Disorder effects in topological insulator thin films
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Disorder effects in topological insulator thin films
Yi Huang ( 黄 奕 ) ∗ and B. I. Shklovskii School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA (Dated: February 16, 2021)Thin films of topological insulators (TI) attract large attention because of expected topological effects fromthe inter-surface hybridization of Dirac points. However, these effects may be depleted by unexpectedly largeenergy smearing Γ of surface Dirac points by the random potential of abundant Coulomb impurities. We showthat in a typical TI film with large dielectric constant ∼ sandwiched between two low dielectric constantlayers, the Rytova-Keldysh modification of the Coulomb potential of a charge impurity slows down the potentialdecay in space, and allows a larger number of the film impurities to contribute in Γ . As a result, Γ is large andindependent on the TI film width for films thicker than 2 nm. We also study disorder effect on hybridizationgap, the role of gates and the surface conductivity. I. INTRODUCTION
Topological insulators (TI) continue to generate a stronginterest because of their surfaces host massless Dirac states onthe background of the bulk energy gap. Typically, as-grown TIcrystals are heavily doped semiconductors with concentrationof donors ∼ cm − . (For certainty, we talk about n-typecase where the Fermi level is high in the conduction band).However, to employ Dirac states in transport, one has to movethe Fermi level close to the Dirac point. In bulk crystals, thisis done by intentional compensation of donors with almostequal concentration of acceptors. With increasing degree ofcompensation, the Fermi level shifts from the conduction bandto inside the gap and eventually arrives at the surface Diracpoints.This seemingly easy solution of the Fermi level problem,however, comes with a price [1]. In fully compensated TI,all donors and acceptors are charged, and these charges ran-domly distributed in space create random potential fluctua-tions as large as the TI semiconductor gap. This potential cre-ates equal numbers of electron and hole puddles, and substan-tially reduces the activation energy of the bulk transport [2, 3].At the same time near the surface, the random potential ofcharged impurities smears the Dirac point by the energy Γ self-consistently determined by the surface electrons screen-ing [4, 5]. This smearing was observed by the scanning tun-nel microscopy in Ref. 6. It also should determine the widthof Landau levels of Dirac electrons and quantum relaxationtime τ q = ~ / Γ as measured by Shubnikov-de-Haas oscilla-tions and other methods.Recently TI research shifted to thin TI films of thickness d in the 4-20 nm range [7–13]. This interest is related to obser-vations of the inter-surface hybridization leading to the Diracpoints hybridization gaps ∆( d ) and related topological effects,including quantum spin Hall effect [10]. However, such ob-servations are obscured by unexpectedly large effects of dis-order. One might think that the role of disorder in thin TIfilms should be smaller than in the bulk TI. Indeed, at a giventotal 3D concentration of charged impurities N , the 2D con-centration of them N d in a thin TI film is quite small. In a ∗ Corresponding author: [email protected]
