Monte Carlo simulations of electron photoemission from cesium antimonide

Alkali antimonides have emerged as one of the most promising families of photocathode materials for ultrafast electron diffraction and microscopy and for producing electron beams with record high brightnesses. This is due to their high quantum efficiency (QE) in the visible spectrum, sub-picosecond response times, and reasonably low intrinsic emittance. Moreover, experiments¹ show that a reduction in photoemitted electrons' mean transverse energies (MTEs) can be achieved by cooling alkali antimonide photocathodes to cryogenic temperatures. However, the exact mechanism behind their high photoemission yield at near-threshold wavelengths is not fully understood. The lack of this knowledge has stood in the way of exploiting their near-threshold photoemission properties to their fullest potential. Unfortunately, many of the material properties that are well-known for other relevant scientific and industrial materials, viz., optical absorption and surface electronic properties, are still experimentally unknown or not unanimously agreed upon in the case of alkali antimonides. Hence, any detailed attempt at modelling photoemission from alkali antimonides is plagued with many undetermined parameters.

In this paper, we present a consistent simulation model for photoemission by using parameters from the existing literature,^2,3 which primarily focuses on photoemission away from threshold, and seek to include new aspects that can have a bearing on near-threshold photoemission, like the presence of dopant impurities,³ interfacial effects,⁴ and change in the band gap as a function of temperature. Realizing the need for further experimental studies aimed at determining physical quantities associated with these aspects, we also present our findings in the form of predictions that will scrutinize our assumptions and can act as directions for further experimental investigation.

Cs₃Sb is a popular high QE photocathode material, with experimental data¹ for QE and MTE available at both room and cryogenic (90 K) temperatures. Several research groups in the quest of ultra-low electron beam emittance show interest in operating photocathodes at ultracold temperatures. Given the well-known p-type nature⁴ of Cs₃Sb photocathodes, it is important to look at how this cathode would behave as we cool down to cryogenic temperatures. For instance, a poor cathode electrical conductivity can impose a limit on the amount of extractable charge. Moreover, it is necessary to consider doping-dependent interfacial effects like band bending, which may play an important role in determining the actual effective work function. Cs₃Sb is an ideal playground to pursue and understand these pressing questions.

Our model, described in the section Method in detail, includes these effects, namely, band bending, work function variation as a function of cathode temperature, and the p-type nature of Cs₃Sb. We use Monte Carlo simulations to simulate the photoemission process and to estimate QE, MTE, and the electron response time at three different temperatures: 20 K, 90 K, and 297 K, under two different modes of operation: “reflection mode” (front illumination) and “transmission mode” (back-illumination). In the “reflection mode,” the vacuum side of the photocathode is illuminated, whereas in the “transmission mode,” light is incident on the photocathode via the substrate (required to be transparent). Transmission mode operation is of great interest because of the observed MTE reduction in multialkali antimonide photocathodes⁵ when compared to the reflection mode operation in photoinjectors and the ability to focus the incident laser light to much tighter spots when illuminating from the back.

A 3-step model,⁶ when combined with the semiclassical approach to electron dynamics,⁷ is a simple yet effective way for describing the photoemission properties of semiconductor photocathodes. In this framework, we consider a flat film of Cs₃Sb on a flat, ideal, conductive, and transparent substrate (see Fig. 1). In the first step (excitation), electrons are excited from the valence band to the conduction band of Cs₃Sb. In the second step (transport), they move within the photocathode and scatter. In the third step (emission), electrons with sufficient momentum perpendicular to the surface tunnel out into vacuum. All physical quantities of interest (QE, MTE, and response time) are determined from these outcoming electrons. Next, we describe these steps and their underlying assumptions in detail and elaborate further on the key features of our model.

In our model, we use the valence band density of states obtained from density functional theory (DFT) calculations⁸ for our simulations. Furthermore, disordered semiconductors are known to have an exponential tail (Urbach tail) in their valence band density of states.⁹ Other alkali antimonide photocathodes have shown polycrystalline structures, and while we cannot address polycrystallinity and disorder in detail, we attempt to include their effect by adding an effective exponential tail of width 0.05 eV in the valence band density of states. 0.05 eV is a typical value of the Urbach tail width for semiconductors.⁹ Lacking data specific for Cs₃Sb, we assumed the said value in our simulations.

