AC Response of the Edge States in a Two-Dimensional Topological Insulator Coupled to a Conducting Puddle
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n AC Response of the Edge States in a Two-Dimensional Topological Insulator Coupledto a Conducting Puddle
K. E. Nagaev
Kotelnikov Institute of Radioengineering and Electronics, Mokhovaya 11-7, Moscow, 125009 Russia
We calculate an AC response of the edge states of a two-dimensional topological insulator, whichcan exchange electrons with a conducting puddle in the bulk of the insulator. This exchange leadsto finite corrections to the response of isolated edge states both at low and high frequencies. Bycomparing these corrections, one may determine the parameters of the puddle.
I. INTRODUCTION
A signature of two-dimensional (2D) topological insu-lators is the existence of helical edge electronic statesthat propagate in clockwise and counterclockwise direc-tions. As the projection of electron spin is locked to thedirection of its motion, the electron can only change bothof them simultaneously and cannot be backscattered bynon-magnetic impurities or phonons as in conventionalconductors. Therefore it was theoretically predicted thatin the absence of spin-flip scattering, a pair of helicaledge states should have a universal value of conductance e /h , no matter how long they are . However experi-ments revealed that actual values of conductance weremuch smaller. In papers reporting measurements onHgTe/CdTe quantum wells, the conductance of 1 µ m-long edge states was 10% smaller than expected. Someother paper reported a decrease of conductance by twoorders of magnitude . A similar suppression of conduc-tance was found in InAs/GaSb/AlSb heterostructures .In all experiments, it had a very weak temperature de-pendence.So far, there was no satisfactory explanation of thesefacts despite a large number of theoretical papers inthis field. First the conductance suppression was at-tributed to spin-flip scattering of electrons by magneticimpurities , but it appeared shortly that axially sym-metric impurities do not contribute to the dc resistancebecause of conservation of total spin of the electrons andimpurities . To avoid this conservation, the authors ofRef. assumed that the magnetic impurities have a ran-dom anisotropy in the plane of the insulator. They ob-tained that such impurities would lead to the Andersonlocalization of the edge states and an exponential de-crease of the conductance with the length of the sample,while its experimental values are inversely proportionalto its length like in diffusive conductors. This could takeplace if along with anisotropic impurities there were suf-ficiently strong dephasing processes. However the esti-mates show that the dephasing length is much larger thanthe distance between the probes , which makes thismechanism unlikely.Another possibility is that conducting puddles are for-med in the bulk of the insulator because of potentialfluctuations due to randomness of impurity doping andelectrons from the edge states are captured into thesepuddles, as suggested by Vayrynen et al. . The authors FIG. 1. (color online). A pair of edge states tunnel-coupledto a conducting puddle in the bulk of the 2D topological in-sulator. The electron spin in the edge states is locked to thedirection of motion, but the electrons in the puddle can flipit without restrictions. found that together with Coulomb interaction, this re-sulted in a suppression of the conductance, but its strongtemperature dependence did not agree with experiments.Nevertheless scanning-gate experiments suggest thatthe suppression arises from well-localized discrete objectsnear the edges.Recently, it was suggested that the suppression of con-ductance may arise from the tunnel coupling between theedge states and conducting puddles of relatively large sizethat have a continuous energy spectrum and allow a two-dimensional motion of electrons in them . The impu-rity scattering in the puddles combined with spin-orbitcoupling may result in a temperature-independent spinrelaxation of electrons via the Elliott–Yafet or Over-hauser mechanism , see Ref. for a review. The exis-tence of these puddles will lead to an effective backscatt-tering of electrons. In particular, it was shown in thateven one puddle could reduce the conductance by half ifthe tunnel coupling and spin-flip scattering in the puddleare sufficiently strong. However the conductance dependson both of these quantities and therefore it is difficult toextract them from measurements of dc current. In thispaper, we present calculations of a frequency-dependentresponse of a pair of edge states coupled to a conduct-ing puddle. By comparing the low- and high-frequencyconductances, one can determine the parameters of thepuddle and judge upon the applicability of this model. II. MODEL AND GENERAL EQUATIONS