Figure 1. TI thin film of thickness d with dielectric constant κ f de-posited on the substrate with dielectric constant κ . The top metallicgate is separated from the film by a spacer of thickness D with dielec-tric constant κ . The topological surfaces are shown by blue lines. Inthe case κ f ≫ κ , a typical charge impurity is shown by a red circlewith its electric field E (black) channeling through the film for a dis-tance λ before exiting outside. In similar topologically trivial films,the electric field exits at a larger distance r . thin film, the Fermi level can be shifted to the Dirac point alsoby the parallel to the TI film gate (see Figure 1). Therefore,one might expect that the compensation by acceptors can beavoided or reduced to get a much smaller Γ( d ) . However, adistant gate can only compensate the average charge densityof donors. Local fluctuations of the donor concentration andcharge density still create a large random potential which, af-ter self-consistent screening by surface electrons, results in alarge Dirac point smearing energy Γ( d ) .In this paper, we concentrate on the calculation of Γ( d ) , be-cause it can interfere with the Dirac points hybridization gaps ∆( d ) [14]. In contrary to the bulk case, what happens in thethin TI film strongly depends on the average dielectric con-stant of the film environment κ = ( κ + κ ) / , where indexes1 and 2 are related to two sides of the film (see Figure 1). Be-low we consider three different cases κ f ≫ κ , κ f = κ , and κ f ≪ κ .In Section II we start from the most interesting first casewhen the potential of a charge impurity was described by Ry-tova and Keldysh [15, 16]. We show that such a long distantpotential leads to an enhancement of Γ( d ) which, as a result,stays practically equal to its bulk value [4] for all d & nm.In section III we study the cases κ f = κ and κ f ≪ κ . In Sec-tion IV we comment on the role of the gate when it is close tothe film surface. In Section V we calculate the conductivity ofTI film. In Section VI we discuss the effect of hybridizationgap on the results from Section II to V. II. THIN TI FILM IN SMALL DIELECTRIC CONSTANTENVIRONMENT
In this section, we calculate Γ( d ) in the case of κ f ≫ κ .For example [10, 13], a BiSbTeSe (BSTS) thin film with κ f ∼ can be sandwiched between two h-BN layers with κ , ∼ [17]. In this case κ ∼ is 10 times smaller than κ f . If κ f ≫ κ , the electric field of a charged impurity in-side the thin film is trapped inside the film for a distance r = ( κ f / κ ) d , and only after r > r the electric field ex-its to the environment. This leads to the effective Coulombinteraction with asymptotic expressions [15, 16, 18] v ( r ) ≈ ( e κr , r > r , − e κr [ln( r/ r ) + γ ] , d < r < r , (1)where r is a 2D vector in the plane of TI film, γ = 0 . isthe Euler constant. At r < d , the Coulomb interaction is backto conventional form e /κ f √ r + z , where z is the distancefrom the impurity to the TI surface (because we are interestedin the random potential for the surface Dirac electrons). TheFourier transform of v ( r ) is v ( q ) = 2 πe κq (1 + qr ) , (2)valid for q < /d .In a TI film, the electric field of a charged impurity expe-riences additional screening by Dirac electrons living on thesurfaces of the film. To describe this screening, we start fromthe equation for the electric potential of screened charged im-purities φ ( r ) µ [ n ( r )] − eφ ( r ) = E F , (3)where E F = const . is the Fermi level (electro-chemical po-tential), µ [ n ( r )] = ~ v F k F [ n ( r )] is the (local) chemical poten-tial, v F is the velocity near the Dirac cone, and k F [ n ( r )] = p π | n ( r ) | is the local Fermi wave vector. If the averagechemical potential µ is large enough, so that µ ≫ e φ , then µ [ n ( r )] can