A Non-Topological Approach to Understanding Weyl Semimetals
Antonio Levy, Albert F. Rigosi, Francois Joint, Gregory S. Jenkins
AA Non-Topological Approach to Understanding Weyl Semimetals
Antonio Levy , ∗ Albert F. Rigosi , Francois Joint , and Gregory S. Jenkins National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA Department of Physics, University of Maryland, College Park, MD 20742, USA and Laboratory for Physical Sciences, College Park, MD 20740, USA (Dated: February 2, 2021)In this work, chiral anomalies and Drude enhancement in Weyl semimetals are separately discussedfrom a semi-classical and quantum perspective, clarifying the physics behind Weyl semimetals whileavoiding explicit use of topological concepts. The intent is to provide a bridge to these modern ideasfor educators, students, and scientists not in the field using the familiar language of traditional solid-state physics at the graduate or advanced undergraduate physics level.
I. INTRODUCTION
Weyl fermions (defined below) have historically beenof interest in answering fundamental questions aboutthe universe, particularly the observation of the matter-antimatter imbalance. The family of elementary parti-cles classified as fermions, or particles of half-integer spin,are important in the Standard Model that unifies threeof the four known forces of nature. Within the modelare twenty-four families of fermions. Almost all of themare massive
Dirac fermions . Within the family of
Diracfermions lies a subset class known as
Weyl fermions , theset of fermions that are massless. Those well-versed inthe physics of the weak nuclear force will recall that thoseinteractions are stronger for both left-chiral matter andright-chiral antimatter than their counterparts of oppo-site chirality (chirality summary shown in Fig. 1).Weyl and 3D Dirac semimetals are the only systems inwhich signatures of Weyl fermions have been observed.Although they are not fundamental particles, the exci-tations in these material systems offer a unique play-ground to study Weyl fermion physics like the chiralanomaly.
Novel properties predicted for Weyl semimet-als (WSMs) like significantly reduced scattering, Drudeenhancement along applied magnetic fields, and long-lived spin-polarized currents in the presence of magneticfields could lead to myriad applications in fields like spin-tronics and quantum computing.Most introductions to WSMs and Weyl Fermions aremathematically intense, making it difficult to develop anintuitive understanding of their physics. There are afew works that try to explain the concepts behind Weylfermions in an educational context.
This paper aimsto provide a conceptual introduction accessible to educa-tors, students, and scientists who have some understand-ing of traditional condensed matter physics. ∗ [email protected] II. BACKGROUND ON WEYL FERMIONS
The concept of Weyl fermions originated from the fu-sion of two familiar topics. The first is the energy-momentum relation from relativity expressed as: E =( pc ) + (cid:0) mc (cid:1) . The second is the time-dependentSchr¨odinger equation from quantum mechanics expressedas: − (cid:125) (cid:79) ψ (2 m ) = i (cid:125) ∂ψ∂t .To incorporate relativity into a quantum mechani-cal formula, Dirac treated each quantity in the energy-momentum relation as an operator acting on a wavefunc-tion. Using p = − i (cid:125) (cid:79) and E = i (cid:125) ∂∂t , the resulting equa-tion is: (cid:18) − c ∂ ∂t + (cid:79) (cid:19) ψ = m c (cid:125) ψ (1)This is known as the Klein-Gordon equation. It doesnot include spin and is therefore applicable to zero-spinbosons.Dirac transformed this equation by taking the squareroot of the operators, which requires consideration of allthree spatial dimensions and the time dependence. Thetransformation involves 4 × −→ σ that describe half-spin fermions. The Diracequation can be written in matrix block form with aparticle-hole ( p-h ) – or particle-antiparticle – basis: (cid:20) (cid:0) εc − mc (cid:1) −−→ p · −→ σ −−→ p · −→ σ (cid:0) − εc + mc (cid:1) (cid:21) (cid:20) ψ p ψ h (cid:21) = 0 , (2)where ε is the energy of the particle. On the otherhand, rather than using a particle-hole basis, the Weylequation uses left- and right-chiral particles as the basis: (cid:20) mc εc − −→ p · −→ σ εc + −→ p · −→ σ mc (cid:21) (cid:20) ψ L ψ R (cid:21) = 0 , (3)Massive particles in the Weyl equation prevent purechiral eigenstates from emerging out of Equation (3).Pure chiral particles must be massless. The eigenstates a r X i v : . [ c ond - m a t . o t h e r] J a n σ p σ p Right-handedLeft-handed
