Coulomb disorder in three-dimensional Dirac systems
In three-dimensional materials with a Dirac spectrum, weak short-ranged disorder is essentially irrelevant near the Dirac point. This is manifestly not the case for Coulomb disorder, where the long-ranged nature of the potential produced by charged impurities implies large fluctuations of the disorder potential even when impurities are sparse, and these fluctuations are screened by the formation of electron/hole puddles. In this paper I present a theory of such nonlinear screening of Coulomb disorder in three-dimensional Dirac systems, and I derive the typical magnitude of the disorder potential, the corresponding density of states, and the size and density of electron/hole puddles. The resulting conductivity is also discussed.
Due to its long-ranged nature, Coulomb disorder often has dramatic consequences even in situations where short-ranged disorder does not. Thus, for example, Coulomb disorder has always deserved special consideration in the physics of semiconductors,Shklovskii and Efros (1984) and such studies have revealed a great number of diverse and interesting scientific phenomena over the preceding half-century.
The large qualitative difference between short-ranged and Coulomb disorder is particularly pronounced for three-dimensional (3D) Dirac materials, in which the electron kinetic energy is linearly proportional to the momentum according to . To see this difference between short-ranged and Coulomb disorder qualitatively, one can compare the behavior of long-wavelength (small-) electron states in a 3D Dirac system (3DDS) for the two cases. Suppose, for example, that the 3DDS has some concentration of positive and negative impurities per unit volume, each with random position and random sign, and that these are taken to be either short-ranged, with finite range and typical potential , or Coulomb, with charge . In the short-ranged case, an electron wavepacket with size experiences disorder from impurities. The average value of the disorder potential created by these impurities is zero, since impurities with opposite signs are equally plentiful, but statistical fluctuations in the impurity concentration create a typical excess of impurities with one of the two signs. Thus, the volume-averaged disorder potential experienced by the electron is . The electron kinetic energy, on the other hand, scales as . One can therefore conclude that short-ranged disorder has a perturbatively small effect on the electron energy for large-wavelength electron states (i.e., for states close to the Dirac point). This robustness of the 3D Dirac point against short-ranged disorder has long been understood theoretically,Goswami and Chakravarty (2011); Hosur et al. (2012); Kobayashi et al. (2014); Syzranov et al. (2014); Sbierski et al. (2014); Ominato and Koshino (2014) and consequently “Dirac semimetal” phases with vanishing density of states (DOS) are generally predicted to survive short range disorder. (In fact, a very recent paper Nandkishore et al. (2014) has shown that rare resonances between short-ranged impurities can create an exponentially small DOS at the Dirac point.)
Now consider the case of disorder produced by long-ranged Coulomb impurities. As before, an electron wavepacket with large size encloses many impurities of both signs, and the net charge of these is . If one naively calculates the potential energy created by these impurities, one finds that the typical Coulomb potential energy experienced by the electron is (in Gaussian units), where is the dielectric constant. Thus, the disorder potential grows with increasing wavelength as , rather than falling off quickly and becoming irrelevant. Clearly, such Coulomb impurities must have a large and nonperturbative effect near the Dirac point at any finite concentration. As one might expect, the growth of the Coulomb potential at large length scales is in fact truncated by the formation of electron and hole puddles that screen the disorder potential, as is the case with narrow band gap semiconductorsShklovskii and Efros (1972, 1984); Rossi and Das Sarma (2011); Skinner et al. (2012) and two-dimensional Dirac systems like grapheneShklovskii (2007); Galitski et al. (2007) and topological insulators. Beidenkopf et al. (2011); Skinner et al. (2013) This disorder-induced puddling is shown schematically in Fig. 1. It is the purpose of this paper to calculate the typical size and density of these puddles, as well as the corresponding disorder potential amplitude, DOS, and conductivity.
The question of disorder effects in 3DDSs has acquired a particular relevance in recent months, following the experimental discovery of two different 3D Dirac materials Borisenko et al. (2013); Neupane et al. (2013); Liu et al. (2014); Jeon et al. (2014) not long after their theoretical prediction.Murakami (2007); Wan et al. (2011); Burkov and Balents (2011); Turner and Vishwanath (2013); Young et al. (2012) While the existence of a Dirac dispersion in these materials has been established, largely by photoemission experiments, it remains to be thoroughly understood how closely the Dirac point can be probed and to what extent its behavior is masked by disorder. As shown below, the presence of Coulomb impurities has the effect of “smearing” the Dirac point via the creation of electron and hole puddles, and this smearing typically occurs over tens of meV.
