Radio Frequency Tunable Oscillator Device Based on Micro-crystal
Radio frequency tunable oscillators are vital electronic components for signal generation, characterization, and processing. They are often constructed with a resonant circuit and a ”negative” resistor, such as a Gunn-diode, involving complex structure and large footprints. Here we report that a piece of , micron in size, works as a current-controlled oscillator in the frequency range. is a strongly correlated Kondo insulator that was recently found to have a robust surface state likely to be protected by the topology of its electronics structure. We exploit its non-linear dynamics, and demonstrate large AC voltage outputs with frequencies from to by adjusting a small DC bias current. The behaviors of these oscillators agree well with a theoretical model describing the thermal and electronic dynamics of coupled surface and bulk states. With reduced crystal size we anticipate the device to work at higher frequencies, even in the THz regime. This type of oscillator might be realized in other materials with a metallic surface and a semiconducting bulk.
pacs:72.15.Qm, 71.20.Eh, 72.20.Ht
Samarium Hexaboride is a mixed-valence Kondo insulator Aeppli and Fisk (1992) with many unusual properties, perhaps most well-known for its resistance-temperature dependence that resembles both a metal and an insulator Cooley et al. (1995). Recently, was proposed to be a topological Kondo insulator Dzero et al. (2010, 2012), with a Kondo insulating gap in the bulk and a gapless (metallic) Dirac state on the surface, which is protected by time-reversal symmetry (see Ref. [Dzero et al., 2016] for a recent review). This metallic surface has been verified in by both transport Kim et al. (2013); Wolgast et al. (2013) and ARPES Neupane et al. (2013); Xu et al. (2013) experiments, with unusual spin texture Xu et al. (2014). The surface state survives non-magnetic perturbations such as electric gating Syers et al. (2015) and mechanical abrasion Kim et al. (2013); Wolgast et al. (2013); Syers et al. (2015), but is destroyed by magnetic dopants that break time-reversal symmetry Kim et al. (2014). Quantum oscillation Li et al. (2014) suggests that the dispersion of the surface state is Dirac-like, similar to that of graphene. The bulk of is extremely insulating Kim et al. (2013); Wolgast et al. (2013) and free from impurity conductions, unless under extremely high pressure Cooley et al. (1995) or magnetic field Tan et al. (2015), when the density of states of the bulk Fermi surface starts to emerge. The combination of a truly insulating bulk, a robust surface state, and strong electronic correlation Alexandrov et al. (2015); Efimkin and Galitski (2014); Iaconis and Balents (2015); Nikolić (2014) makes a promising candidate to search for useful properties Dzero et al. (2016). Up to date, most research has been focused on the equilibrium properties of ; its transient dynamics could also be interesting.
When biased with a few mA of DC current, self-heating in causes a large nonlinear resistance Kim et al. (2012), which could lead to oscillation behavior. In a prior paper Kim et al. (2012) we found that mm-sized crystals coupled to an external capacitor generate AC voltages at frequencies up to kHz with a few mA of DC bias current. While we speculated that this oscillation behavior might be related to the coupled electric and thermal dynamics of , the exact mechanism was not understood. This was in part because the surface state had not been discovered at that time Kim et al. (2012), making correct modeling impossible. It was unclear how and if the intricate oscillations could be pushed to higher frequencies. With the recent advancements in the understanding of , we are now able to develop a model with both the surface and bulk states. In this model, the origin of the oscillations lies in the strong coupling between the thermal and electrical phenomena in this surface-bulk system. Sufficient Joule heating, induced by an external DC current, can heat the bulk into a less insulating state, and trigger coupled temperature and current oscillations in both bulk and surface states. This model describes various aspects of the oscillations well and predicts that the frequency will rise sharply with reduced surface area, which we found to be true in micro-crystals.
