ(ω)=1$⁠, $Λph$ is quantized⁹ as $Λph≈M(π2kB2T/3h)$ where $M$ is the number of acoustic modes. For $L=1$$μ$ m and $W=20$ nm one, roughly gets $Λph=(50−6000)$ W/(m $⋅$ K). Since in real junctions always there are reflections at the interfaces, either due to mismatch of sound velocities or because interface roughness, one obtains $ζ(ω)<1$⁠.

Electron transport through the metallic multilayer is determined by the corresponding electrons (⁠$Tel$⁠) and phonons (⁠$ζ$⁠) transmission probabilities. We compute $Tel$ and $ζ$ in terms of the S-matrix method.¹⁰ The electron S-matrix of the whole ABA…B multilayer is composed of elementary blocks ABA. Here, we assume that there are no interface A/B-barriers separating the A and B layers. The electron (hole) transmission (⁠$tABA$⁠) and reflection (⁠$rABA$⁠) coefficients which are the $SABA$ matrix elements are obtained from the A/B-interface boundary conditions as (see, e.g., Ref. 8)

where the denominator is $DkA, kB=(kB2+kA2) sin(LkB)+2ikAkB cos(LkB)$⁠, $kA$ and $kB$ are the electron wave vectors in A and B. Since the main contribution to $S$ and $G$ comes from the electron states near the Fermi level, one may use the linear dispersion law $E−EF≃ℏvF(k−kF)$⁠. Since the electron energy is conserved during the interlayer transmission, $EA=EB$⁠, it gives $ℏvFA(kA−kF)=ℏvFB(kB−kF)$ or $vFAkA≃vFBkB$⁠. This allows rewriting the above equation simply as

where $Dα=(1+α2) sin α+2iα cos α$ and $α=vFA/vFB$⁠. The corresponding phonon transmission and reflection coefficients are derived in a similar way. A very interesting consequence of the above formula takes place when $α≃1$ for electrons and while simultaneously $α≫1$ for phonons. Such situation takes place when one of the metals in the elementary bilayer is lead (i.e., A = Pb) while another metal is aluminum (B = Al). The Fermi velocities for lead and aluminum are very close (⁠$vFPb=1.83×106$ m/s and $vFAl=2.3×106$ m/s, respectively), whereas the sound velocities in the two metals are remarkably different (i.e., $sPb=1158$ m/s and $sAl=6420$ m/s). For such reasons, the ratio of Fermi velocities is close to unity, $αel=vFAl/vFPb=0.8$⁠, whereas the ratio of the sound velocities $αph=sAl/sPb=5.5$⁠, i.e., is much larger than 1. Such a big difference between the two ratios, $αel=0.8$ and $αph=5.5$⁠, serves as a reason why the multilayer in the perpendicular direction is fairly transparent for electrons, whereas the phonon propagation across the multilayer is obstructed. The whole ABA….AB multilayer is then composed as a sequence of bilayers with randomly changing thickness. The electrons correspond to kL_A,B ≪ 1 while phonons to qL_A,B ≃ 10. For such reason (see Fig. 5(c)), the effective electron transmission probability is practically ideal, i.e., T_el ≃ 1 while the averaged phonon transmission probability ⟨ζ⟩_q ≃ 0.3 (see the green line in Fig. 5(c)). Further suppression of the thermal phonon flow is accomplished with forming of a sequence of the N bilayers with a close but non-equivalent thickness around L = 50 nm (for T = 10 K). The phonon transmission through such a randomized Pb/Al-multilayer is not phase-coherent, thus the total phonon transmission probability is composed of corresponding probabilities for individual elementary bilayer blocks, i.e., ζ_tot ≃ ζ/N which already for, e.g., N = 6 gives ζ_tot ≃ 0.3/6 = 0.05 while T^tot_el ≃ (T_el)⁵ ≃ 1, i.e., it remains nearly ideal. The last example illustrates the principle of the heat flow filtering: The electron and hole carry the heat freely while the phonon thermal conductance almost vanishes across the Pb/Al bimetal multilayer.

Since the lattice constants of the two metals A and B differ, there is a lattice strain built into the layer's A/B-interfaces. However, the strained region involves just a few atomic layers at the Pb/Al or Pb/Sn interfaces. Therefore, the strain affects just on a tiny fraction of the whole multilayer's volume. Contributions of the strained region to the electron and phonon transport across the metallic multilayer is negligible if the length of the strained region in lateral direction is much shorter than the metallic layer thickness and/or the electron and phonon mean free paths. Similar metallic multilayers have been widely used in the superconducting junction technology and it is well known that the interface strains do not impact the lateral electron transport. This happens because it acts just like a scatterer with size is less than 1 nm, i.e., it is much narrower than the phonon's wavelength of interest which exceeds 10 nm for $T>$ 300 K. The strained region is also too narrow to change the electron band structure on the local scale $<1$ nm, and the local short scale lattice deformation $<1$ nm is also unable to generate an electrostatic potential barrier for the electrons propagating in the lateral direction.