The structure and primary results of this paper are as follows. In Sec. II, a self-consistent theory is developed to describe the disorder potential based on the Thomas-Fermi (TF) approximation. Corresponding expressions are derived for the magnitude of the disorder potential [Eq. (7)], the DOS [Eq. (8)], the size of electron/hole puddles [Eq. (9)], and the concentration of electrons/holes in puddles [Eq. (10)]. While the primary focus of this section is on the behavior near the Dirac point, results are also presented for the case when the chemical potential is away from the Dirac point. In Sec. III, the zero-temperature conductivity is discussed, and a result is presented for the conductivity [Eq. (16)] and its minimum value [Eq. (17)]. Sec. IV concludes with a summary and some discussion of recent experiments.
Ii Self-consistent theory of disorder and screening
This paper focuses on a model of disorder in which monovalent Coulomb impurities per unit volume are randomly distributed throughout the bulk of a 3DDS. Such impurities create a random Coulomb potential, and this potential induces an electron/hole concentration , where for electrons and for holes. The value of at a given spatial coordinate is related to the self-consistent magnitude of the Coulomb potential . In this paper, the primary tool for describing this relationship is the TF approximation:
Here, is the local Fermi energy, where is the Fermi wave vector and is the chemical potential of the 3DDS measured relative to the Dirac point. Assuming, generically, that the Dirac point has a degeneracy (which for Weyl semimetals is equal to the number of degenerate Dirac points), the Fermi wave vector is , so that .
If the chemical potential is large enough in absolute value that , one can think that the electron density is relatively uniform spatially, and the corresponding electron DOS is also uniform. In this case, one can straightforwardly define a TF screening radius
Here, is the effective fine structure constant. The TF approximation is valid in cases where the Fermi wavelength is much shorter than the typical scale over which the potential varies, . As can be seen in Eq. (2), this corresponds to . For comparison, the 3DDS CdAs has – eVÅ,Neupane et al. (2013); Borisenko et al. (2013); Jeon et al. (2014) , and , so that – and this approximation is justified. As shown below, the same criterion justifies the use of the TF approximation for the case of .
If is not large, so that the 3DDS is close to the Dirac point, then one cannot consider the electron density to be uniform and the typical screening radius must be found self-consistently. In particular, one can assume that the disorder potential is screened with some unknown screening radius and then calculate analytically the corresponding magnitude of the disorder potential and the resulting average density of states . Inserting the result for into Eq. (2), one arrives at a self-consistent relationship for , which can be solved to give a result for , , and the magnitude of the disorder potential.Stern (1974) This procedure is carried out explicitly in the remainder of the present section.
In a medium with screening radius , the screened potential produced by a single impurity with charge is the Yukawa-like potentialMahan (1990)
If one assumes that impurity positions are uncorrelated, then the mean squared value of the electron potential energy, , can be found by integrating the square of the potential created by a single impurity, , over all possible impurity positions. This gives
For cases where (justified below), the potential at each point in space is the sum of the potentials produced by many independently-located impurities. By the central limit theorem, then, one can assume that the distribution of values of the potential across the system is Gaussian with variance . Within the TF approximation, the value of the DOS at a point with potential is , so that the spatially-averaged DOS is
It is perhaps worth noting that this value for the disorder potential amplitude is smaller than the corresponding result for the surface of a disordered 3D topological insulatorSkinner and Shklovskii (2013) by a factor . This smaller disorder for 3DDSs is a consequence of stronger screening in three dimensions.
In this case, the screening radius , which is generically equal to the correlation length of the disorder potential, defines the typical size of electron and hole puddles (as illustrated in Fig. 1). By Eq. (2), its value is given by
The typical concentration of electrons/holes in puddles is found by equating with , which gives
so that the corresponding number of electrons/holes per puddle is
When , there are many electrons per puddle: . Intriguingly, this value for is independent of the impurity concentration, so that the number of electrons per puddle is independent of the details of the disorder. This universality is reminiscent of the problem of a single supercritical nucleus in a 3DDS, where the maximum observable “nuclear charge” also obtains a universal value .Kolomeisky et al. (2013)
Notice also that at , the correlation length of the potential is much longer than the typical Fermi wavelength , so that the TF approximation is justified. This same condition also guarantees , which validates the assumption of a Gaussian-distributed potential.
It is worth noting that Eqs. (7)–(11) can be derived qualitatively using the following very simple argument (which for simplicity uses ). Consider a volume of size within the 3DDS; this volume is effectively a single electron/hole puddle. Those impurities within the volume can be said to contribute to the potential within it, while others are effectively screened out. The net charge of impurities in the volume is (with a random sign), and this impurity charge is compensated by the charge of electrons/holes, which have total number . Equating with gives . Now one can note that the typical kinetic energy of electrons within the volume, , must be similar in magnitude to the typical Coulomb energy . This equality gives . Combining the two equations for gives , as in Eq. (9), and the other relevant quantities can be found by substitution.