be linearized in the local carrier density variation δn ( r ) µ [ n ( r )] ≈ µ + δn ( r ) /ν ( µ ) . (4)where ν ( µ ) = d n /d µ = µ/ (2 π ( ~ v F ) ) is the thermody-namic density of states (TDOS) at zero temperature. Intro-ducing the effective fine structure constant α = e /κ f ~ v F ,we can write the TDOS as ν ( µ ) = κ f α πe µ. (5) Figure 2. Log-log plot of the screened interaction v ( r ) for differ-ent q s . The gray lines are obtained by the Fourier transform of πe /κ ( q + q s ) , which shows ∼ r − in large distance. In the Thomas-Fermi (TF) approximation [19], the screeningby surface electrons can be described by the dielectric func-tion ǫ ( q ) = 1 − v ( q )Π T F , (6)where the TF polarization bubble is Π T F = − ν ( µ ) , and thebare interaction v ( q ) is given by Eq. (2). We arrive at thescreened potential of one charge impurity within a thin TI film v ( q ) = v ( q ) ǫ ( q ) = 2 πe κ [ q (1 + qr ) + q s ] , (7)where q s = 2 πe ν/κ and q < /d .We see that if q s r ≫ then, unlike in uniform 3D di-electrics, inside the TI film a strong screening happens at thedistance λ = ( r /q s ) / . (8)Indeed, the behavior of v ( q ) changes at q = λ − : v ( q ) ≈ ( πe κq s , q < λ − , πe κq r , λ − < q < d − . (9)The behavior of v ( r ) for different values of q s is shown inFigure 2. At large distance r ≫ λ , we get v ( r ) ≃ e /κq s r like for a quadruple. The difference between topological andtopologically trivial films is also schematically illustrated inFigure 1.Assuming that impurities are randomly distributed insidethe film, the mean squared fluctuation of the potential is givenby [20] (cid:10) φ (cid:11) = 1 e Z d r v ( r ) N d = 2 πN de κ f ( q s r ) , (10)where the function f ( x ) reads f ( x ) = 24 x − ( − x ) / tanh − √ − x, < x < / , − x − / tan − √ x − , x > / . (11)We are interested in two limiting cases of the dimensionlessparameter q s r = ( r /λ ) : (cid:10) φ (cid:11) = 2 πN de κ ( (2 q s r ) − , λ ≪ r , − − ln( q s r ) , λ ≫ r . (12)There is a simple qualitative interpretation of the limiting ex-pression of (cid:10) φ (cid:11) . In the case when λ ≪ r (or q s r ≫ ),surface electrons screening cuts off the impurity potential atdistance λ from the impurity center. The fluctuation of num-ber of impurities inside radius λ is equal to ( N dλ ) / . Sinceeach charge impurity of this area contributes to the potential ∼ e/κr [see Eq. (1)], we get (cid:10) φ (cid:11) ∼ ( N dλ )( e/κr ) ,namely the first line of Eq. (12). On the other hand, at λ ≫ r (or q s r ≪ ) the potential of impurity v ( r ) fol-lows Eq. (1) with effective screening length r . Taking intoaccount that fluctuation of number of impurities inside radius r is ∼ p N dr , we arrive at the second line of Eq. (12).We are interested in the charge neutrality point where E F =0 , and φ has the Gaussian distribution function with h φ i = 0 and (cid:10) φ (cid:11) = Γ /e . Next we want to calculate the averagedensity of states h ν i using the Gaussian distribution functionof φ : h ν i = Z ∞−∞ d ( eφ )2 ν ( eφ ) e − e φ / √ π Γ = 2 α κ f Γ(2 π ) / e . (13)Here we multiply the density of states by a factor of 2 be-cause the potential at each surface is screened by electronsof both the top and bottom TI surfaces. The above use ofRytova-Keldysh electric field inside the the TI film at dis-tance d < r < λ from a Coulomb impurity apparentlyis valid only for d < λ . This condition is equivalent to d . d c = α − / N − / [21]. For thicker films, d > d c , oneshould think about two separate surfaces like in a bulk sam-ple where each surface screens