Coulomb disorder in three-dimensional Dirac systems
CCoulomb disorder in three-dimensional Dirac systems
Brian Skinner
Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA (Dated: June 9, 2014)In three-dimensional materials with a Dirac spectrum, weak short-ranged disorder is essentiallyirrelevant near the Dirac point. This is manifestly not the case for Coulomb disorder, where thelong-ranged nature of the potential produced by charged impurities implies large fluctuations ofthe disorder potential even when impurities are sparse, and these fluctuations are screened by theformation of electron/hole puddles. In this paper I present a theory of such nonlinear screening ofCoulomb disorder in three-dimensional Dirac systems, and I derive the typical magnitude of thedisorder potential, the corresponding density of states, and the size and density of electron/holepuddles. The resulting conductivity is also discussed.
I. INTRODUCTION
Due to its long-ranged nature, Coulomb disorder of-ten has dramatic consequences even in situations whereshort-ranged disorder does not. Thus, for example,Coulomb disorder has always deserved special considera-tion in the physics of semiconductors, and such studieshave revealed a great number of diverse and interestingscientific phenomena over the preceding half-century.The large qualitative difference between short-rangedand Coulomb disorder is particularly pronounced forthree-dimensional (3D) Dirac materials, in which theelectron kinetic energy ε is linearly proportional to themomentum (cid:126) (cid:126)k according to ε = (cid:126) v | (cid:126)k | . To see thisdifference between short-ranged and Coulomb disorderqualitatively, one can compare the behavior of long-wavelength (small- k ) electron states in a 3D Dirac sys-tem (3DDS) for the two cases. Suppose, for example,that the 3DDS has some concentration N of positiveand negative impurities per unit volume, each with ran-dom position and random sign, and that these are takento be either short-ranged, with finite range a and typ-ical potential ± V , or Coulomb, with charge ± e . Inthe short-ranged case, an electron wavepacket with size λ (cid:29) N − / (cid:29) a experiences disorder from ∼ N λ impurities. The average value of the disorder poten-tial created by these impurities is zero, since impuritieswith opposite signs are equally plentiful, but statisticalfluctuations in the impurity concentration create a typ-ical excess of ∼ √ N λ impurities with one of the twosigns. Thus, the volume-averaged disorder potential ex-perienced by the electron is ∼ V √ N λ / ( λ/a ) ∝ /λ / .The electron kinetic energy, on the other hand, scalesas ε ∝ k ∝ /λ . One can therefore conclude thatshort-ranged disorder has a perturbatively small effect onthe electron energy for large-wavelength electron states(i.e., for states close to the Dirac point). This robust-ness of the 3D Dirac point against short-ranged disor-der has long been understood theoretically, and conse-quently “Dirac semimetal” phases with vanishing densityof states (DOS) are generally predicted to survive shortrange disorder. (In fact, a very recent paper has shownthat 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 electronwavepacket with large size λ encloses many impuri-ties of both signs, and the net charge of these is ∼± e √ N λ . If one naively calculates the potential energycreated by these impurities, one finds that the typicalCoulomb potential energy experienced by the electronis ∼ e √ N λ /κλ (in Gaussian units), where κ is thedielectric constant. Thus, the disorder potential grows with increasing wavelength as λ / , rather than falling offquickly and becoming irrelevant. Clearly, such Coulombimpurities must have a large and nonperturbative effectnear the Dirac point at any finite concentration. Asone might expect, the growth of the Coulomb poten-tial at large length scales is in fact truncated by theformation of electron and hole puddles that screen thedisorder potential, as is the case with narrow band gapsemiconductors and two-dimensional Dirac systemslike graphene and topological insulators. Thisdisorder-induced puddling is shown schematically in Fig.1. It is the purpose of this paper to calculate the typicalsize and density of these puddles, as well as the corre-sponding disorder potential amplitude, DOS, and con-ductivity. r eϕ r s Γ FIG. 1. Schematic illustration of electron/hole puddles in a3DDS at the Dirac point. The Coulomb potential energy eφ is shown as a function of some coordinate r . Coulomb im-purities create potential fluctuations with typical size r s andamplitude Γ. Regions of positive eφ correspond to electronpuddles, while negative eφ implies hole puddles. a r X i v : . [ c ond - m a t . d i s - nn ] J un The question of disorder effects in 3DDSs has ac-quired a particular relevance in recent months, fol-lowing the experimental discovery of two different 3DDirac materials not long after their theoreticalprediction.
