Induced electronic anisotropy in bismuth thin films

Bismuth (Bi) has recently received renewed interest because it is a key ingredient of many thermoelectric materials,^1–3 topological insulators,⁴ and valleytronic materials.⁵ Bi is a semimetal with a small band overlap of ∼38 meV at cryogenic temperatures, and has highly anisotropic electron effective masses^6,7 with a ratio of ∼200. The large band mass anisotropy makes Bi a good material platform for studying very heavy and light electrons simultaneously as well as anisotropic electrodynamics. When alloyed with small concentrations of antimony (Sb), a bandgap opens, causing Bi_1−xSb_x to become a very good low temperature thermoelectric material, with its peak figure of merit (ZT) occurring near the boiling temperature of liquid nitrogen.¹ Therefore, Bi and Bi_1−xSb_x remain promising candidates for cryogenic Peltier and Nernst coolers.

While previous studies have focused on bulk single crystalline samples, nano-structures offer a route to create higher ZT materials.⁸ Nano-structuring has been shown to be effective in increasing phonon scattering, while having a lesser impact on degrading electronic transport.^3,9 As a result, the previous literature on Bi and Bi_1−xSb_x has investigated nanowires,¹⁰ thin films,¹¹ nanoporous thin films,¹² and nano-composites.³ Despite numerous studies on nanostructured Bi, there have been few works on investigating the effects of crystalline texture and order on electrical and thermal conduction. Recently, it was theoretically shown that different band structures could be derived from Bi or Bi_1−xSb_x films grown in different crystal orientations.¹³ Some of the resulting band structures were highly anisotropic, a desirable property for thermoelectrics. Therefore, it is expected that the degree of anisotropic electrical conduction in a thin polycrystalline film of Bi will depend on the collective crystalline orientation of its grains.

In this paper, we characterize and compare electronic transport in Bi thin films with texture (i.e., films having grains with preferential crystalline orientations) versus thin films without texture. We control the texture of Bi thin films, deposited using thermal evaporation or molecular beam epitaxy, by using different supporting substrates. In this study, we primarily focus on films grown on top of crystalline mica (001) or amorphous SiO₂ substrates. Films were deposited via thermal evaporation at ∼1 × 10⁻⁸Torr, at a rate of 0.1–0.2 Å/s and a substrate temperature of ∼120 °C. We monitor the texture and crystallinity of the films using X-ray diffraction (XRD) and we check the surface roughness and grain shape with atomic force microscopy (AFM). The AFM image in Fig. 1(a) shows that each grain grows into a triangular shape suggesting that the film has texture. This is confirmed through the XRD spectrum shown in Fig. 1(c), where all of the observed peaks correspond to the (0001) direction, indicating growth of a textured polycrystalline film where the trigonal axis points out of the plane of the film. From transmission electron microscope (TEM) measurements, we measure the grain sizes to be ∼200–500 nm, and we observe that while all grains are grown normal to their trigonal axes, their in-plane grain structures are rotated randomly with respect to each other.¹⁴ Thickness fringes are observed, which confirm that the grains maintain the same preferred trigonal crystalline orientation resulting from the growth substrate interface to the top surface of the film.¹⁴ In contrast, the AFM image in Fig. 1(b) shows that the surface of Bi deposited on SiO₂ now contains several different shapes in addition to triangular ones. The corresponding XRD spectrum in Fig. 1(d), exhibits many more peaks than Fig. 1(c), indicating that the polycrystalline film no longer has a preferred trigonal axis texture, but is made up of grains with randomly oriented crystalline directions. In this study, we focus on thin films with thicknesses (<80 nm), much smaller than the average grain size in order to ensure that the longest electron mean free path is in the plane of the film. We also note from AFM measurements that the textured samples tend to have a root mean square (RMS) intra-grain surface roughness of ∼1 nm. By comparison, non-textured samples have an RMS intra-grain surface roughness of ∼1.5 nm.