Shown in Fig. 1 is a oscillator device based on a --sized micro-crystal. DC bias currents () up to are applied through two platinum wires in diameter that are spot-welded onto the crystal surface. And the output voltage is measured via the same platinum wires using an oscilloscope. The crystal itself is placed at low temperature in a cryostat, while other electronic components are held at room temperature outside the cryostat. If not explicitly stated, the measurements were performed when the cryostat is at . For mm-sized crystals an external capacitor is required to generate oscillation Kim et al. (2012), such an external capacitor is found to be unnecessary for micro-crystals described here, likely due to the self-capacitance Kim et al. (2012) we found in . The exact origin of the self-capacitance is still unclear, but it seems to scale with the surface area, suggesting its relevance to the surface state. The oscillation behavior doesn’t depend on the exact geometry of the crystal: as shown in Fig. 1 inset this oscillator is based on a rather irregularly shaped sample. The output of this oscillator can be continuously tuned from to by varying between and . Shown in Fig. 1-a are outputs for three representative , and . And the Fourier transformations are shown in Fig. 1-b, showing a typical full width at half maximum (FWHM) spectral width of only . We note that this is achieved without a phase-locked loop circuit.
We find that the center frequency, which is the frequency where maximum oscillation amplitude occurs, rises quickly with smaller crystals. Plotted in Fig. 2-a are the center frequencies for a few representative oscillators of various sizes and geometries, versus their surface areas, which we found to show the highest correlation to center frequency, compared to volume or any single dimension. Projecting the frequency-surface area scaling further, we speculate that THz oscillations might occur for --sized crystals. Operation above is unlikely due to the bulk activation gap in . For each device, a range of external capacitors can be used to generate oscillations, as illustrated in Fig. 2-f, with no need for an external capacitor for the two highest frequency devices ( and ).
Both surface and bulk states are found to be essential for oscillation to occur. The oscillation amplitude diminishes at temperatures above , when bulk conduction dominates; or below , when surface conduction prevails. Optimal operation occurs at around when both the bulk and surface contribute to the electric conduction. This trend can be seen in the supplemental SM () Fig. S2-c. It is known that in , magnetic dopants such as Gd destroy the conductive surface state Kim et al. (2014), while inducing little change to the bulk insulating gap. We fabricated several devices using crystals from the same Gd doped growth batch as described in Ref.Kim et al. (2014). These Gd: samples are insulating to the lowest measurement temperature with no sign of a conductive surface. And the measured activation gap is found to be , which is very close to the bulk gap in (supplemental SM () Fig. S4), suggesting its bulk is very similar to pure . With the destruction of the surface state, no oscillation was observed from these crystals, despite testing a wide range of parameters such as temperature, bias current and external capacitance. We have also performed control experiments using another well-known Kondo insulator, (supplemental SM () Fig. S4, Fig. S5) Hundley et al. (1990); Cooley et al. (1997), which has a Kondo gap of similar to that of but without a surface state at least down to our lowest measurement temperature of . We couldn’t find any oscillation from samples.
Taking into account both surface and bulk states we have built a model to describe the oscillations process depicted in Fig. 3-d. It is based on charge and heat conservation equations and can be casted as follows
Here and are surface and total currents through the sample; is combined internal and external capacitance;, where and are the surface and bulk resistances with insulating gap ; and are the heat capacity dominated by phonons and heat transfer through external leads with temperature . and are heat capacity and bulk resistance in the thermal equilibrium. The detailed analysis of the model is presented in supplemental materials SM () and here we outline our main results.
In dimensionless units the dynamics of the model depend on four parameters , , , and , where and are time scales of thermal and electrical processes. The latter two can be easily tuned in the experiment by the DC bias current or capacitance , and they control the behavior of the model. For the first two we use and , which correspond to a 0.7-mm-sized sample (supplemental SM () Fig. S3) at . The system of equations (1) has only one fixed point (at which and ), which is not allowed to be a saddle one. The regime diagram of the model, presented in Fig.3-a, has the steady state regime, corresponding to a stable fixed point. The regime is separated by the Hopf bifurcation line from the limit-cycle regime, supporting nonlinear time-dependent oscillations of the current and the temperature and corresponding to an unstable fixed point. Fig. 3-b and -c present the phase curves for the system (1), which illustrates the fate of the unstable fixed point, and the result of its explicit numerical integration. The oscillations, illustrated in Fig. 3-d, can be separated into four phases: 1) Joule heating. The surface current achieves maximum, while temperature is minimum. 2) Discharging of capacitor. The energy flows from electrical to thermal. 3) Cooling phase. Energy dissipates to wire leads. The current is minimized, while the temperature is maximized. 4) Charging of capacitor. The energy flows from thermal to electrical. The system is open and non-equilibrium, but during the second and fourth phases the energy of the system is approximately conserved.