Also, it is important to consider the majority carrier concentration as a function of temperature, as Cs₃Sb is a p-type semiconductor. The literature suggests two possible routes for dealing with the majority carrier concentration in p-type Cs₃Sb. The first route, suggested by Spicer,⁴ claims acceptor levels being close to the valence band maximum and contributing an effective tail to the valence band density of states. In such a scenario, temperature will not strongly affect the hole concentration and hence electrical conductivity. The second route, suggested by Sakata,¹⁰ posits a sharply defined, isolated acceptor level well into the band gap. His fit values for doping density and acceptor level energy (⁠$NA0∼1021cm−3$⁠, E_A ∼ 0.4 eV above the valence band maximum) imply miniscule electrical conductivity at ultracold temperatures¹¹ and an excess Sb atom every few (⁠$∼5$⁠) primitive unit cells, thereby questioning the soundness of the assumption of an isolated acceptor level. Moreover, an isolated acceptor level well into the band gap contributing to photoemission should provide a clear signature of hot electrons coming from it by causing a notable increase in MTE at room temperature and near-threshold exciting wavelengths. As of now, experimental results do not report this increase in MTE. However, electrical conductivity measurements,^12,13 showing a reduction of many orders of magnitude at cryogenic temperatures, support Sakata's view. Figure 2 shows the schematic band structures in the 2 scenarios.

Band gap variations of other popular semiconductors over the temperature range of interest, namely, 20 K–300 K, are of the order of 0.1 eV.¹⁴ This suggests the possibility of band gap change causing the observed change¹ in the Cs₃Sb work function from 90 K to room temperature. Lacking experimental measurements of the band gap of Cs₃Sb as a function of temperature, we resort to empirical models that work well for well-known semiconductors such as Si, Ge, and GaAs. A central feature of these models is electron-phonon coupling and hence the longitudinal optical phonon energy (⁠$ℏωLO$⁠), something that we also use in our description of electron scattering. We use the following formula¹⁵ to calculate band gap changes with respect to temperature:

Here, $Egcn$ is chosen so that E_g = 1.6 eV at 297 K,¹⁶ whereas $ΔEZP$⁠, the electron-phonon coupling strength parameter, is assumed to be 0.1 eV, in agreement with the generic value for semiconductors.¹⁵ 

Interface effects, on the other hand, make the band structure inhomogeneous with respect to the longitudinal coordinate perpendicular to the photocathode surface (z). This inhomogeneity is known as band bending and has been discussed in detail¹⁷ for Cs₃Sb.

Thermodynamically, in the Sakata model, the photocathode structure has three systems in equilibrium, namely, the substrate, the cathode (consisting of the valence band and an acceptor level), and the surface. E_F refers to the common chemical potential achieved by charge transfer among these systems. The shift in the VBM (actually in the entire band structure; but we focus on the VBM for simplicity) is related to the equilibrium charge density in the photocathode. Therefore, a Poisson equation can be written to describe the band bending profile.^18,19 As the Poisson equation is a second-order ordinary differential equation, we need two boundary conditions to solve it. In our case, the Cs₃Sb-vacuum interface and the Cs₃Sb-substrate interface contribute a boundary condition each. We assume the substrate interface to be a metal-semiconductor contact, with the work function of the substrate (⁠$ϕS$⁠, taken to be 4.4 eV here) higher than the work function of the semiconductor, that is, the sum of the band gap (E_g) and the bulk electron affinity (χ). If the semi-conductor is p-type, the VBM energy is located $S=ϕsubstrate−Eg−χ$ above E_F.²⁰ Hence, the positions of the VBM at the vacuum interface (B) and at the substrate interface (S) with respect to E_F are the boundary conditions to the Poisson equation given below:

Here, T denotes the photocathode temperature. $y=EV(z)−EFkBT, λ=h2πmΓ,v*kBT$⁠. Here, $F12$ denotes the Fermi-Dirac integral of order $12$⁠.

Solving Eq. (2), we obtain upward band bending close to the Cs₃Sb-substrate interface and downward band bending close to the Cs₃Sb-vacuum interface (see Fig. 3). Band bending profiles for the photocathode thickness we consider also show some remarkable characteristics as a function of temperature: at room temperature, we can break the band bending profile into two isolated band bending regions, approximately 10 nm in width each, near the substrate and vacuum interfaces, separated by a flat, bulk-like region. On the contrary, at cryogenic temperatures, there is an internal electric field of at least ∼1 $MVm$ throughout the semiconductor thickness.