While the existence of a Dirac disper-sion in these materials has been established, largely byphotoemission experiments, it remains to be thoroughlyunderstood how closely the Dirac point can be probedand to what extent its behavior is masked by disorder.As shown below, the presence of Coulomb impurities hasthe effect of “smearing” the Dirac point via the creationof electron and hole puddles, and this smearing typicallyoccurs over tens of meV.The structure and primary results of this paper are asfollows. In Sec. II, a self-consistent theory is developedto describe the disorder potential based on the Thomas-Fermi (TF) approximation. Corresponding expressionsare derived for the magnitude of the disorder potential[Eq. (7)], the DOS [Eq. (8)], the size of electron/hole pud-dles [Eq. (9)], and the concentration of electrons/holesin puddles [Eq. (10)]. While the primary focus of thissection is on the behavior near the Dirac point, resultsare also presented for the case when the chemical po-tential is away from the Dirac point. In Sec. III, thezero-temperature conductivity is discussed, and a resultis presented for the conductivity [Eq. (16)] and its mini-mum value [Eq. (17)]. Sec. IV concludes with a summaryand some discussion of recent experiments.
II. SELF-CONSISTENT THEORY OFDISORDER AND SCREENING
This paper focuses on a model of disorder in which N monovalent Coulomb impurities per unit volume are ran-domly distributed throughout the bulk of a 3DDS. Suchimpurities create a random Coulomb potential, and thispotential induces an electron/hole concentration n ( (cid:126)r ),where n > n < n ( (cid:126)r ) at a given spatial coordinate (cid:126)r is related tothe self-consistent magnitude of the Coulomb potential φ ( (cid:126)r ). In this paper, the primary tool for describing thisrelationship is the TF approximation: E f [ n ( (cid:126)r )] − eφ ( (cid:126)r ) = µ. (1)Here, E f ( n ) = (cid:126) vk f ( n ) sgn( n ) is the local Fermi en-ergy, where k f is the Fermi wave vector and µ is thechemical potential of the 3DDS measured relative tothe Dirac point. Assuming, generically, that the Diracpoint has a degeneracy g (which for Weyl semimet-als is equal to the number of degenerate Dirac points),the Fermi wave vector is k f = (6 π | n | /g ) / , so that E f ( n ) = (6 π /g ) / (cid:126) v | n | / sgn( n ).If the chemical potential µ is large enough in abso-lute value that | eφ | (cid:28) | µ | , one can think that the elec-tron density is relatively uniform spatially, and the corre-sponding electron DOS ν = dn/dE f = gE f / (2 π (cid:126) v ) (cid:39) gµ / (2 π (cid:126) v ) is also uniform. In this case, one canstraightforwardly define a TF screening radius r s = (cid:114) κ πe ν = (cid:114) π αg k − f . (2)Here, α = e /κ (cid:126) v is the effective fine structure constant.The TF approximation is valid in cases where the Fermiwavelength ∼ k − f is much shorter than the typical scaleover which the potential varies, r s . As can be seen in Eq.(2), this corresponds to αg (cid:28)
1. For comparison, the3DDS Cd As has (cid:126) v ≈ · ˚A, κ ≈
36, and g = 2, so that αg ≈ . .
16 and this approximationis justified. As shown below, the same criterion αg (cid:28) µ = 0.If | µ | is not large, so that the 3DDS is close to theDirac point, then one cannot consider the electron den-sity to be uniform and the typical screening radius r s must be found self-consistently. In particular, one canassume that the disorder potential is screened with someunknown screening radius r s and then calculate analyt-ically the corresponding magnitude of the disorder po-tential and the resulting average density of states (cid:104) ν (cid:105) .Inserting the result for ν into Eq. (2), one arrives at aself-consistent relationship for r s , which can be solved togive a result for r s , (cid:104) ν (cid:105) , and the magnitude of the disor-der potential. This procedure is carried out explicitlyin the remainder of the present section.In a medium with screening radius r s , the screenedpotential produced by a single impurity with charge ± e is the Yukawa-like potential φ ( r ) = ± eκr exp[ − r/r s ] . (3)If one assumes that impurity positions are uncorrelated,then the mean squared value of the electron potentialenergy, Γ , can be found by integrating the square of thepotential created by a single impurity, ( eφ ) , over allpossible impurity positions. This givesΓ = (cid:90) ( eφ ( r )) N d r = 2 πe N r s /κ. (4)For cases where r s (cid:29) N − / (justified below), the po-tential at each point in space is the sum of the potentialsproduced by many independently-located impurities. Bythe central limit theorem, then, one can assume that thedistribution of values of the potential across the systemis Gaussian with variance Γ /e . Within the TF approx-imation, the value