Consider a pair of helical edge states with linear disper-sion ε p = | p | v that connect the electron reservoirs, whichare kept at externally controllable voltages. Each of thetwo directions of the electron momentum is locked to adefinite spin projection, which is labeled by σ = ±
1. Theedge states are tunnel-coupled with electron or hole pud-dles that are formed in the bulk of the insulator becauseof large-scale potential fluctuations. We also assume thatthese puddles are sufficiently large to have a continuousspectrum and that the electrons in the puddles are alsosubject to a spin relaxation because of spin-orbit pro-cesses.For simplicity, the interaction between the electrons inthe edge states is neglected, as well as their interactionwith the electrons in the puddle.Hence the distribution functions f σ ( x, ε, t ) in the edgestates obey the equation (cid:18) ∂∂t + σv ∂∂x (cid:19) f σ ( x, ε, t )= − Γ( x ) [ f σ ( x, ε, t ) − F σ ( ε, t )] − e ∂u∂t ∂f σ ∂ε , (1)where Γ( x ) is the rate of electron tunneling from point x to the puddle, F σ ( ε, t ) is the spin-dependent distribu-tion function of electrons in the puddle, and u ( x, t ) is theelectric potential. As the conductance of the puddle ismuch higher than that of the edge states, this distribu-tion functions is spatially uniform inside it and obeys theequation ∂F σ ∂t + 1 hν p Z dx Γ( x ) [ F σ ( ε, t ) − f σ ( x, ε, t )]+ 12 τ s ( F σ − F − σ ) = − e dUdt ∂F σ ∂ε , (2)where ν p is the number of states in the puddle per unit en-ergy, τ s is the spin-relaxation time, and U is the electricalpotential of the puddle. In its turn, the time derivativesof u and U may be obtained through electric capacity ofthe edge state per unit length c , the puddle capacity C and the charge-balance equations ∂u∂t = − ec X σ Z dεhv (cid:20) σv ∂f σ ∂x + Γ ( f σ − F σ ) (cid:21) , (3) dUdt = 1 C dQdt = eC X σ Z dε Z L dx Γ( x ) hv ( f σ − F σ ) . (4)As ν p and Γ may be considered as energy-independentnear the Fermi level, it is convenient to introduce theintegrated quantities n σ ( x, t ) = Z dεhv [ f σ ( x, ε, t ) − f ( ε )] , (5) N σ ( t ) = Z dε ν p [ F σ ( ε, t ) − f ( ε )] , (6)where f ( ε ) is the equilibrium Fermi distribution. Notethat these are not the total electron concentrations be-cause they take into account only the changes of electron number near the Fermi level and do not include the shiftsof the bottom of the conduction band that result from theoscillating electric potential. One may exclude the quan-tity u from Eq. 1 to obtain the equation for n σ in theform hv ∂n σ /∂t − ( e /c ) ∂n − σ /∂thv + e /c + σv ∂n σ ∂x + Γ n σ = Γ hvν p N σ . (7)Furthermore, it is convenient to separate N σ into thecharge and spin parts N Q = N + + N − and N S = N + − N − . By adding and subtracting Eqs. 6 for δN + and δN − and making use of Eq. 4, one obtains the equations forthese quantities in the form (cid:20) ∂∂t + (cid:18) e ν p C (cid:19) ϕ L hν p (cid:21) N Q = (cid:18) e ν p C (cid:19) Z L dx Γ( x ) ( n + + n − ) (8)and (cid:18) ∂∂t + 1 τ s + ϕ L hν p (cid:19) N S = Z L dx Γ( x ) ( n + − n − ) , (9)where the notation ϕ L = Z L dx Γ( x ) /v (10)denotes the dimensionless tunnel-coupling strength. Thissystem of equations must be solved together with the bo-undary conditions n + (0) = eu (0) hv , n − ( L ) = eu ( L ) hv , (11)and the current at point x can be calculated as I ( x, t ) = ev [ n + ( x, t ) − n − ( x, t )] . (12) III. AC RESPONSE
Calculate now the linear response of the system. Weassume that the voltage drop with frequency ω is sym-metrically applied to the terminals, i. e. u (0) = V exp( − iωt ) and u ( L ) = − V exp( − iωt ). Estimatesshow that for the edge states in HgTe quantum wells, hv and e /c are of the same order of magnitude. There-fore the terms with time derivatives in Eqs. 7 are muchsmaller than the ones with spatial derivatives and maybe omitted if we restrict ourselves to ω ≪ v/L . Using theboundary conditions Eq. 11, one may write the solutions FIG. 2. (color online). The real part (red curve) and imag-inary part (blue curve) of the conductance vs. frequency for ϕ L = 20 and hν p /τ s = 1. of these equations in the form n + ( x ) = eV hv K ( x, N Q + N S hvν p Z x dx ′ v Γ( x ′ ) K ( x, x ′ ) , (13) n − ( x ) = − eV hv K ( L, x )+ N Q − N S hvν p Z Lx dx ′ v Γ( x ′ ) K ( x ′ , x ) , (14)where K ( x, x ′ ) = exp (cid:26) − Z xx ′ dx ′′ v Γ( x ′′ ) (cid:27) . (15)A substitution of these solutions into Eqs. 8 and 9 givesa system of equations h − iω + (cid:18) hν p + e C (cid:19) (1 − e − ϕ L ) i N Q = 0 , (16) h − iω + 1 τ s + 1 − e − ϕ L hν p i N S = eVh (1 − e − ϕ L ) . (17)which suggests that N Q = 0. By substituting the valueof N S from Eq. 17 into Eqs. 14 and 12, one obtains theexpression for the current at the left and right terminals.Using the dimensionless spin-flip time η = τ s − e − ϕ L hν p , (18)it may be presented in the form I ω = e V h (cid:20) e − ϕ L + (1 − e − ϕ L ) η η − iωτ s (cid:21) . (19) FIG. 3. (color online). Contour plot of the dc conductanceEq. 