Basal-plane thermal conductivity of nanocrystalline and amorphized thin germanane

Two-dimensional (2D) layered materials have received renewed interest in recent years because of potential applications in electronic, optoelectronic, and thermoelectric devices.^1–3 Compared to graphene, a material with a vanishing electronic bandgap, the presence of a sizable bandgap in other 2D materials can help achieve a large on-off ratio of field effect transistor devices fabricated from these materials. Among different known 2D layered materials that can either be exfoliated from a bulk crystal or synthesized via chemical vapor deposition (CVD), transition metal dichalcogenides (TMDs), including MoS₂, MoSe₂, Bi₂Te₃, and Bi₂Se₃, have been investigated actively because of their good chemical stability, high electron mobility, the presence of topological surface electronic states, and thermoelectric applications.^4–8 Recently, relatively high electron mobility has also been reported for 2D phosphorene exfoliated from black phosphorus, although the material can degrade quickly in air or by moisture.⁹ Besides graphene and phosphorene, which were made from elemental carbon and phosphorus, respectively, there have been significant efforts in realizing 2D materials from elemental silicon (Si) and germanium (Ge), which typically form sp³ instead of sp² bonds.^10,11 In addition, SiGe is an efficient, high-temperature thermoelectric material because of its relatively low thermal conductivity compared to either Si or Ge.¹² 

Besides the electronic properties, the thermal transport properties of these different 2D materials also play an important role in their device performance. A high thermal conductivity (κ) is desirable for dissipating the Joule heat in nanoelectronic devices, where power dissipation density has increased with decreasing feature size and increasing packing density.¹³ Meanwhile, low thermal conductivity is needed to achieve a high thermoelectric figure of merit. Although there have been several studies of thermal transport in graphene, hexagonal boron nitride (h-BN), TMDs, and phosphorene, their size-dependent and environmentally sensitive thermal conductivities still require better understanding.^1,14–20 Moreover, the thermal transport properties of 2D Si and Ge, including silicene, germanene, and germanane (GeH), are mostly unknown.^11,21–24 Besides relevance to device applications, the lattice thermal conductivity can serve as an indicator of the crystallinity of these 2D materials that have been synthesized by different methods.

In this letter, we report on basal plane thermal conductivity measurements of thin germanane (GeH) flakes. Based on a four-probe thermal transport measurement method, the measured thermal conductivity of the as-exfoliated GeH flakes increases with increasing temperature, indicating that extrinsic phonon scattering processes dominate over intrinsic lattice anharmonicity in the measured samples. For one GeH sample, the measured room-temperature value is reduced from 0.53 ± 0.09 W m⁻¹ K⁻¹ to 0.29 ± 0.05 W m⁻¹ K⁻¹ after the sample is heated to 195 °C. Transmission electron microscopy (TEM) and electron diffraction measurements further reveal a transition from a nanocrystalline structure to an amorphous structure due to the heating process. The values obtained from the nanocrystalline and amorphized samples are in general agreement with the calculated results for defective GeH with a grain size on the order of 20 nm and an amorphous GeH structure, respectively. Comparison between the measured and calculated results for crystalline and amorphous GeH can help to evaluate the crystallinity of the existing GeH materials as well as to determine the potential for further enhancement of the structure and thermal transport properties.

GeH is a germanium-based hydrogen-terminated graphene analogue. Its layered structure consists of a hexagonally arranged, honeycomb sp³-bonded germanium framework, in which each germanium is terminated with a hydrogen alternately above and below every atom in the framework (Figures 1(a) and 1(b)). Unlike graphene, its honeycomb structure buckles in the basal plane. In this work, GeH samples were synthesized via the topochemical deintercalation of CaGe₂ with HCl.¹¹ Thin GeH flakes were mechanically exfoliated onto a polydimethylsiloxane (PDMS) stamp. With the use of a custom micro-manipulator with a very sharp tungsten tip, the GeH flakes were transferred from the PDMS onto the top of the four suspended Pd/Cr/SiN_x thermometer lines of a micro-fabricated measurement device, as seen in Figs. 1(c) and 1(d). The lateral dimensions of the flakes were measured from the top-view scanning electron micrograph (SEM). Upon the completion of the thermal measurements, the thickness of one sample (sample 1) was measured from the tilted SEM to be 519 ± 26 nm. For another sample (sample 2), the flake was transferred to a silicon substrate and measured using atomic force microscopy (AFM) to obtain a thickness of 274 nm. The presence of a through-substrate hole under sample 1 assembled on the suspended device allows for transmission electron microscopy (TEM) and selected area electron diffraction (SAED) characterization of that sample during different stages of the transport measurement, as shown in Figs. 1(e)–1(h).

A recently reported four-probe thermal transport measurement method was employed for in-plane thermal transport measurements of the GeH sample.²⁵ Compared to a prior two-probe measurement method, the four-probe method provides the ability to directly measure both the contact thermal resistances (R_c,2 and R_c,3) and the intrinsic thermal resistance (R₂) of the middle suspended segment of the sample.²⁶ This capability is necessary for eliminating the contact thermal resistance errors from the obtained thermal conductivity. Unlike a different four-probe thermoelectric transport measurement method reported previously,²⁷ this four-probe method does not require electrical contact between the sample and the measurement device.