of the DOS at a point with potential φ is ν ( φ ) = ( g/ π (cid:126) v )( µ + eφ ) , so that the spatially-averaged DOS is (cid:104) ν (cid:105) = ∞ (cid:90) −∞ ν ( φ ) exp (cid:2) − e φ / (cid:3)(cid:112) π Γ /e dφ = g π (cid:126) v (Γ + µ ) . (5)Inserting this expression for ν in Eq. (2) and pluggingthe resulting expression for r s into Eq. (4) gives the fol-lowing self-consistent expression for the amplitude Γ ofthe disorder potential:Γ (Γ + µ ) = 2 π gα (cid:18) e N / κ (cid:19) . (6)Equation (6) can be solved for generic values of thechemical potential µ , but it is worth considering specifi-cally the cases of µ = 0 (when the 3DDS is at the Diracpoint) and large | µ | . When µ = 0, Eq. (6) givesΓ ≡ Γ = (cid:18) π gα (cid:19) / e N / κ . (7)It is perhaps worth noting that this value for the disor-der potential amplitude is smaller than the correspond-ing result for the surface of a disordered 3D topologicalinsulator by a factor ∼ α / . This smaller disorder for3DDSs is a consequence of stronger screening in threedimensions.Inserting the value of Γ from Eq. (7) into Eq. (5) givesthe corresponding DOS at µ = 0: ν = (cid:18) α g π (cid:19) / N / (cid:126) v . (8)In this case, the screening radius r s , which is genericallyequal to the correlation length of the disorder potential,defines the typical size of electron and hole puddles (asillustrated in Fig. 1). By Eq. (2), its value is given by r s = (cid:18) gα (cid:19) / N − / . (9)The typical concentration of electrons/holes in puddles n p is found by equating Γ with E f ( n p ), which gives n p = (cid:114) gα π N, (10)so that the corresponding number of electrons/holes perpuddle is M p ≈ π r s n p = (cid:115) π/ gα . (11)When gα (cid:28)
1, there are many electrons per puddle: M p (cid:29)
1. Intriguingly, this value for M p is independentof the impurity concentration, so that the number of elec-trons per puddle is independent of the details of the dis-order. This universality is reminiscent of the problem ofa single supercritical nucleus in a 3DDS, where the maxi-mum observable “nuclear charge” also obtains a universalvalue ∼ /α / . Notice also that at gα (cid:28)
1, the correlation length ofthe potential r s is much longer than the typical Fermi wavelength k f ( n p ) − ∼ N − / g / /α / , so that theTF approximation is justified. This same condition alsoguarantees r s (cid:29) N − / , which validates the assumptionof a Gaussian-distributed potential.It is worth noting that Eqs. (7)–(11) can be derivedqualitatively using the following very simple argument(which for simplicity uses g ∼ ∼ r s within the 3DDS; this volume is effectivelya single electron/hole puddle. Those impurities withinthe volume can be said to contribute to the potentialwithin it, while others are effectively screened out. Thenet charge of impurities in the volume is Q ∼ e (cid:112) N r s (with a random sign), and this impurity charge is com-pensated by the charge of electrons/holes, which havetotal number M p ∼ n p r s . Equating M p with Q/e gives n p ∼ N/r s . Now one can note that the typical kineticenergy of electrons within the volume, ∼ (cid:126) vn / p , mustbe similar in magnitude to the typical Coulomb energy ∼ Qe/κr s . This equality gives n p ∼ α N r s . Combiningthe two equations for n p gives r s ∼ N − / /α , as in Eq.(9), and the other relevant quantities can be found bysubstitution.As the chemical potential µ is moved away from theDirac point, the magnitude of the disorder potential de-creases, as dictated by Eq. (6), and correspondingly thescreening radius r s shrinks. At | µ | (cid:29) Γ , puddles ofelectrons (for µ <
0) or holes (for µ >
0) dry up, andthe system is well-described by linear screening with aspatially-uniform DOS. In this case the disorder poten-tial magnitude becomesΓ (cid:39) (cid:18) π gα (cid:19) / (cid:18) e N / /κ | µ | (cid:19) / e N / κ . (12)The corresponding DOS approaches that of the non-disordered system, ν (cid:39) g π µ ( (cid:126) v ) (13)and the correlation length of the disorder potential is r s (cid:39) (cid:114) π αg (cid:126) v | µ | . (14)Equations (7)–(11) and (12)–(14) describe the systemin the limits of µ = 0 and | µ | (cid:29) Γ , respectively. Thecrossover between these two regimes can be described byevaluating Eqs. (5) and (6). The result of this process isshown in Fig. 2, where the variance of the disorder poten-tial and the DOS are plotted as a function of the chemicalpotential µ . As one can see, these self-consistent equa-tions predict a smooth, monotonic crossover from thepuddle-dominated µ = 0 result to the linear screeningregime at large µ . It is worth noting, however, that Γmay in fact exhibit weakly nonmonotonic behavior as afunction of µ , achieving a weak maximum at | µ | / Γ ∼ and arises because at | µ | / Γ ∼ d i s o r d e r , ( Γ / Γ ) chemical potential, | µ |/ Γ DO S , ν / ν a)b) FIG. 2. a) The variance in the disorder potential as a func-tion of the chemical potential µ . The dashed line correspondsto the linear screening regime of Eq. (12). b) The spatially-averaged density of states as a function of the chemical po-tential. The dashed line corresponds to Eq. (13), which is theDOS for a non-disordered system. Γ and ν are given byEqs. (7) and (8), respectively. III. CONDUCTIVITY