As the chemical potential is moved away from the Dirac point, the magnitude of the disorder potential decreases, as dictated by Eq. (6), and correspondingly the screening radius shrinks. At , puddles of electrons (for ) or holes (for ) dry up, and the system is well-described by linear screening with a spatially-uniform DOS. In this case the disorder potential magnitude becomes
The corresponding DOS approaches that of the non-disordered system,
and the correlation length of the disorder potential is
Equations (7)–(11) and (12)–(14) describe the system in the limits of and , respectively. The crossover between these two regimes can be described by evaluating Eqs. (5) and (6). The result of this process is shown in Fig. 2, where the variance of the disorder potential and the DOS are plotted as a function of the chemical potential . As one can see, these self-consistent equations predict a smooth, monotonic crossover from the puddle-dominated result to the linear screening regime at large . It is worth noting, however, that may in fact exhibit weakly nonmonotonic behavior as a function of , achieving a weak maximum at . Such behavior is predicted theoretically for topological insulators,Skinner and Shklovskii (2013) and arises because at the distribution of values of the Coulomb potential becomes skewed toward those values that bring the system locally closer to the Dirac point, where screening is poorer.
The previous section discussed the disorder potential produced by random Coulomb impurities using a self-consistent theory of screening. In this section I briefly discuss the implications of this screening for the low-temperature conductivity.
In situations where the mean free path for electron scattering is relatively large, , the electron transport is well-described by the Boltzmann equation. In particular, the momentum relaxation time satisfies
(see, e.g., Ref. Burkov et al., 2011). Here, is the momentum change resulting from scattering by an angle and is the Fourier transform of evaluated at wave vector . The final factor of in Eq. (15) arises when backscattering is suppressed as a consequence of the spin texture at the Dirac point, as in Weyl semimetals; omitting this factor does not change the results to leading order in . Evaluating the integral in Eq. (15) gives
assuming , which is a condition of validity for the screened potential used here, and, again, corresponds to .
One can arrive at an expression for the conductivity by combining the expression for with the Einstein relation for conductivity, .Abrikosov, Alexei A (1988) If one assumes a relatively large and uniform electron density (i.e., a large chemical potential ), then and
As the electron density is reduced (the chemical potential is brought closer to the Dirac point), the number of carriers is reduced and the conductivity declines. At , the conductivity achieves a minimum whose value is determined by the concentration of electrons and holes in puddles. The value of this conductivity minimum, , can be estimated by inserting into Eq. (16), which gives
Finally, one can check that the derived expressions indeed correspond to the Boltzmann semiclassical limit . As expected, the value of corresponding to the minimum conductivity, where , gives . Thus, , provided that . At larger the mean free path only increases, so that the Boltzmann equation is a good description everywhere.
Iv Concluding remarks
This paper has presented a simple picture of self-consistent screening of Coulomb impurities in 3DDSs through the formation of electron and hole puddles. Such effects are manifestly not perturbative near the Dirac point, regardless of the impurity concentration, and they have a prominent effect on both the observed DOS and the conductivity. The DOS, for example, vanishes only as the power of the impurity concentration, which suggests that a true Dirac semimetal phase with vanishing DOS may remain frustratingly elusive experimentally.
While the experimental study of 3DDSs is still very young, one can get a sense of the typical scales of the disorder potential by using parameters for the recently-discovered bulk Dirac material CdAs , which has and . Thus far, experimentally studied samples are -type, apparently resulting from uncontrolled doping by As vacancies.Jeon et al. (2014); Neupane et al. (2013); Borisenko et al. (2013) For example, Ref. Jeon et al., 2014 reports relatively large meV, with a corresponding carrier concentration cm; one can expect that the concentration of donor impurities is similar in magnitude. Inserting these parameters into Eqs. (12) and (14) gives an estimated disorder potential of meV and a screening radius nm. The latter seems consistent with the scale of disorder fluctuations seen by scanning tunneling microscopy measurements.Jeon et al. (2014) By Equation (16), this level of disorder corresponds to a mobility cm/Vs, which also closely matches the value seen in experiment.Neupane et al. (2013)
Future efforts to bring the bulk chemical potential of 3DDSs to the Dirac point will presumably require compensation of donors by acceptors. By Eqs. (10) and (11), the resulting disorder landscape can be expected to have a typical concentration cm of electrons in puddles and electrons/holes per puddle, with a disorder potential of magnitude meV, assuming the impurity concentration remains of order cm. Equation (17) suggests a corresponding minimum conductivity S/cm.
Finally, one can note that existing 3DDS materials seem to have anisotropic Dirac cones, with a Dirac velocity in one particular direction, , that is as much as ten times smaller than the velocity in the transverse directions, .Neupane et al. (2013) This anisotropy can be accounted for at the level of the present theory by substituting for the geometric mean velocity , so that the fine structure constant is also modified.
Acknowledgements.I am grateful to R. Nandkishore, S. Gopalakrishnan, J. C. W. Song, B. I. Shklovskii, and E. B. Kolomeisky for helpful discussions and comments. Work at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Basic Energy Sciences under contract no. DE-AC02-06CH11357.
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