We measure magneto-resistance (MR) from 3 to 300 K in a Quantum Design Physical Properties Measurement System cryostat while rotating the films in a 9 T magnetic field for the two different configurations, shown schematically in Fig. 2(a). Films were also measured in a Hall bar geometry, but no appreciable difference in these measurements were observed relative to those done in the Van der Pauw geometry. In what we denote as configuration 1, the current is applied such that as the film is rotated, the angle between the current flow direction and the magnetic field changes. In configuration 2, the current is applied such that it is always kept perpendicular with respect to the magnetic field. In these measurements, we rotate samples by 360° with an incremental step size of 3°, to ensure that there is no misalignment between the film and the magnetic field. The resistance in configuration 1, R_C1, shown in Fig. 2(b), reaches a minimum when the film is parallel to the field (when the rotator is oriented at 90° and 270°). At these angles, R_C1 is approximately equal to the zero-field resistance at each respective temperature, indicating that there is negligible MR when the current and field are aligned. As we tilt the sample, and the angle between the field and the plane of the film increases, the component of the field perpendicular to the film also increases. Thus, R_C1 reaches its maximum value when the field is normal to the plane (i.e., when the rotator is at 0° or 180°). Our observation of a large change in MR is consistent with other measurements in bulk Bi,¹⁵ thin films,¹⁶ and nanowires.¹⁷ We note that quantum oscillations have not been observed in the present work, because of the polycrystalline nature of the films. In configuration 2, the magnitude of the magnetic field normal to the current flow remains constant during rotation. Instead, changing the angle between the film plane and the magnetic field tunes the degree of contribution between surface and grain boundary scattering. When the film is oriented perpendicular to the field, the cyclotron orbit is in the plane of the film so that the dominant scattering mechanism is grain boundary scattering. When the film is oriented parallel to the field, the carriers are pushed towards the surface of the film, thereby increasing surface scattering. Therefore, the observed resistance in configuration 2, R_C2, reaches its maximum when the plane of the film is parallel to the field, in agreement with expectations.

We use the Van der Pauw method to extract the resistivity, ρ, of films using R_C1 and R_C2, as shown in Fig. 2(d). It should be noted that in a quantitative comparison between Figs. 2(b) and 2(c), R_C1 has a much larger angular dependence than R_C2. Therefore, the angular dependence of ρ is dictated by R_C1. As a result, plots of ρ versus θ very closely resemble Fig. 2(b). We plot, in Fig. 3(a), the ratio of the maximum over the minimum value of resistivity, ρ_max/ρ_min, taken from the rotation experiment, as a function of temperature for various film thicknesses (3–80 nm). At room temperature, electron-phonon scattering is the dominant scattering mechanism and keeps ρ_max/ρ_min small. As the temperature is lowered, ρ_max/ρ_min reaches a maximum, as electron-phonon scattering is reduced in comparison with other scattering mechanisms, such as impurity scattering. We find that samples with a sharper peak in Fig. 3(a), tend to have higher mobility (see discussion below about using Eq. (1) to extract mobility), indicative of a purer sample. For all the films, below ∼80 K, no temperature dependence is observed in ρ_max/ρ_min. This is the temperature range where we will focus the discussion, as it is also the range most relevant to cryogenic Peltier coolers. Focusing on the low temperature regime, we observe that as the thickness of the film decreases, ρ_max/ρ_min also decreases in a near linear fashion, as seen in Fig. 3(b). To understand this trend, the resistivity can be broken up into two components: ρ = ρ_i + ρ_s, where ρ_i is the “intrinsic” resistivity that may vary with magnetic field strength, and ρ_s is the surface scattering component of resistivity which varies with film thickness. As the film thickness decreases, the contribution of surface scattering increases, thereby decreasing the values of ρ_max and ρ_min. This is evidenced in Fig. 3(b) as ρ_max/ρ_min approaches a value of one at very small thicknesses, indicating that ρ_max ∼ ρ_min ∼ ρ_s. Even though ρ_max/ρ_min decreases, we are still able to observe curves that qualitatively look the same as Fig. 2(c) for films down to thicknesses of 3 nm.

When non-textured films are rotated in a magnetic field, the detected change in resistivity is very small compared to the results for the textured films of similar thicknesses. Figure 4(a) shows the temperature dependence of ρ_max/ρ_min for textured films that are 20 and 40 nm thick and for non-textured films that are 25 and 40 nm thick. At temperatures above 200 K, the resistivity of our Bi thin films does not depend on texture, since phonon scattering, which is isotropic, strongly reduces any anisotropic transport. However, we observe a strong deviation in resistivity versus angular orientation dependence for textured versus non-textured films at temperatures < 200 K. At cryogenic temperatures, ρ_max/ρ_min of non-textured films is smaller than that of textured films.

We first discuss the possibility that different scattering mechanisms are responsible for the observed texture dependent trend. While, the non-textured films do have slightly smaller grains, we keep the film thicknesses about an order magnitude smaller than the grain size in both the textured and non-textured case. In these thin films, surface scattering is the main scattering mechanism, given the strong dependence of resistivity on film thickness. It is worth noting that the low temperature values of ρ_max/ρ_min for the non-textured 40 nm thick sample are even less than that of the 20 nm thick textured sample. Based on Fig. 3(b), we would expect the slightly rougher non-textured 40 nm thick film's curve in Fig. 4(a) to fall between that of the 40 nm and 20 nm thick textured films. To more quantitatively compare the transport between textured and non-textured films, we extract the mobility as a function of film thickness for both types of films in Fig. 4(b). To extract the mobility, we fit the ordinary magneto-resistance R_C1 with the following equation at low field (<0.5 T) where the data are well represented by a quadratic equation:^14,18

Using this method, we observe that the mobility, μ of textured films and non-textured films do not differ appreciably, as seen in Fig. 4(b). Therefore, it is very unlikely that the drop in ρ_max/ρ_min for the non-textured samples can come from a change in scattering mechanisms.