its own random potential [4].Then one also can find the lower limit of applicability of large d theory [4], d c , as the screening radius r s of a single surfacefound in Ref. 4.At d . d c replacing ν by h ν i in q s = 2 πe ν/κ , we have q s = r π α κ f Γ κe . (14)Now one can solve for Γ and q s self-consistently usingEqs. (12) and (14). If λ ≪ r , then Γ = (cid:18) π (cid:19) / e N / κ f α / , (15) q s = 2 / α / κ f κ N / , (16) λ = 2 − / α − / ( N d ) − / d. (17) Figure 3. Schematic log-log plot of Γ as a function of d . The bluesolid line corresponds to the scenario (a): ( κ f /κ a ) > α − > (inthe plot we choose κ f /κ a = 10 and α − = 7 ), while the red dashedcurve corresponds to the scenario (b): < ( κ f /κ b ) < α − (forthis case we used κ f /κ b = 2 and α − = 7 ). On the horizontalaxis we show characteristic dimensionless TI film widths ˜ d a,b =( κ a,b /κ f ) α − / , ˜ d a,b = ( κ a,b /κ f ) / , ˜ d = α / and ˜ d c = α − / . The result for Γ is independent on d , and up to a numericalfactor is the same as the results earlier obtained for a bulksamples [4]. Therefore, our Γ easily matches that of Ref. [4]at d = d c . To ensure the self-consistency, one should checkwhether the assumption q s r ≫ with q s given by Eq. (16)is correct. We find that, Eqs. (15), (16) and (17) are validif d ≫ d = ( κ/κ f ) α − / N − / . Note that at the neu-trality point Eq. (17) provides the typical size of puddles,while the concentration of electrons and holes in puddles is n p ∼ ( N dλ ) / /λ ∼ ( αN ) / . This concentration doesnot depend on d and is the same as the puddle concentrationat the surface of a bulk sample [4].In the other limiting case λ ≫ r , in the first approximationwe have Γ ≈ ( πN de κ ln "(cid:18) κκ f (cid:19) α ( N d ) / / , (18) q s ≈ (cid:18) ακ f κ (cid:19) √ N d ( ln "(cid:18) κκ f (cid:19) α ( N d ) / / . (19)Eqs. (18) and (19) are valid if d ≪ d , i.e., the arguments oflogarithms are much larger than unity.In order to derive the above results we assumed that elec-tric potential fluctuations follow Gaussian distribution. Thisassumption is valid if the number of substantially contribut-ing to the potential impurities M ≫ . If λ > r , or q s r < ,then Eq. (19) yields M = N dr ∼ N d ( κ f /κ ) ≫ when d ≫ d = ( κ/κ f ) / N − / . On the other hand, if λ < r ,or q s r > , using Eq. (17) we get that M = N dλ ∼ ( N d ) / α − / ≫ when d ≫ d = α / N − / .In Figure 3 we schematically summarize the dependenceof Γ( d ) for two scenarios (a) and (b) different by the ratio oftwo dimensionless parameters α − and κ f /κ . In scenario (a), κ f /κ > α − > , the energy Γ is a constant given by Eq. (15)for d > α / N − / , while for d < α / N − / the Gaussianapproach fails. On the other hand, in scenario (b), α − >κ f /κ > , the energy Γ is a constant given by Eq. (15) for d > ( κ/κ f ) α − / N − / , and it crosses over to Eq. (18) for ( κ/κ f ) / N − / < d < ( κ/κ f ) α − / N − / . In this sce-nario, the Gaussian approach fails at d < ( κ/κ f ) / N − / .Let us see how these two scenarios work for TIs based onBSTS-like systems with v F ∼ × m/s, κ f ∼ , α − ∼ and N ≃ cm − . If such a TI film is sliced between h-BNlayers, then κ ≃ κ a = 5 [17], κ f /κ ≃ > α − bringing usto scenario (a). If the same film is sliced between two layers ofHfO with κ ≃ κ b = 25 [22], then κ f /κ ≃ < α − and wefind ourselves in scenario (b). These two examples are usedin Figure 3 to plot functions Γ( d ) for both scenarios. In bothscenarios, d c ∼ α − / N − / ≫ d , , is the largest lengthscale.In the first example, Eq. (15) obtained for bulk