One can also calculate the band bending profile in Spicer's scenario (see Fig. 4), where acceptor impurities form a separate tail in the valence band density of states. It is important to note that this tail is not the Urbach tail, even though both of them look similar in form. Urbach tail comes from the possible disorder in the structure of Cs₃Sb, whereas Spicer's tail of acceptor levels is deduced from the p-type conductivity of Cs₃Sb. Due to the absence of further details on the nature of Spicer's tail and eventually finding the Urbach tail to have only a minor effect on the simulated photoemission properties, we neglect photoemission from Spicer's tail.

Band bending in Spicer's scenario is different from that in Sakata's scenario in 2 ways. In Spicer's scenario, the ionized acceptor concentration is independent of z and T, which is not the case for Sakata's scenario. The electric field in the depth of the photocathode beyond the band bending region in Spicer's scenario at 20 K, 90 K, and room temperature is negligible when compared to the electric field in the band bending regions close to the interfaces formed by the photocathode, whereas the same electric field in Sakata's scenario is not negligible at 20 K and 90 K.

The absorption of light incident on a semiconductor depends on the underlying band structure. If the valence band maximum is bent upwards so that it is well above E_F, no excitation of electrons by light can occur because practically no electrons occupy the states from where they can be excited into the conduction band. This reduces the absorption of light in the photocathode near the substrate. To describe the absorption of light by including band bending effects, we first need reference attenuation coefficients for the bulk (α₀).

In our model, for Spicer's scenario, we derive α₀ from a Lorentz oscillator fit² to experimental optical data,¹³ where the optical constants are described as follows:² 

ω and ω_T denote the angular frequency of incident light and the resonance angular frequency, respectively. n and $κ=α0c2ω$ denote real and imaginary parts of the refractive index, respectively, and ϵ_s and $ϵ∞$ denote static and high frequency dielectric constants, respectively. Γ denotes the damping coefficient or the width of the resonance due to absorption of light at ω_T. Intuitively, absorption starts at $ℏ(ωT−Γ)$ and peaks at $ℏωT$⁠. So, $ℏ(ωT−Γ)$ should roughly correspond to the band gap. In the present case, $ℏ(ωT−Γ)$ = 1.74 eV is quite close to the band gap values we consider.

However, due to an extra acceptor level in Sakata's scenario, we should also consider absorptions from the acceptor level to the conduction band and from the valence band to the acceptor level (E_A) in order to calculate α₀. Absorption from the acceptor level (E_A) to the conduction band is negligible due to the ionized acceptor concentration being much lower than the valence band density of states. Hence, we only need to add another Lorentz oscillator corresponding to electrons going from the valence band to the acceptor level (E_A). As seen in Fig. 2, the original Lorentz oscillator peak has a higher Γ (resonance width) due to states available around both the initial and the final states. The new oscillator should have a smaller Γ, as the electrons excite into a state with sharply defined energy (E_A). We approximate this new resonance width to $Γ2$⁠, to avoid the complexity of a detailed, ab initio calculation of modified optical constants. As described earlier, we define $ωT′$⁠, the new resonance angular frequency, to be at $EA−EVℏ+Γ2$⁠. Moreover, we assume the 2 oscillators to be equal in strength. This results in the following expression for the optical constants in Sakata's scenario:

Figure 5 describes the optical constants in Spicer's and Sakata's scenarios based on the Lorentz oscillator model. Note that this additional oscillator modifies ϵ_s. So, ϵ_s is different for Sakata's and Spicer's scenarios.

After finding α₀ in the above-mentioned manner, we calculate a new, position-dependent attenuation coefficient by appropriately scaling with the number of excitable valence band electrons. Mathematically, we write

Here, BB(z) denotes the valence band maximum with respect to E_F as a function of z and g(E) the valence band density of states. Note that we neglect any substrate light reflection and further absorption of light reflected back into the photocathode.