Since scattering cannot explain the discrepancy in behavior between textured and non-textured samples, we investigated the MR in the films more carefully: we monitored R_C1 as we incrementally increased the magnetic field. Here, the experiment was conducted in two different film orientations, parallel and perpendicular to the field for both textured and non-textured samples. For the case of a textured film, shown in Fig. 4(c), when the field is oriented perpendicular to the field, we observe the textbook case of an increasing resistance as a function of field strength. In agreement with previous measurements made on polycrystalline Bi films,¹⁹ we observe a much weaker field dependence of the resistance when the textured film is oriented parallel to the field. However, when we measure the non-textured sample in the same manner, we observe the presence of MR in both the perpendicular orientation as well as the parallel orientation, as seen in Fig. 4(d). The existence of MR for both orientations indicates that current flow is anisotropic in the non-textured film, a result of preferential current flow along certain crystallographic orientations.

To fit the high field dependence of the MR up to 9 T, we use a simplified Hikami-Larkin-Nagaoka (HLN) formula,²⁰ in the limit of strong spin-orbit coupling, given as

where ψ is the digamma function, α is a prefactor, L_ϕ is the phase coherence length, and β is the quadratic coefficient which accounts for additional scattering terms. We use Eq. (2) to extract the phase coherence length of both films in both the parallel and perpendicular orientations. Fits to data are shown in Figs. 4(c) and 4(d). For the textured sample, the perpendicular orientation gives a phase coherence length of ∼11.4 nm. Interestingly, the parallel orientation exhibits weak localization as the MR slowly decreases as a function of field strength, and we are therefore not able to use Eq. (2) to fit the data. The extracted phase coherence lengths of the non-textured sample are ∼121.5 and 15.7 nm for the perpendicular and parallel cases, respectively. Again, the out-of-plane roughness is not interfering with the in-plane conduction, as shown from the fact that the phase coherence length in the textured sample is smaller or close to those extracted from the non-textured sample. Finally, the large difference in L_ϕ for the non-textured samples indicates that the transport in non-textured films is highly anisotropic.

The conductivity of a material depends on the effective mass, density of states, and scattering rates. Compared to the anisotropy of the effective mass, the anisotropy that could be introduced through scattering mechanisms is small. Therefore, we hypothesize that the preferential current flow direction in a crystal grain is dominated by the large mass anisotropy in Bi. To test this hypothesis, we compare our results on Bi with the same measurements that we performed on Sb thin films (see the inset of Fig. 4(d)). Unlike Bi, which has a mass anisotropy of ∼200, Sb has a mass anisotropy of 6.^6,7 We use Sb as our control material, as opposed to another metal or semiconductor, because similar to Bi, Sb has the same rhombohedral lattice structure and is also a semimetal.⁷ In the inset of Fig. 4(d), we observe that the non-textured Sb thin film exhibits a weaker MR when it is oriented parallel to the field. Therefore, it seems that the anisotropic conduction that we observe is a direct result of the large mass anisotropy of the electron effective mass in Bi.

In conclusion, we have demonstrated that removing texture from Bi thin films can induce anisotropic conduction, giving rise to preferential current flow directions in each grain depending on its crystallographic orientation. Our results show that this preferential current flow occurs for T < 80 K, but not at higher temperatures. At higher temperatures, scattering between carriers and phonons destroys these preferential pathways, thereby making electrical conduction in textured and non-textured films nearly identical. However, for applications operating at low temperature (e.g., cryogenic Peltier coolers operating at 77 K or below), mass anisotropy will give rise to texture dependent effects on electrical transport, especially in materials that have very large mass anisotropy like Bi and Bi_1−xSb_x. Looking forward, control over the arrangement and alignment of the crystalline orientation of individual grains could prove to be technologically important since texturing would allow control over current flow within a material with a single chemical constituent, without the use of external fields.

The authors would like to thank Professor Elena Rogacheva and Professor Joseph Heremans for fruitful discussions and experimental advice. A.D.L., S.T., and M.S.D. acknowledge the AFOSR-MURI Grant No. FA9550-10-1-0533 for funding. F.K. and J.S.M. would like to thank the support they received from NSF Grant No. DMR-1207569, 0907007, ONR Grant No. N00014-13-1-0301. Part of this work was carried out at the MIT-CMSE shared experimental facilities.