The previous section discussed the disorder potentialproduced by random Coulomb impurities using a self-consistent theory of screening. In this section I brieflydiscuss the implications of this screening for the low-temperature conductivity.In situations where the mean free path (cid:96) for electronscattering is relatively large, k f (cid:96) (cid:29)
1, the electron trans-port is well-described by the Boltzmann equation. Inparticular, the momentum relaxation time τ satisfies (cid:126) τ = πN ν ∞ (cid:90) dθ sin θ (cid:12)(cid:12)(cid:12) (cid:101) φ ( q ) (cid:12)(cid:12)(cid:12) (1 − cos θ ) 1 + cos θ q = 2 k f sin( θ/
2) is the mo-mentum change resulting from scattering by an angle θ and (cid:101) φ ( q ) = 4 πe / [ κ ( q + r − s )] is the Fourier transformof φ ( r ) evaluated at wave vector q . The final factor of(1 + cos θ ) / k f r s (cid:29)
1. Evaluating the integral in Eq. (15) gives (cid:126) τ (cid:39) π N νe κ k f ln(2 k f r s )assuming k f r s (cid:29)
1, which is a condition of validity forthe screened potential used here, and, again, correspondsto αg (cid:28) σ by combining the expression for τ with the Ein-stein relation for conductivity, σ = e ν ( v τ / If oneassumes a relatively large and uniform electron den-sity n (i.e., a large chemical potential | µ | (cid:29) Γ ), then k f (cid:39) (6 π n/g ) / and σ = (cid:18) π (cid:19) / α g / ln(2 π/αg ) nN e n / (cid:126) . (16)As the electron density is reduced (the chemical potentialis brought closer to the Dirac point), the number of car-riers is reduced and the conductivity declines. At µ = 0,the conductivity achieves a minimum whose value is de-termined by the concentration of electrons and holes inpuddles. The value of this conductivity minimum, σ min ,can be estimated by inserting n ∼ n p into Eq. (16), whichgives σ min ≈ π (2 g ) / ln(2 π/αg ) e N / (cid:126) . (17)Finally, one can check that the derived expressionsindeed correspond to the Boltzmann semiclassical limit k f (cid:96) (cid:29)
1. As expected, the value of τ correspond-ing to the minimum conductivity, where n ∼ n p , gives k f (cid:96) = k f vτ ∼ g / /αg (cid:29)
1. Thus, k f (cid:96) (cid:29)
1, providedthat αg (cid:28)
1. At larger n the mean free path only in-creases, so that the Boltzmann equation is a good de-scription everywhere. IV. CONCLUDING REMARKS
This paper has presented a simple picture of self-consistent screening of Coulomb impurities in 3DDSsthrough the formation of electron and hole puddles. Sucheffects are manifestly not perturbative near the Diracpoint, regardless of the impurity concentration, and theyhave a prominent effect on both the observed DOS andthe conductivity. The DOS, for example, vanishes only asthe 2 / As , which has α ≈ .
04 and g = 2. Thus far, experimentally studied sam-ples are n -type, apparently resulting from uncontrolleddoping by As vacancies. For example, Ref. 19 re-ports relatively large µ ≈
200 meV, with a correspondingcarrier concentration n ∼ × cm − ; one can expectthat the concentration of donor impurities N is similar inmagnitude. Inserting these parameters into Eqs. (12) and(14) gives an estimated disorder potential of Γ ≈
20 meVand a screening radius r s ≈
20 nm. The latter seemsconsistent with the scale of disorder fluctuations seenby scanning tunneling microscopy measurements. ByEquation (16), this level of disorder corresponds to amobility σ/ ( en ) ∼
30 000 cm /Vs, which also closelymatches the value seen in experiment. Future efforts to bring the bulk chemical potential of3DDSs to the Dirac point will presumably require com-pensation of donors by acceptors. By Eqs. (10) and (11),the resulting disorder landscape can be expected to havea typical concentration n p ∼ − N ∼ − cm − ofelectrons in puddles and ∼
10 electrons/holes per pud-dle, with a disorder potential of magnitude Γ ∼
45 meV,assuming the impurity concentration N remains of order 10 cm − . Equation (17) suggests a corresponding min-imum conductivity σ min ∼ v z , that is as much as tentimes smaller than the velocity in the transverse direc-tions, v ⊥ . This anisotropy can be accounted for at thelevel of the present theory by substituting for v the geo-metric mean velocity ( v ⊥ v z ) / , so that the fine structureconstant α ∝ /v is also modified. ACKNOWLEDGMENTS
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