In the semi-classical approach, an electron in the conduction band moves with a momentum equal to the crystal momentum of the state it occupies. The electron mass is replaced by the conduction band effective mass. Furthermore, we assume a single, spherical, parabolic Γ-valley to determine the crystal momenta of excited electrons and the effects of further scattering. We neglect the band non-parabolicity, as the non-parabolicity of the conduction band Γ valley band structure suggests $<25%$ increase in the polar optical phonon scattering rate at electron kinetic energies less than 0.15 eV, which correspond to our primary region of interest, namely, photon energies near the photoemission threshold. As a consequence, crystal momenta are chosen isotropically from the surface of a sphere. Also, we assume that any valence band electron with sufficient energy from the exciting photon can always transition into the conduction band, irrespective of whether there are states available in the conduction band at the required value of the crystal momentum. Therefore, we use the valence band density of states rather than the joint density of states (JDoS) for determining the distribution of initial kinetic energies of the electrons in this step of the 3-step model. Further discussion regarding crystal momentum conservation can be found in the section Results.

Band bending acts as an external electrostatic potential because the crystal momentum of an electron changes as it moves across parts of the photocathode with relative shifts in the position of the conduction band minimum (CBM). We include this in the form of an external electric field acting on electrons.²¹ 

The next important aspect is electron scattering—a phenomenon key to explaining observed QE and MTE. A no-scattering assumption overestimates QE and MTE for semiconductor photocathodes.⁵ There are a number of electron scattering mechanisms described in the literature²² for semiconductors. Electron-electron scattering occurs when excited electrons have kinetic energy greater than the band gap, well outside the range of the experiments we aim to describe here. We also neglect in our treatment elastic and nearly elastic scattering mechanisms, as the rates of these scattering mechanisms in Cs₃Sb are much smaller when compared to polar optical phonon scattering.² Moreover, effects due to other valleys above the Γ valley become significant, in both the excitation and the transport step, when incident photon energy becomes greater than the sum of the work function and an inter-valley gap. In Cs₃Sb, the smallest intervalley gap is the $Γ→X$ valley gap, that is, 0.4 eV.⁸ Therefore, we expect intervalley scattering to become important at photon energies greater than ∼2.2 eV. By neglecting inter-valley phonon scattering, we overestimate the QE and the MTE at incident photon energies greater than ∼2.2 eV.

Therefore, we consider only polar optical phonon scattering in our simulations. The polar optical phonon scattering rate,²³ between initial crystal momentum $k→$ and final crystal momentum $k′→$⁠, is given as follows:

$W±$ correspond to absorption and emission of longitudinal optical phonons, respectively. Here, $ℏωLO$ denotes the longitudinal optical phonon energy. Lacking experimentally measured Cs₃Sb Raman/infrared spectra, we assume $ℏωLO$ = 0.022 eV, from Raman spectra of multialkali antimonide photocathodes.^5,24 Scattering rates calculated using Eq. (6) at 297 K and 90 K are reported in Fig. 6. The scattering rate at 20 K, not shown here, has a phonon emission part comparable to 90 K and 297 K, whereas the phonon absorption part is ∼4 orders of magnitude below 90 K.

Apart from the scattering rate, we also need to calculate the scattering angles. The probability distribution for the polar angles θ and $ϕ$ between $k→$ and $k′→$ is given as²² (see also Fig. 7)

For further details regarding the theory of this scattering mechanism, we refer the reader elsewhere.^22,23 Between scattering events, electrons experience the force from internal electric fields due to band bending. Additionally, electrons relaxing to the bottom of the conduction band are discarded from the simulation.

In a semiconductor photocathode, without interfacial effects, one would expect the surface barrier faced by outcoming electrons to be χ. But due to charge transfer between surface states and a p-type conducting bulk, the VBM at the surface can shift down drastically, causing a lowering of the surface barrier. Due to lack of information about the surface states, we cannot determine this shift in Cs₃Sb. The position of the Cs₃Sb-vacuum interface VBM with respect to E_F is the only ad-hoc parameter in our treatment.

Important parameters used in our simulations are given in Table I.

Figures 8 and 9 report the plots of QE, MTE, and response time (defined as the time taken for the first 68% of outcoming electrons to emit into vacuum, starting from the beginning of the transport step) for a 150 nm thick photocathode operated in reflection and transmission modes. We use wavelength-dependent optical constants (see Fig. 5) to calculate the reflectivity of the photocathode for the purpose of calculating QE. Based on the experimental conditions typically used to grow Cs₃Sb for use in photoinjectors, the thickness value has been chosen to be 150 nm.²⁶ We compare our simulation results with experimental data obtained from photocathodes operated in the reflection mode at room temperature (297 K) and 90 K.¹ We also present simulated photoemission properties corresponding to operating the photocathode at 20 K. All of these simulations neglect post-emission effects such as space charge repulsion and assume cathode conductivity to be sufficient at all temperatures, aligning well with the experimental conditions of low electron beam currents.