Chirality
FIG. 1. A left-handed chiral particle has a spin σ that isantiparallel to its momentum p . The spin and momentum areparallel for a right-handed chiral particle. of massless Weyl particles only involve the momentumand a parallel or antiparallel spin as shown in Figure 1.In solid state physics, the dispersion relation is a math-ematical description of the available energies for an elec-tron moving about in a material with a given momentum.It is frequently plotted as energy versus momentum (alsoreferred to as k -space). Further details on the basics ofsolid state physics can be found in standard textbooks. The relativistic energy-momentum dispersion relation formassless particles is E = pc . For particle-like excita-tions (or quasiparticles) moving through solid materials,the velocity can be a constant other than the speed oflight while still obeying the Weyl equation. Casting themomentum in terms of wavenumber −→ p = (cid:125) −→ k , the oper-ative Weyl dispersion relation becomes E = (cid:125) kv , wherethe v is the Fermi velocity which plays a role similar to c in the relativistic equations. The zero-energy is definedat zero momentum and is called the Weyl point , and itspresence in Weyl semimetals is responsible for the novelphysics they are predicted to exhibit.
III. CHIRAL ANOMALIES
To appreciate how a chiral anomaly emerges, it willhelp to recall a few concepts from statistical and solid-state physics. Within any material, electrons and otherquasiparticles only have access to a limited number ofmomenta. Mapping out these allowable momenta cre-ates what is commonly referred to as k -space (closelyrelated to the reciprocal lattice of a crystalline solid).The highest-energy electrons, in some sense, define thebounds of achievable momenta, and these bounds sketchout an object in k -space called a Fermi surface. TheFermi surface for a Weyl particle is a sphere centered on the zero-momentum Weyl point.Due to quantum mechanics, the allowable energies atwhich electrons are permitted to exist are discrete (andby extension, are also discrete in k -space via Fouriertransformation). These quantized energies make up theband structure of a crystal. For the remainder of thispaper, it is assumed that the Fermi energy exists solelywithin the Weyl band and that other bands are suffi-ciently separated in energy so as not to perturb the Weylstate. With this last concept recalled, we are ready todescribe the anomaly.Consider the divergence theorem as applied to a charge Q , charge density ρ and electrical current density j : dQdt = (cid:121) (cid:18) ∂ρ∂t + −→ (cid:79) · −→ j (cid:19) d x (4)Since charge is conserved, the integral extends overthe entire volume of the system, leaving both sides ofEq. (4) equal to zero. This expression indicates that thetotal amounts of charge entering and leaving the systemare equal. Combining Eq. (4) with Eq. (3) for m = 0under non-zero electromagnetic fields (which is done byreplacing −→ p with −→ p − e −→ Ac ), the difference between thenumber of right- and left-chiral particles, n R − n L can beobtained after considerable effort: ddt ( n R − n L ) ∝ (cid:121) −→ E · −→ B d x (5)Equation (5) shows that if a population of Weylfermions is subjected to non-zero applied electric andmagnetic fields, the chirality of the population willchange with time. This is equivalent to saying that Weylfermions of a single chirality will be annihilated and re-placed with Weyl fermions of the opposite chirality. Thisis the chiral anomaly.The chiral anomaly in a crystal is best understood in-tuitively by considering the effects of applied magneticand electric fields on the Fermi surface with linear dis-persions illustrated in Fig. 2. The momentum directionof any quasiparticle on a spherical Fermi surface is ra-dially outward. In the depicted generic WSM, there aretwo locations in k -space around which the band struc-ture obeys the Weyl equations. Each location hosts onespecies of chiral particle where the spin is either radiallyinward opposite the momentum (orange Fermi surfacepocket on the left side of each subfigure) or purely alignedradially outward with the momentum (green Fermi sur-face pocket on the right side of each subfigure). As willbe discussed later, including only a single chiral Fermipocket violates the conservation of energy.So what happens to the Fermi