samples [4]gives Γ ∼ meV which remains valid till very small filmwidths d ∼ d ≃ nm, in spite of smaller concentration ofimpurities N d . Such an unexpectedly strong role of disorderin thin BSTS-like TI films sandwiched between two low- κ layers is a result of the dielectric constant contrast between theTI film and its environment leading to the large contributionfrom distant Coulomb impurities into potential fluctuations.Above we ignored the concentration of charged impuritiesin the environment outside the TI film N e . Let us now eval-uate the role of such impurities. To save electrostatic energy,the electric field lines of an impurity at distance z . r fromthe film surface first enter inside the TI film and then radiallyspread inside the film to distance ∼ r before exiting outsidethe film to infinity. Thus, one can think that effectively eachoutside impurity is represented inside the film by a charge e disk, with radius z and thickness d . In the presence of screen-ing, only small minority of the outside impurities with z < λ contribute in fluctuating charge of the volume dλ . As a result,total effective concentration of impurities projected from out-side the film is N e λ/d . If N e λ/d < N , where λ is given byEq. (17), outside impurities can be ignored and our results arevalid. For example, in our scenario (a), for the BSTS TI filmwith d ∼ nm on silicon oxide substrate with [23] N e ∼ cm − , λ/d ∼ and our results are valid [24]. III. THIN TI FILM IN THE SAME OR LARGERDIELECRIC CONSTANT EVIRONMENT
In this section, we first consider the case when κ f = κ andthe Coulomb interaction with a charged impurity is v ( r ) = e /κr . In the TF approximation, the interaction screened bythe surface electrons is given by v ( r, z ) = e κ Z ∞ dq J ( qr )1 + q s /q e − qz , (20) where J ( x ) is the zeroth Bessel function of the first kind. Thepotential fluctuation squared reads (cid:10) φ (cid:11) = 1 e Z N d r Z d dzv ( r, z )= 2 πN de κ [ − e q s d Ei( − q s d )] , (21)where Ei( x ) is the exponential integral function. Eq. (21) hasthe following limits (cid:10) φ (cid:11) = 2 πN de κ ( (2 q s d ) − , q s d ≫ , − γ − ln(2 q s d ) , q s d ≪ . (22)Next, we solve Γ and q s self-consistently similarly to previoussections. If q s d ≫ one obtains the results for Γ and q s givenby Eqs. (15) and (16) with κ f = κ and smaller by / incoefficients. On the other hand, if q s d ≪ one gets the resultsof Γ and q s given by Eqs. (18) and (19), with κ f = κ .For BSTS film with κ f ∼ , α − ∼ and the impuritiesconcentration N ∼ cm − surrounded by the dielectricswith κ ∼ κ f , we get Γ ∼ meV and q − s ∼ nm.Let us now briefly consider the case of large- κ environment,when κ f ≪ κ , κ . For example, we can imagine thin TIfilms sandwiched between two STO layers which have verylarge dielectric constant. They should screen the random po-tential of impurities Γ and Γ on both side 1 and 2 surfacesand make Γ , ≪ e N / /κ f .If STO is only on side 2 of the TI film, it dramaticallyreduces Γ of this side, while on the other side potential isscreened by STO only at the distance r > d . The num-ber of impurities contributing to Γ ( d ) is ∼ √ N d , so that Γ ( d ) ∼ ( e /κ f d ) √ N d .For BSTS film with κ f ∼ , α − ∼ , and impuritiesconcentration N ∼ cm − sitting on top of STO, we have Γ ( d ) ∼ √ d meV where d is measured in units of nm. Forexample, if d = 10 nm, then Γ ∼ meV. IV. METALLIC GATE