In principle, conservation of transverse momentum should be applied to estimate the MTE of photoemitted electrons in vacuum. Hence, the MTE of photoemitted electrons should scale by a factor $mΓ,c*me$ with respect to their MTE in the bulk just before photoemission.²¹ This factor, in the case of Cs₃Sb being $∼0.23$⁠, should result in MTEs ∼0.23 times the MTEs in the bulk, thus severely underestimating the experimentally measured MTE values.¹ The interaction of outgoing electrons with the photocathode surface is an active topic of research²⁷ and outside the scope of our work. Effects inconsistent with transverse momentum conservation have also been observed at other material-material interfaces. For example, photon-assisted electron tunneling measurements on hydrogen-annealed metal-SiO₂-Si contacts²⁸ and ballistic electron emission microscopy (BEEM) spectra from the Au-Si(111) interface²⁹ do not turn out as expected from transverse momentum conservation. Moreover, we have neglected any effect on the MTE due to surface roughness. A quantitative description of such effects requires detailed information about the cathode surface morphology and structure²⁷ that is beyond our present knowledge. Similarly, we neglect image charge based effects³⁰ on the effective energy barrier at the photocathode surface. For these reasons, the MTEs reported in Figs. 8 and 9 are calculated using the transverse energies of the electrons just before emission into vacuum.

The qualitative agreement between QE in simulations and experiments at 297 K and 90 K in Figs. 8 and 9 is quite convincing, indicating that the relevant physical processes have been captured by our model. Our simulations also replicate the reduction in MTE and QE in the transmission mode, again qualitatively matching the trend experimentally observed in multialkali antimonide photocathodes.⁵ 

Simulations also reproduce the expected saturation of MTE to the equivalent cathode lattice temperature (⁠$kBT$⁠) for temperatures as low as 20 K, aligning with the prediction of ultra-low MTE values at ultracold temperatures, and confirm the experimental observation of sub-picosecond response times in Cs₃Sb photocathodes.³¹ While it is challenging to perform response time measurements at sub-picosecond time scales, we can resort to the fastest streak camera resolutions,³² around 200 fs, as approximate indicators of response time. These values match with our response time predictions. An interesting feature captured by our simulations is the shortening of response time at longer wavelengths of incident light. This behaviour comes from long tails in the outcoming electron bunches being cut off at longer wavelengths of incident light due to polar optical phonon scattering and the surface energy barrier (energy difference between the vacuum level and the Cs₃Sb-vacuum interface CBM). At shorter exciting wavelengths of incident light, these delayed electrons can still come out eventually, due to their high initial kinetic energies.

The above-stated observations hold true for both Spicer's and Sakata's scenarios. The key difference between Sakata's and Spicer's scenarios is the increase in MTE at room temperature around the 740 nm wavelength of incident light because of photoemission from the isolated acceptor level becoming comparable with photoemission from the valence band of Cs₃Sb. Another difference is the slight increase in QE because of the higher escape probability of electrons coming from the acceptor level, when compared to valence band electrons that are lower in kinetic energy. Furthermore, at cryogenic temperatures, due to the internal electric field, the MTE in Sakata's scenario is higher than the MTE in Spicer's scenario. At room temperature in Sakata's scenario close to the photoemission threshold, the MTE in the transmission mode is higher than the MTE in the reflection mode due to a higher proportion of electrons coming from regions near the substrate with upward band bending. This effect is not seen in Spicer's scenario due to weaker absorption of light (see Fig. 5). The weaker absorption in Spicer's scenario causes a lower proportion of electrons with higher kinetic energy coming from near-substrate regions in the transmission mode.

Despite the satisfactory description our model provides, there are a number of unaddressed points needing further experimental investigations. For example, we assume a crystalline, single valley model of Cs₃Sb, whereas microscopic images of other multialkali antimonides suggest polycrystallinity/amorphousness.³³ X-ray diffraction/spectroscopy can provide better insights into the crystal structure of Cs₃Sb photocathodes and its variability, which is in turn closely related to the surface morphology and the electronic structure.