surface if you apply amagnetic field? It will distort, expanding or contractingby an amount proportional to the dot product of the spinand the magnetic field (Zeeman effect). To intuitively un-derstand these distortions, consider the left-handed chi-ral (orange) Fermi surface pocket in Figure 2(e-g) at thepoint where the spin of a quasiparticle is purely antipar-allel with applied magnetic field. The quasiparticle’s en-ergy will decrease with applied field. This decrease in en-ergy, through the linear energy dispersion, causes the mo-mentum to decrease. The decrease in momentum shiftsthis point of the Fermi surface toward the center of theFermi pocket sphere. Likewise, the opposite point, wherethe field and spin are parallel, shifts to higher momen-tum away from the center. All the points where the spinis perpendicular to the field do not change their energyor momentum. Visualizing the smooth interpolation be-tween these points results in a Fermi surface distortedinto an egg-like shape with the major axis aligned withfield as shown in Figure 2(e).Similar effects on the Fermi sea occur from the chargeof the quasiparticle subjected to an applied electric field.In this case, all negatively charged quasiparticles, regard-less of their spin or location on the Fermi surface, de-crease their (vector) momentum along the direction ofthe electric field causing a net momentum shift of theFermi surface.With either an applied electric or magnetic field, thechanges in a singly-chiral Fermi pocket induces a net mo-mentum and therefore an electric current. This currentalso transports a net spin current. However, when both(oppositely) chiral spherical Fermi surface pockets areconsidered, the application of a magnetic field (with noelectric field) causes two equal and oppositely-orienteddistortions that negate the total current. In fact, thiscan be understood as the reason that all Weyl semimet-als must include pairs of oppositely-chiral Fermi pockets;a static magnetic field should not be able to generatea perpetual current in a material system with a finitescattering rate like Weyl semimetals, as this would vio-late the conservation of energy. Furthermore, in ordernot to violate conservation of energy, this current wouldhave to be superconducting. At high fields, there wouldbe a strong current that would flow consistently in onedirection, which would persist independent of the direc-tion in which an external electric field was applied tothe material. This would mean that carriers could flowin the direction opposite to the bias applied to the mate-rial, leading to negative power dissipation in the materialas a result of an applied external electric field (providedthat field was not large enough to fundamentally alterthe band structure).To formulate an expression for the generated currents,the quasiparticles removed and added at various mo-menta with applied fields must be properly counted. Re-call the density of states, which is the derivative of thenumber of states with respect to energy in the system.The density of electronic states (g, which we define asthe number of states per unit energy at a specific pointin k -space) is energy dependent and therefore becomesdependent on k-space location ( θ , defined as the polarangle between the k -vector and the applied fields) with the application of fields.Consider for simplicity the case with both fields point-ing in the − x direction (as in Fig. 2). Then, for the right-chiral location, we track the difference between right- andleft-chiral particles n R ( θ ) − n L ( θ ), which is proportionalto integrated subtraction of the density of states, ex-pressed as g R ( L ) ( ε F ) = g ( ε F +( − ) a Bcos ( θ )+ a Ecos ( θ ))for the right and left pockets, where a and a are con-stants. The difference in sign in front of the a Bcos ( θ )is due to the opposite spin of the two pockets, high-lighting the antiparallel nature of the chirality. Ap-proximating for small fields, g R ( L ) ( ε F ) can be rewritten: g R ( L ) ( ε F ) = g ( ε F ) + ( − ) a Bcos ( θ ) g (cid:48) ( ε F + a Ecos ( θ ))or, equivalently, we could say: g R ( ε F ) − g L ( ε F ) =2 a Bcos ( θ ) g (cid:48) ( ε F + a Ecos ( θ ). The difference in the pop-ulation of right- and left- chiral Weyl fermions is givenby: (cid:90) π ( g R ( ε F , θ ) − g L ( ε F , θ )) dθ ≈ a B (cid:90) π cos ( θ ) g (cid:48) ( ε F + a Ecos ( θ )) dθ (6)This subtraction is reduced to a derivative of the den-sity of states for