In this section, we return to the case κ f ≫ κ , and discussthe effect on Γ from the metallic gate on top of the low di-electric constant layer with thickness D (see Figure 1). To getsome intuitions, we will start from the question how such agate affects the electric field of a point charge inside the film,namely how the gate modifies the Rytova-Keldysh potentialEq. (2) for the case of topologically trivial semiconductor filmwithout surface electrons and their screening. This questionwas carefully studied in Ref. 25. The main result is that, atsmall enough separation D < dκ f κ /κ ∼ r , large dis-tance part of the potential Eq. (2) is truncated (screened) atthe distance Λ = p Ddκ f /κ . r . This happens becauseelectric field lines exits the film in the direction to the gate atthe distance r & Λ [26].Let us now recall what surface electrons screening does tothe point charge potential in a TI film without gate. We saw inSection II that TI surface electrons screening length is givenby λ = p r /q s . Now for a TI film with the gate we have bothgate and surface electron screening working together. Com-paring the expressions of Λ and λ we see, as one could expect,that the distance D (2 κ/κ ) ∼ D , which is essentially the dis-tance to the gate, should play the role of q − s [27]. This meansthat if D & q − s the gate plays only a perturbative role, whilefor in the case D . q − s the distance D should replace q − s inthe final result for Γ . Replacing q − s by D (2 κ/κ ) in the caseof λ ≪ r in Eq. (12) yields Γ = 2 (cid:18) πe N Dκ f κ (cid:19) / . (23)This result is valid when it is smaller than Eq. (15), i.e. at D . D c = N − / α − / ( κ /κ f ) .In most experiments, D > D c , so the screening by gateis negligible compared to surface electrons screening. For ex-ample, in Ref. 10, the gate separation is D ≃ nm, while D c ∼ nm assuming κ f /κ ∼ and N ∼ cm − . V. CONDUCTIVITY
In this section, we calculate the conductivity of the surfacefor the scenario (a) in section II assuming that
D > D c . Inthe linear screening region µ ≫ e φ , where the electrondensity is weakly perturbed by impurities, using Boltzmannkinetic equation for Dirac electrons, one has the expression ofthe conductivity for a single surface [28]: σ = e h µτ ~ . (24)Here τ is the transport relaxation time whose inverse is givenby τ = ακ f N dk F π ~ e Z π dθv ( q )(1 − cos θ ) 12 (1 + cos θ ) , (25)where v ( q ) is given by Eq. (7) with q = 2 k F sin θ/ and q s = k F ακ f /κ . The factor (1 + cos θ ) / in Eq. (25) ariseswhen the backscattering is suppressed as a consequence ofthe spin texture at the Dirac point, as in Weyl semimetals [29].Changing the integral variable from θ to q , Eq. (25) can berewritten as τ = 4 πe κ f αN dκ ~ k F Z k F dq k F q p − ( q/ k F ) [ q (1 + qr ) + q s ] (26)Using x = q/ k F the integral in Eq. (26) can be expressed ina dimensionless form I = Z dx x √ − x [ x (1 + 2 k F r x ) + ( αr /d )] (27)Since we are considering the scenario (a) in section II, where d & d = α / N − / , d . d c = (2 k F α ) − [30] and k F > Γ / ~ v F ∼ ( αN ) / , we have the product k F d & α . Thereforewe are interested in the result of Eq. (25) in the limit k F d ≫ α . In this case k F r ≫ ακ f /κ > , so the integral in Eq. (27) isapproximated by I ≈ Z dx x √ − x [2 k F r x + ( αr /d )] (28)The integral kernel peaks at x ≃ (2 k F λ ) − , which corre-sponds to a momentum transfer q max ≃ λ − ≪ k F . Thepeak value is (4 q s r ) − = (2 k F dακ f /κ ) − . The width ofthe peak is ∆ x ∼ ( k F λ ) − . As a result, the integral in limit k F d ≫ α is given by I ≈ π √ κ /κ f ( k F d ) / α / . (29)and Eq. (26) is τ ≈ √ π α / e Nκ f ~ k / F d / . (30)Substituting Eq. (30) into Eq. (24) with k F = √ πn , we havethe