During the four-probe thermal transport measurement, one of the four suspended Pd/Cr/SiN_x lines (line i = 1, 2, 3, or 4) was electrically heated by a DC current (I) and voltage (V) at a heating rate of (IV)_i. The four-probe electrical resistances of the heater line i and the other three thermometer lines (line j = 1, 2, 3, or 4, j ≠ i) were measured from the I–V curve and with lock-in amplifiers, respectively. This measurement was repeated four times with a different Pd/Cr line serving as the heater line each time. From this dataset of four measurement configurations per temperature, the thermal resistance of each of the four thermometer lines, R_b,j, can be obtained together with R_2, R_c,2, R_c,3, the combined thermal resistance of the left suspended segment and the contact thermal resistance at the left contact (R₁ + R_c,1), and the combined thermal resistance of the right suspended segment and the contact thermal resistance at the right contact (R₃ + R_c,4), as shown in Fig. 2. Because the heat flow through the outer contact and adjacent suspended segment of the sample is the same, R_c,1 cannot be separated from R₁, and R_c,4 cannot be separated from R₃ in this measurement. In comparison, the heat flows across the two middle contacts are different from that in the middle suspended segment. The ratio between the heat flows depends on which of the four thermometer lines is used as the heater line. This feature allows for independent determination of R_2, R_c,2, and R_c,3. For the low-thermal conductivity GeH flakes measured here, R_c,2 and R_c,3 are found to be negligible compared to R₂, as shown in Fig. 2.

The thermal conductivity of the GeH flake is calculated as κ = L/R₂A_c, where L and A_c are the length and cross-sectional area of the middle suspended segment. As shown in Fig. 3, the obtained thermal conductivity of two samples increases with increasing temperature. The temperature dependence reveals that extrinsic scattering, rather than intrinsic lattice anharmonicity, dominates phonon transport in the sample. In addition, the room temperature values are 0.53 ± 0.09 and 0.74 ± 0.13 W m⁻¹ K⁻¹ for the two samples. These values are much smaller than the 60 W m⁻¹ K⁻¹ value for a bulk, diamond-type structured germanium crystal.

The TEM image in Fig. 1(e) was obtained on sample 1 after the first set of thermal transport measurements were completed on this sample, and reveals the presence of crystalline grains in the sample. The polycrystalline nature of the sample is also evident in the corresponding SAED pattern (Fig. 1(f)), which consists of discernible peaks embedded in the concentric ring pattern. After the TEM image and SAED pattern were obtained, this sample was annealed in an evacuated tube furnace at a temperature of 195 °C and 10⁻³ Torr pressure of an argon atmosphere for four hours. As shown in Fig. 3, the measured thermal conductivity of the annealed sample was lower than the values obtained before annealing. The TEM image (Fig. 1(g)) and SAED pattern (Fig. 1(h)) obtained after the annealing and second thermal measurement cycle do not contain either crystalline lattice fringes or discrete diffraction reflections, indicating that the amorphization of the sample has occurred during the annealing step, which is in agreement with previous studies that suggest GeH amorphizes above 75 °C because the crystalline phase is metastable.^11,28

The measured lattice thermal conductivity in the electrically insulating, undoped GeH samples can be used to evaluate the crystal quality of the sample. The basal-plane thermal conductivity of bulk crystalline GeH from first principles calculations is found to decrease with the increasing temperature from 12.8 W m⁻¹ K⁻¹ at 200 K to 8.8 W m⁻¹ K⁻¹ at 300 K, as shown in Fig. 3. The calculation method is similar to those reported in prior works.^29,30 The temperature dependence is typical for intrinsic thermal conductivity that is limited only by phonon-phonon scattering. These calculation results are considerably higher than the measurement data and present an upper limit for a perfect bulk GeH crystal. In comparison, the lower limit of the basal plane thermal conductivity of GeH is expected when the structure becomes amorphous.³¹ 

To provide an additional comparison to the measured thermal conductivity, we have calculated the amorphous limit in the lattice thermal conductivity of bulk GeH. For a thin layer of thickness t along the OZ axis and large lateral dimensions in the OXY plane of the GeH crystal, the relaxation time approximation can be used to express the in-plane thermal conductivity as

Here, BZ denotes the two-dimensional Brillouin zone with area A_BZ, k_z runs over the N_z possible values of the through-plane component of the wave vector, $d2k||$ is a differential area in the in-plane wave vector plane, and p runs over different phonon polarizations. For each phonon mode identified by a tuple (k,p), τ, v_x, and c_v denote its relaxation time, group velocity along the OX axis, and contribution to the isochoric specific heat, respectively.