In Fig. 4 we compare the modeling results with experimentally measured oscillation behavior in sample . The raw experimental data can be found in supplemental SM () Fig. S3. As shown in Fig. 4-a and -b, oscillations appear if the capacitance is larger than the minimal value and only in a finite interval of currents, , which corresponds to conditions and in our model (1) according to the regime diagram in Fig. 3-a. Illustrated in Fig. 4-c and -d, between and the dependence of the frequency on the current is linear, while the dependence of the amplitude has a bell-shaped dependence. For a fixed DC bias current, the amplitude of the oscillations increases with capacitance until saturation, while the frequency smoothly decreases. The critical values of currents and capacitance differ from sample to sample, but the behavior is general for all of them. According to our model, the oscillation frequencies are given by the inverse time scale , which drastically decreases with sample surface area, in agreement with the experimental trend (Fig. 2-a).
While the major focus of this paper is on oscillators operating at low temperature based on proposed topological Kondo insulator , the model developed here, in fact, describes a general system of a semiconductor and a metallic channel thermally and electrically coupled together. It is therefore in principle possible to realize such a tunable oscillator in other materials and at ambient temperatures. Candidate systems are / topological insulators Qi and Zhang (2011); Hasan and Kane (2010), or less-exotic semiconductor quantum well heterostructures with two-dimensional electron gas. Consider a quantum well embedded in undoped narrow-band semiconductor InAs sample with length , width and height . At room temperatures undoped InAs has resistivity , heat capacitance , heat conductivity , and density . As a result for a typical resistance of two-dimensional electron gas , the condition is satisfied. The parameters of our model, given by equations (1), can be estimated as and . The first condition is satisfied if the time of thermal processes matches the time of electrical processes , which can be achieved for a capacitance . The second condition, , is satisfied for electric current . The observation of oscillation in this nanostructure may demand fine-tuning of parameters; nevertheless we are optimistic that the conditions can be satisfied at room temperature.
Acknowledgements: This material is based on research sponsored by Air Force Research Laboratory (AFRL) and the Defense Advanced Research Agency (DARPA) under agreement number FA8650-13-1-7374.
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Supplemental Material: ”Radio Frequency Tunable Oscillator Device Based on Micro-crystal”
Appendix A Device Fabrication
We used Al flux growth in a continuous Ar purged vertical high temperature tube furnace with high purity elements to grow all single crystals. The samples are leached out in sodium hydroxide solution. The surfaces of these crystals were carefully etched using an equal mixture of hydrochloric acid and water for one hour to remove possible oxide layer or aluminum residues. Samples used in the experiments were selected from a batch of samples based on their size. The exposed surfaces are planes. Platinum wires are attached to the samples using a micro spot welding. Measurements were carried out in a cryostat () using a standard DC voltage supply and oscilloscope. Only the crystal was held at low temperature, while all other components were held at room temperature. Low resistance wires connect the crystal to the current supply, capacitor, and oscilloscope in parallel.