The difference in transmission probabilities of the electrons and phonons through the metallic multilayer can be exploited to filter the heat transport components: The electrons and holes propagate through the metallic multilayer almost free while the phonon transport is considerably reduced. Using the bimetal multilayers allows to reduce the overall high thermal conductance Λ_ph due to propagating the phonons in CNTs. This eliminates another obstacle to achieving a noticeable field effect transistor (FET) efficiency. We strongly reduce Λ_ph, e.g., by connecting the CNT in a sequence with a material which Λ_ph is very low whereas the electron part, Λ_e, is high. Here, eliminating the phonon part of heat conductance is accomplished by connecting CNT in a sequence with a multilayered valve pad. A multilayer with random layer thickness had been also considered by the authors of Ref. 4, who studied a set of semiconducting layers with random thickness where energy gaps in the phonon spectrum are formed. However, there are inherent controversies in that approach.⁴ In semiconducting multilayers with random thickness, energy gaps in the phonon spectrum are formed. Since the phonon gaps in distinct layers are positioned at different phonon energies, the resulting phonon heat conductivity is strongly suppressed. Although it is supposed to improve the figure of merit, in reality the idea⁴ fails to work. The contradiction originates from two opposing inherent properties. For example, if the semiconducting layers are strongly coupled, the electric conductivity and Seebeck coefficient are large. On the other hand, a strong interlayer coupling eliminates the phonon gaps since series of strongly coupled layers behave like a monolithic material where the phonon gaps vanish. As a result, the phonon heat conductivity Λ_ph is not suppressed. In the opposite limit, when the layers are weakly coupled, the phonon gaps are indeed formed, hence the phonon heat conductance is suppressed. But an adverse effect is that the electric conductance and Seebeck coefficient are diminished as well, because they are proportional to the interlayer coupling strength. In either case, weak or strong interlayer coupling, the figure of merit cannot be improved considerably by using the method of Ref. 4. Our approach is based on a different idea which is more suitable for the TEG optimization. First, we consider metallic multilayer with ideally transparent interfaces. The metals have a much higher DOS than the semiconductors do and the metal/metal interfaces are very transparent for electrons. Therefore, the electric conductance and Seebeck coefficient of bimetal multilayered structure is much higher than of the semiconducting multilayer stack with decoupled layers, studied in Ref. 4. A critical distinction is that we use two metals A and B with similar Fermi velocities v_AF ≃ v_BF but with very different sound velocities s_A ≪ s_B (e.g., Pb/Al or Pb/Sn). It yields a strong reduction of the phonon thermal conductivity Λ_ph without sacrificing the electron transport coefficients. In this way, we are considerably improving the TEG figure of merit. Another benefit of using the metallic multilayer over the semiconducting multilayer is that there is better consistency between the higher DOS of metals and the spectral singularities of the electron spectrum of the carbon nanotube. Because the electron part of the heat flow Q_el is strongly increased while the phonon part of the thermal flow Q_th is diminished, the figure of merit is improved by orders of magnitude. For instance, using the Pb/Al-multilayer with random thickness d_A,B of individual layers, whose average value is ⟨d_A,B⟩ = 50 nm (see Fig. 5(c)), one obtains ζ ≃ 0.05 corresponding to the reduction of Λ_ph by the factor of 20. In turn, this results in 20 times increase of the figure of merit ZT_cold = S²G_eT_cold/Λ_ph, where S is Seebeck coefficient and G_e is the electric conductivity. The suppression of the phonon part of heat flow becomes stronger when the number of bimetal layers is increased.

The phonon part of the thermal transport through the TEG had been examined as follows. We describe the non-equilibrium heat flow through the CNT in the presence of multiple scattering on lattice defects, boundaries, and electrons. A finite temperature difference $δT$ between the opposite ends of each CNT induces the thermal flow given as a sum of contributions of the individual phonon subbands. The phonon density of states $Fβ(ω)$ related to the phonon subband $β$ is mismatched in adjacent layers of the H electrode sketched in Fig. 5(a). Inside the CNT, the phonon distribution function $N(ω)$ is non-equilibrium which means that $N(ω)$ deviates from the Bose-Einstein distribution in the hot (H) and cold (C) ends. For a “clean” CNT, the phonon mean free path exceeds the CNT length $L$⁠. Therefore, the non-equilibrium effect does not influence the final results. The equilibrium phonon distribution at the CNT ends is established due to a free phonon diffusion into the bulk of attached metallic contacts and dielectric substrate. The thermal conductivity Λ_ph of the CNT had been computed by using the phonon density of states $Fβ(ω)$ preliminary obtained for each phonon subband $β$⁠.

The thermoelectric characteristics are found by solving the Dirac equation for chiral fermions in the carbon nanotube (see above and also Ref. 8). The analytical model is verified by the numeric calculations based on the density functional theory. The electron and phonon excitation spectra are obtained considering influence of the inelastic electron-phonon and elastic electron-impurity scatterings. They are taken into account along the processes of the electron tunneling through the interface barriers. The scattering of electrons on impurities and on phonons are included by using the Keldysh-Green function technique which allows deriving of the quantum kinetic equations.

We conclude that the monitoring of the intrinsic temperature combined with the heat flow management has a great potential when countering the problem of overheating of the nano-circuits. The double tunnel junction thermometer, which was designed and tested in the course of this work, performs the real time monitoring of the local temperature on the nano-scale. Furthermore, in this work, we have suggested that the bi-metal multilayer structure minimizes the phonon component of the heat flow, while preserving the electronic part of it. These approaches, if implemented, can greatly improve the overall efficiency (the figure of merit) of thermoelectric devices converting the heat energy to electricity and vice versa or when implemented in the electronic nano-circuits to minimize of overall Joule Lentz energy losses. Besides, similar method can be used to counter the Joule Lenz overheating of large nano-circuits.

We wish to thank P. Kim, M. S. Dresselhaus, I. P. Nevirkovets, and G. Chen for fruitful discussions. We are also grateful D. Dikin for technical help. This work had been supported by the AFOSR Grant FA9550-11-1-0311.