In the minimum thermal conductivity model of Cahill et al., the phonon lifetime is taken to be approximately half of the phonon oscillation period, namely, $τ≈π/ω$⁠.³¹ Using the full phonon dispersion calculated for bulk GeH, this model is used to calculate the amorphous limit in the in-plane thermal conductivity of bulk GeH. The result calculated for bulk amorphous GeH is in good agreement with the results measured for the annealed sample with a thickness of 519 ± 26 nm, as shown in Fig. 3.

The higher thermal conductivity measured in this sample prior to the annealing process indicates the benefit of the nanocrystalline domain size, although the thermal conductivity is an order of magnitude lower than the calculated result for the defect-free GeH bulk crystal as a consequence of the presence of grain boundaries and defects. In the small grain limit approximation, the grain boundaries in the nanocrystalline GeH samples provide the dominant phonon scattering mechanism. The phonon-boundary scattering mean free paths (l_b) are assumed to be independent of frequency (f) and comparable to the grain size. However, for GeH, whose phonon dispersion has different phonon bundles separated by substantial energy gaps, as demonstrated in Fig. 4, the thermal conductivity calculations within the small grain limit approximation are not able to capture the correct experimental trend with temperature, and yield higher values than the measurement results at near room temperature when l_b is set to be a constant value to fit the low-temperature measurement result, as shown in Fig. 3. This finding suggests that the scattering rate for the high frequency phonon bundles (f > 5 THz) needs to be increased further relative to that for the low frequency bundle (f < 5 THz) in the nanocrystalline GeH samples. This result can be attributed to the competing point defect scattering contributions in these samples along with the grain boundary scattering, because defect scattering is more effective for high-frequency phonons than for low-frequency phonons.

Vacancy scattering can be accurately calculated using a Green's function approach, as shown in a prior work.³² However, this calculation requires prior knowledge of the atomic structure of the vacancies. In the current case, many different possible vacancy types may exist, and it is not feasible to determine their structures and concentrations from theory alone. Instead, phonon scattering by vacancies can be approximated by that of substitutional defects presenting a large mass difference Δm = m_subs - m_host, with typical Δm values around 3 or 6 times that of the host atom.^32,33 The point defect scattering rates for GeH can then be calculated within the Born approximation and are proportional to a factor F = c(Δm/m_host)² within the small defect concentration approximation where c is the point defect concentration. Using F as a fitting parameter can yield the correct temperature dependence and good agreement between the calculated thermal conductivities and the experiments for nanocrystalline GeH flakes for two different samples with l_b equal to 16 nm and 34 nm, respectively, when F = 0.024. In comparison, X-ray diffraction (XRD) measurements (Fig. 5) of these GeH flakes ground into powders give a basal plane grain size of 17.6 ± 2.5 nm from fitting the full width at half maximum of the {hk0} reflections to the Debye-Scherrer equation. Considering only Ge vacancies in the samples and using Δm = 3m_Ge for vacancies, the obtained F would correspond to a c of 0.27%, and using Δm = 6m_Ge yields a c of 0.07% in these samples.^32,33 Furthermore, the obtained defect concentrations could vary depending on the presence of other point defects in these GeH samples. As an alternative to point defects, the strong suppression of the contribution from high-frequency phonon bundles between 5 and 20 THz to the thermal conductivity might be due to the low grain boundary phonon transmission for high-frequency phonons,³⁴ in addition to defect scattering.

The experiments and calculations reported here have addressed the information gap in the thermal conductivity of GeH. Both the temperature dependence and room-temperature values of the basal plane thermal conductivity obtained for nanocrystalline GeH are similar to those reported for misfit layered WSe₂³⁵ and lower than those reported for nanograined SiGe, which is a state-of-the-art thermoelectric material. The analysis shows the effects of extrinsic grain boundary and defect scattering of phonons in the basal-plane thermal conductivity of thin GeH samples. The decrease in the room-temperature value from 0.53 ± 0.09 to 0.29 ± 0.05 W m⁻¹ K⁻¹ upon annealing at 195 °C is consistent with the observed sample amorphization in the TEM measurements, as well as the basal plane thermal conductivity calculated for a GeH bulk crystal with a grain size on the order of 20 nm and an amorphous structure, respectively. The deviation between the observed thermal conductivities and the calculated values for single-crystal and amorphous GeH indicates that further control of defect concentrations and grain size can be utilized to further tune the thermal conductivity in this 2D material near and above the amorphous regime.

A. Weathers contributed to the early stage of the experimental works. B. Smith, E. Fleming, and Y. Zhou contributed to the derivation of an analytical solution to the four-probe measurement method of a sample with a finite contact width. The TEM measurements were carried out with the help of M. Palard. The authors thank D. Cahill for critical discussions. This work was primarily supported by National Science Foundation Award No. #1433467. E. Ou was supported by U.S. Department of Energy, Office of Science Award No. # DE-FG02-07ER46377. J. Kim was supported by Office of Naval Research Award No. #N00014-16-1-2293. L. Lindsay acknowledges the support from the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. A.K., J.C., and N.M. acknowledge support from the Air Force Office of Scientific Research, USAF under award No. FA9550615-1-0187 DEF, and the European Union's Horizon 2020 Research and Innovation Programme [grant number 645776 (ALMA)].