Appendix B A Low Frequency Oscillator
Appendix C Details of Theory Model
To describe voltage oscillations on the surface of we have developed a phenomenological model based on charge and energy conservation equations, which can be casted as follows
Here and are the surface and the total currents through the sample; is internal or/and external capacitance; , where and are the surface and bulk resistances; the former is assumed to be temperature independent, while the dependence of the latter is crucial; is the heat capacity, which is supposed to be dominated by the bulk phonons; is the heat capacity and is the bulk resistance in the thermal equilibrium at temperature , while is the energy gap in the insulating bulk; is the temperature independent heat transfer rate trough external leads, which plays the role of the bath, and is their temperature. The system S1 needs to be supplemented by initial conditions. We assume that the surface current settles down without any delay, while the temperature is equal to the temperature of leads. The equations for the model (S1) are nonlinear, have reach behavior and closely capture oscillations observed in the experiment.
In dimensionless units the dynamics of the model is controlled by four parameters , , , and ). The former two are not tuned in experiment, and weekly influence the behavior of the model. The latter two can be easily tuned by the current or capacitance , and they control the behavior of the model. They can be rewritten as and , where and are the time scales of the electrical and thermal process, while is the current sufficient to change the temperature to . For numerical calculations we fix the first two parameters as (it corresponds to temperature ), and (it originates from the fitting for the sample 5). It should be noted that the value of cannot be directly extracted from the experimental data, since the depth of sample which temperature is affected by currents is unknown. From the value we estimate the depth as .
with and the corresponding matrix we denote as . According to the general theory of dynamical systems with two degree of freedoms, the behavior of the system in the vicinity of a fixed point is defined by the signs of and . It can be shown explicitly, that and the fixed point of the model is not allowed to be a saddle one. The value of can be positive (unstable spiral or nod), negative (stable spiral or nod), and the transition between these two regimes, , corresponds to the Hopf bifurcation. The stable fixed point corresponds to the steady state to which the system relaxes, while from the unstable fixed point the system flows to the limit-cycle behavior, which supports nonlinear time-dependent oscillations of the current and the temperature. The phase diagram of the model is presented in Fig. 3-a for different values of the parameter . Its area is dominated by the steady state, and only appears for a rather narrow range of parameters for the limit-cycle behavior. It appears only in an interval of currents and only for . The former condition demands Joule heating, induced by the currents, to be enough strong, , to make the bulk conductive. Nevertheless, if the system is overheated, , the bulk current dominates and oscillations do not appear. The latter condition implies that the time scale of thermal processes is smaller than the scale of electrical processes, which makes possible their coupled behavior. Phase boundaries depend explicitly on parameters , while the general structure of the regime diagram is insensitive to them. The phase curves for the system (S1), which illustrate the fate of the unstable fixed point and the appearance of the limit-cycle behavior, are depicted in Fig. 3-b. Results of the explicit numerical solution of the system in the limit-cycle regime are presented in Fig. 3-c.
Nonlinear oscillations appear in a rather narrow range of parameters and need fine-tuning. Redistribution of current between bulk and surface is crucial and temperature oscillations should be strong enough to change the hierarchy of the surface and bulk resistances. They are supplemented by oscillations of energy between electrical and thermal states and the corresponding times should match each other. These conditions are not easy to be satisfied. The bulk of the material does not need to be topologically nontrivial, and Dirac nature of the surface states is unimportant here, nevertheless topological Kondo insulator is the playground where all conditions required for the effect are naturally satisfied.
Appendix D Raw Data of sample 5 for Fig. 4 in the Main Text
Appendix E Non-oscillation in and devices
As described in the main text we have studied two sets of control samples: and . Unlike , no surface conduction was observed in these samples at least down to . And none of them showed any oscillation despite trying a wide range of operating parameters.
Shown in Fig. S4 are the temperature dependence of resistances and the corresponding Arrhenius plots for 3 representative samples: , with 3 Gd doping, and . Only the sample shows clear resistance saturation at low temperatures due to surface state. From Arrhenius plots in the high temperature region, the activation gap of these samples are found to be rather close: (), (), () respectively. A number of and devices, with sizes range from 0.3 mm to 1 mm have been studied at various conditions. Unlike none of these control samples showed any sign of oscillation. Fig. S5 shows the voltage output of two such samples at a representative temperature of with a few representative DC drive currents. No oscillation can be discerned within experimental resolution.