small fields. With additional symmetryconsiderations, the bounds of the integral are halved aslong as we take into account E → − E for the respectivehalves of the Fermi pockets:2 a B (cid:90) π cos ( θ ) g (cid:48) ( ε F + a Ecos ( θ )) dθ ≈ a B (cid:90) π (cid:16) g (cid:48) ( ε F + a E cos ( θ )) − g (cid:48) ( ε F − a Ecos ( θ )) (cid:17) cos ( θ ) dθ ∝ a a EB (cid:90) π g (cid:48)(cid:48) ( ε F ) cos ( θ ) dθ (7)where the approximation g (cid:48) ( ε F ± a E cos ( θ )) = g (cid:48) ( ε F ) ± a E cos ( θ ) g (cid:48)(cid:48) ( ε F ) is used to go from the secondline of Eq. (7) to the third.With these considerations, the origin of Eq. (5)becomes apparent. Since the first time derivative of( n R − n L ) is a nonzero quantity under the experimen-tal conditions, a net chiral current, the “chiral anomaly,”emerges and begins pumping particles from the left-chiralFermi pocket to the right-chiral Fermi pocket leading to anet accumulation of right-chiral particles. Furthermore,the total number of right-chiral particles increases fasterthan the corresponding decrease seen in the left-chiralparticles. For static fields, this ever-growing chirality im-balance is eventually counteracted by quasiparticle scat-tering between and within the two chiral Fermi pockets.A chirality-imbalanced equilibrium value is establishedthat depends on both pumping rate and (current relax-ation) scattering rate. (c)(b) (f)(e) (i)(h) k x k y k x k y k x k y B BE E = ≠ E ≠ ≠ E = = (a) k x k y k z k x k y k z A = A (d) (g) (j) k x ε k x ε k x ε B BE
Left HandChiral (L) Right HandChiral (R)(L) (R) (L) (R) (L) (R)(L) (R) (L) (R) (L) (R)(L) (R) (L) (R) (L) (R)
FIG. 2. The chiral anomaly in Weyl semimetals is understoodby distortions and shifts of the Fermi surface caused by theapplication of electric and magnetic fields. (a,b) With no ap-plied field, the Fermi surface is two spheres: one Weyl pocketis left-handed chiral (orange) with spins perpendicular to theFermi surface (pointing inward) and the other right-handedchiral (green) with spins (red arrows) perpendicular to theFermi surface (pointing outward). (c) The k x − k y planar cutthrough the center of the Fermi pocket spheres results in twocircles that demarcate allowable quasiparticle momenta. (d)The energy of the quasiparticles, with momenta taken as a k x − line cut through the centers of the circles, is given by thelinear dispersion E = (cid:125) k x v (where velocity v is constant). (g)Application of an external magnetic field in the - x directionraises or lowers the energy of the quasiparticle depending onthe direction of the spin. Changes in energy are connectedto changes in momenta through the linear energy dispersionresulting in (e,f) k -space shifts and distortions of the Fermisurface. (j) The addition of an applied electric field inter-acts with the (negative) charge of the quasiparticle causinga vectoral increase of the momenta in the - x direction thateither increases or decreases the quasiparticle kinetic energydepending on the initial direction of the momentum. (h,i)The combined effects of the applied fields distort and shiftthe Fermi surface, generating currents that, in the depictedcase, are largest for the (green) right-handed chiral carriers. IV. SEMI-CLASSICAL PICTURE OF DRUDEENHANCEMENT
Another interesting phenomenon regarding WSMs isthe existence of Drude enhancement. To get comfortablewith this topic, one must recall the Drude model of ACconductivity from solid state physics: σ D ( ω ) = ω D π (cid:0) τ − iω (cid:1) (8)As ω → B = 0 ) the DC conductivity appearsas σ DC = ω D τ π , where the term ω D is referred to as theDrude weight. The Drude weight and conductivity of amaterial are properties that are generally well-describedby the equations of motion for the charge carriers.For most metallic and semi-metallic systems in thesemi-classical limit, application of a magnetic field par-allel to the driving electric field does not affect currentsince there is no Lorentz force. In the case of Weyl and3D Dirac semimetals, such an arrangement of appliedfields is expected to increase the Drude weight and thiswas recently verified experimentally. Most explanationsof this very unusual effect rely heavily on the mathemat-ical consequences of the Berry’s phase curvature near theWeyl points. Such explanations, though rigorous in thesemi-classical limit, do not provide an intuitive explana-tion of the origin of such an effect.Again, studying the Fermi surface of the