A Cs₃Sb photocathode grown via sequential deposition is expected to have a surface layer of $Cs$ that lowers the work function due to band bending. So, we need to know the surface state energy and its occupancy, in order to know B beforehand rather than treating it as an ad-hoc parameter. Hence, even the effective energy barrier at the photocathode surface may vary significantly with temperature. Similarly, optical constants may vary from sample to sample. By using transmission mode photocathodes, it is possible to check how strongly absorption coefficient, refractive index, etc. depend on the usual fluctuations in sample preparation recipes.

Moreover, the generic electron-phonon coupling we use for calculating the band gap as a function of temperature may not be accurate enough to reproduce experimental results for Cs₃Sb. In our simulations, the variation of band gap with temperature is the crucial knob determining the photoemission threshold, and hence, its accurate experimental determination as a function of temperature through experimental probes such as photoconductivity is necessary. Even the longitudinal phonon energy, $ℏωLO$⁠, is not known experimentally for Cs₃Sb. Raman/infrared spectra of Cs₃Sb can be used to measure $ℏωLO$⁠. This will also help in determining whether electron-phonon coupling is the main reason behind the variation of the band gap and hence the work function of Cs₃Sb as a function of temperature.

The effect of doping on band gap is another major aspect that remains outside the scope of our paper. In the case of silicon, for doping densities similar to the ones we consider for Cs₃Sb, band gap changes significantly.¹⁴ In-situ Hall conductivity measurements at cryogenic and room temperatures of photocathodes can provide an accurate measure of the doping density (⁠$NA0$⁠) and the acceptor level energy (E_A) and also help us better answer the pressing question about the electrical conductivity of Cs₃Sb at cryogenic temperatures.

Modification of absorption coefficients due to an extra, isolated acceptor level in Sakata's scenario has been modelled only preliminarily. A detailed ab initio calculation can help us to describe the effect of an isolated acceptor level in more detail. Moreover, no signs of an isolated acceptor level have been found in experimental absorption data.¹² Therefore, spectroscopic measurements such as photoluminescence are needed to characterize various samples grown via various procedures, to settle the debate between the 2 possible scenarios (Sakata and Spicer).

Contrary to our expectations, the Urbach tail in reproducing the temperature-independent exponential tails in the experimental QE curves^1,34 as a function of photon energy has proven to be of limited importance, for reasons we do not know yet.

An increase in MTE at wavelengths close to the threshold is a key prediction of Sakata's scenario. In particular, precise MTE measurements at wavelengths greater than 730 nm at room temperature for Cs₃Sb photocathodes are necessary. However, experimental results for other alkali antimonides³⁵ suggest no increase in MTE close to the photoemission threshold.

Spicer's scenario contradicts available electrical and Hall conductivity and thermoelectric power measurements. In order to resolve this inconsistency, we need to understand better the reason behind the tail of acceptor levels in the valence band density of states. Stoichiometric information from samples grown via different recipes can help us calculate the properties of these acceptor states in an ab initio manner, for example, with a supercell DFT calculation. We believe that photoemission from such acceptor levels will result in an exponential tail in QE as a function of incident photon energy matching the one observed by Spicer⁴ and others,^1,12,34 instead of the sharper cut-offs we see in near-threshold QE in our simulations.

We have provided simulation studies that reproduce experiments¹ qualitatively in terms of near-threshold MTE and QE at room temperature and 90 K reasonably well. The simulated, sub-picosecond response times bolster the potential of alkali antimonides in ultrafast electron applications. Interestingly, our simulations predict shortening of response time at longer wavelengths, due to scattering and surface barrier, eliminating electrons spending longer times in the photocathode.

Future extensions to our simulations should include polycrystallinity and roughness. Additionally, exploring cathodes with different band bending profiles at the substrate and at the vacuum interfaces can provide a way to engineer photocathodes with desired photoemission properties. Given the generic nature of our simulation method, we can extend our simulations to other existing and prospective photocathode materials and layered structures. We intend to perform experimental studies of QE and MTE of Cs₃Sb photocathodes at 20 K along with other key semiconductor parameters as suggested by our study.

This work was funded by grants from the National Science Foundation (PHY-141631 and PHY-1549132) and the Department of Energy (DE-SC0014338 and DE-SC0011643). We would like to thank Hyeri Lee, Kevin Jensen, Dimitre Dimitrov, Siddharth Karkare, Suresh Vishwanath, Chandra Joshi, and Guru Bahadur Khalsa for useful inputs.