20 as a function of ϕ L and hν p /τ s . Brighter colors corre-spond to smaller values of conductance The real and imaginary parts of the frequency-dependentconductance are shown in Fig. 2. In the low-frequencylimit ω ≪ τ − s , Eq. 19 gives I dc = e V h η + e − ϕ L η , (20)which suggests that the dc conductance varies from e /h to e / h and increase either with decreasing tunnel cou-pling ϕ L or increasing spin-flip rate τ − s . The contourplot of this quantity is shown in Fig. 3 as a function of ϕ L and the dimensionless spin-flip rate hν p /τ s . In thehigh-frequency limit, it follows from Eq. 19 that I hf = e V h (1 + e − ϕ L ) . (21)The high-frequency conductance also varies from e /h to e / h , but is independent of the spin-flip rate andis always smaller than the dc conductance. The high-frequency current is in phase with the ac voltage, andthe phase shift between them appears only at ω ∼ τ − s . IV. DISCUSSION
Though the response is calculated at frequencies muchlower than the inverse time of flight of an electron be-tween the terminals and the pileup of the charge is for-bidden in the system, it still exhibits a dispersion relatedwith spin imbalance in the puddle. At low frequencies,the conductance monotonically decreases as the couplingto the puddle and the spin-flip rate in it increase. Even-tually it becomes equal to one half of the conductance inthe absence of the puddle. This means that the puddlebreaks the system into two independent quantum resis-tors, each with a conductance e /h . When connected inseries, these resistors exhibit the conductance two timeslower, i. e. e / h . Should there be m puddles stronglycoupled to the edge states, the dc conductance would be m + 1 times smaller than e /h . In some sense, increas-ing the frequency is equivalent to increasing the spin-fliprate, and it leads to a similar decrease of conductance.The single-puddle model involves three unknown param-eters, i. e. ϕ L , ν p , and τ s . All of them can be determinedby comparing the experimental dispersion curve with Eq.19. If it is not possible to measure the ac response in thewhole frequency range, it may be possible to measure itin the dc regime and at a frequency well above τ − s , soone still can extract ϕ L and the product hν p /τ s by meansof Eqs. 20 and 21.The estimates show that the Fermi velocity in theedge states of HgTe quantum wells is about 5 × m/s.If the length of the edge state is one micron, the condition ω < v/L will be fulfilled up to the terahertz frequencies.It is more difficult to give reliable estimates of the spin-flip rate in the puddle. In low-temperature experimentson Au and Cu, the spin-flip time was ≈ . . To thebest of our knowledge, so far the ac response in 2D topo-logical insulators was measured at a constant frequencyof 2.5 THz and for several-micron long samples , whichis marginal for testing the obtained results. One could extend the frequency limits for observing the predictedeffects by choosing a shorter distance between the mea-suring probes and making an artificial puddle betweenthem by approaching a charged STM tip or by selectivedoping. This would provide a test for the proposed modelof the conductance suppression in the edge states of 2Dtopological insulators. V. CONCLUSION
We have calculated a current response to an ac volt-age of a pair of edge states in a 2D topological insulatorscoupled by tunneling to a conducting puddle in its bulk,where the electrons can flip their spin. Our goal was toprovide a means of experimental detection of such pud-dles. In a presence of such a puddle, the response exhibitsa dispersion at the inverse spin-flip time in the puddle.Its real part decreases from the zero-frequency value to asmaller value, while its imaginary part exhibits a maxi-mum at this frequency. By comparing the low-frequencyand high-frequency response, one can determine the pa-rameters of the puddle.
ACKNOWLEDGMENTS
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