Weyl stateprovides a non-topological approach to understandingthis effect. Since the Weyl pockets are represented in k- space as asymmetrically distorted (egg-shaped) Fermisurface pockets (Fig. 2 (h)), any applied magnetic fieldwill have additional, distinct effects on both populationsof chiral particles, pushing more of the particles into theportion of the Fermi surface that contributes to the netcurrent, as shown in Figs. 2(h) and 2(i). The Drudeweight is proportional to the current generated by theapplied external electric field. Integrating over the Fermisurface of a pair of Weyl points in the semi-classical small-field limit, with (cid:113) (cid:125) v x v y l B (cid:28) ε F : ω D ∝ (cid:90) d Ω[ g + ( ε F , −→ B ) + g − ( ε F , −→ B )] v (cid:107) F ( θ ) ∝ (cid:90) d Ω (cid:20) g ( ε F ) + 12 g (cid:48)(cid:48) ( ε F ) ( a B cos ( θ )) (cid:21) v (cid:107) F ( θ ) (9)where v (cid:107) F ( θ ) = v F cos ( θ ) is the Fermi velocity compo-nent parallel to the applied fields and l B = (cid:113) (cid:125) ceB is themagnetic length. The second term in (9) yields the B Drude weight enhancement obtained using the Berry’sphase curvature. V. QUANTUM MECHANICAL PICTURE OFTHE CHIRAL ANOMALY AND DRUDEENHANCEMENT
For a more comprehensive picture of Drude enhance-ment, we turn to the quantum limit of the Drude weightof a Weyl pocket in 3D. The energy dispersion of the n th excited Landau level (LL) for B (cid:107) x is given by: k x ε BE ε F n = 0 n = 1n = 2n = 3 FIG. 3. The energy dispersion of the various Landau levelsalong the magnetic field direction is illustrated. Left-handedparticles are pumped from one Weyl pocket to the right-handed pocket. The formation of Landau levels makes thedispersion quasi one-dimensional, due to the discretization ofkinetic energy perpendicular to the magnetic field (i.e., in the y and z directions). ε n ( k x ) = (cid:115) (2 n + (1 + s )) (cid:125) v y v z l B + (cid:125) v x k x (10)where s = ± n = 0 ) has anenergy dispersion containing an effective mass. There-fore, only the n = 0 Landau level is chiral, making itresponsible for all of the nontrivial behavior associatedwith WSMs. Consider the chiral anomaly. For all n (cid:54) = 0LLs, the application of an electric field simply shifts elec-trons within the LL. They do not change the populationof the pocket or even the LL itself.Under an electric field parallel to the magnetic field (see Fig. 3), the number of carriers in one of the two n = 0 LLs will decrease while the number in the other n = 0 LL will increase. The dispersion in the two pock-ets’ n = 0 LLs are given by ε + ( k x ) = + (cid:125) v x k x and ε − ( k x ) = − (cid:125) v x k x for the left- and right-chiral cases,respectively. Applying an electric field along the x -direction will deplete the population of one n = 0 LLwhile increasing the population of the other n = 0 LL bythe same amount.The transfer of charge from one pocket to the other oc-curs entirely through the chiral n = 0 LL of both pockets,since charge will be redistributed within each excited LLunder an electric field. The amount of charge transfer isproportional to the electric field component parallel tothe magnetic field and the degeneracy of the n = 0 LL,which is proportional to l B due to Landau quantizationperpendicular to the magnetic field which, in our case, isthe yz -plane. From this, the chiral anomaly will persistat both low and high field strengths in WSMs.The spectral weight in the chiral n = 0 LL is indepen-dent of the Fermi energy. From the quantum mechani-cal perspective, the increase in spectral weight with fieldcan be understood as transferring more and more carriersfrom slower bands (where they are closer to the vertex) tothe faster, massless n = 0 LL. This quantum mechanicalperspective on Drude enhancement gives a more nuancedunderstanding of the complexity of this effect without theneed to rely on topological descriptions. VI. CONCLUSIONS
It is rather uncommon to find approaches to thephysics behind Weyl fermions in the literature withoutinvolving a discussion on topology. Here, we explain theconcepts of chiral anomalies and Drude enhancement inthe context of Weyl semimetals intended for those notin the field. Non-topological approaches offer a moreintuitive conceptual framework for understanding thesephysical systems. This understanding is useful when de-signing, performing, or analyzing data from experiments.
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