conductivity σ ≈ e h π / n / d / α / N . (31)where d . d . d c . At d ∼ d c ∼ n − / /α or n ∼ ( αd ) − ,our conductivity Eq. (31) becomes of the order of σ ∼ e h n / N α (32)and with logarithmic accuracy crosses over to the bulk one [4].At the charge neutrality point n = n p ∼ ( αN ) / , we get theminimum conductivity σ min ∼ ( e /h )( N d /α ) / (33)which is larger than e /h in the range of its validity d & d = α / N − / . At d ∼ d c = α − / N − / our σ min ∼ e /hα and with logarithmic accuracy crosses over to the bulk one [4].It is remarkable that, in large range of concentrations ( αN ) / . n . ( αd ) − , not only Γ , but also the conduc-tivity Eq. (31) are determined by long range potential with q ∼ λ − ≪ k F . Only at large n and d the conductivityEq. (32) is determined by large momentum q ∼ k F scatter-ing on standard Coulomb potentials of impurities located atdistances smaller than k − F from the TI film surface [4]. VI. COULOMB DISORDER AND HYBRIDIZATION GAP
In a thin enough TI film the surface states of two oppo-site surfaces hybridize and their Dirac spectra acquires the hy-bridization gaps ∆ = ∆ exp( − d/d ) , where ∆ ∼ eVand d ∼ . nm for Bi . Sb . Te . Se . while d ∼ . nm for BiSbTe . Se . [10]. In the absence of disorder whenwe bring the Fermi level into the middle of this gap, the TIfilm surfaces become insulators. A strong long range disor-der potential φ ( r ) with characteristic scale a and with am-plitude Γ ≫ ∆ moves both bands up and down creating atthe Fermi level large electron and hole puddles with diameter ∼ a (Γ / ∆) / [31]. These puddles are separated by thin insu-lating stripes of the width x = a ∆ / Γ , which form insulatinginfinite cluster (see Fig. 4 in Ref. 4) residing at the potential φ ( r ) percolation level φ ( r ) = 0 . At low temperatures, thissystem can conduct only if electrons can easily tunnel acrossthese insulating stripes. So far in previous sections, we werethinking about relatively thick TI films with d ≥ nm wherethe hybridization gap ∆ is small enough to allow easy tunnel-ing, so that the conductivity of surface states is still metallic.Let us find the upper limit of ∆ for such metallic films, ∆ c .The probability of the Zener-like tunneling across a thininsulating stripe of width x is [9] P ∝ exp( − x ∆ / ~ v F ) = exp (cid:0) − a ∆ / Γ ~ v F (cid:1) , (34)Thus, the critical value of the hybridization gap at which P loses its exponentially small factor is ∆ c = (Γ ~ v F /a ) / . (35)In the case when κ f ≫ κ studied in Section II a = λ andusing Eqs. (15) and (17) we find that ∆ c ∼ e κ f d ( N d ) / α / . (36) For BSTS film with d ∼ nm, α − ∼ , κ f ∼ , andthe impurities concentration N ∼ cm − , Eq. (36) gives ∆ c ∼ meV.For the STO case studied in Section III using Γ ( d ) ∼ ( e /κ f d ) √ N d and the characteristic length of the potential a = d . Substituting these values into Eq. (35) we arrive tothe same ∆ c Eq. (36). This universal value is larger than theestimate of Ref. [9], where the bulk TI surface screening ra-dius [4] was used for a . Thus, the results of Section V forconductivity are valid if ∆ < ∆ c . On the other hand, our re-sults of Sections II and III for Γ and λ survive hybridizationgap effects even if ∆ c < ∆ < Γ . ACKNOWLEDGMENTS
We are grateful to S.K. Chong, V.V. Deshpande, B. Skinner,and D. Weiss and for useful discussions. Calculations by Y. H.were supported primarily by the National Science Foundationthrough the University of Minnesota MRSEC under AwardNumber No. DMR-1420013 and DMR-2011401. [1] B. Skinner, T. Chen, and B. I. Shklovskii, Phys. Rev. Lett. ,176801 (2012).[2] Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando, Phys.Rev. B , 165311 (2011).[3] T. Knispel, W. Jolie, N. Borgwardt, J. Lux, Z. Wang, Y. Ando,A. Rosch, T. Michely, and M. Gr¨uninger, Phys. Rev. B ,195135 (2017).[4] B. Skinner and B. I. Shklovskii, Phys. Rev. B , 075454(2013).[5] B. Skinner, T. Chen, and B. Shklovskii, Journal of Experimen-tal and Theoretical Physics , 579 (2013).[6] H. Beidenkopf, P. Roushan, J. Seo, L. Gorman, I. Drozdov,Y. San Hor, R. J. Cava, and A. Yazdani, Nature Physics , 939(2011).[7] Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang,X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y. Shan, et al. , NaturePhysics , 584 (2010).[8] D. Kim, P. Syers, N. P. Butch, J. Paglione, and M. S. Fuhrer,Nature communications , 2040 (2013).[9] D. Nandi, B. Skinner, G. H. Lee, K.-F. Huang, K. Shain, C.-Z.Chang, Y. Ou, S.-P. Lee, J. Ward, J. S. Moodera, P. Kim, B. I.Halperin, and A. Yacoby, Phys. Rev. B , 214203 (2018).[10] S. K. Chong, L. Liu, T. D. Sparks, F. Liu, and V. V. Desh-pande, “Topological phase transitions in a hybridized three-dimensional topological insulator,” (2020), arXiv:2004.04870[cond-mat.mes-hall].[11] D. Chaudhuri, M. Salehi, S. Dasgupta, M. Mondal, J. Moon,D. Jain, S. Oh, and N. P. Armitage, “Ambipolar magneto-optical response of ultra-low carrier density topological insu- lators,” (2020), arXiv:2010.10273 [cond-mat.mes-hall].[12] I. D. Bernardo, J. Hellerstedt, C. Liu, G. Akhgar, W. Wu, S. A.Yang, D. Culcer, S. K. Mo, S. Adam, M. T. Edmonds, andM. S. Fuhrer, “Progress in epitaxial thin-film na3bi as a topo-logical electronic material,” (2020), arXiv:2009.00244 [cond–mat.mes-hall].[13] J. Wang, C. Gorini, K. Richter, Z. Wang, Y. Ando, andD. Weiss, Nano Letters , 8493 (2020).[14] The interplay between ∆ and Γ at the surface of bulk TI is dis-cussed in Ref. [5].[15] N. S. Rytova, Moscow University Physics Bulletin , 18 (1967).[16] L. V. Keldysh, Soviet Journal of Experimental and TheoreticalPhysics Letters , 658 (1979).[17] A. Laturia, M. L. Van de Put, and W. G. Vandenberghe, npj 2DMaterials and Applications , 1 (2018).[18] P. Cudazzo, I. V. Tokatly, and A. Rubio, Phys. Rev. B ,085406 (2011).[19] TF approximation is justified as long as α ≪ . See relateddiscussion in Ref. 4.[20] Here we drop the short distance contribution r < d to the po-tential φ ( r ) which is standard Coulomb potential. This does notchange the result substantially as long as d ≪ λ .[21] The value of d c will be determined after we obtained λ ( d ) self-consistently in Eq. (17).[22] J. Robertson, The European physical journal applied physics ,265 (2004).[23] B. I. Shklovskii, Phys. Rev. B , 233411 (2007).[24] On the other hand, if N e λ/d > N the screening length λ should be recalculated self-consistently together with Γ . Then, instead of Eqs. (15) and (17), we arrive at newresults Γ ∼ ( e N / e /κ f ) α − / ( N e d ) − / and λ ∼ dα − / ( N e d ) − / .[25] S. Kondovych, I. Luk’yanchuk, T. I. Baturina, and V. M. Vi-nokur, Scientific Reports , 42770 (2017).[26] The length Λ can be found also via the following simple vari-ational estimate. Let us assume that electric field lines areconfined inside the TI film within radius Λ ≪ r from thepoint charge and exit from the film to the gate in the area ∼ π Λ . The total electrostatic energy consists of two ma-jor contributions, one is the energy of the field inside the film ∼ ( e /κ f d ) ln(Λ /d ) , and the other one is the field energy inthe gate dielectric ∼ e D/κ Λ . Minimizing the total energyyields Λ = p Ddκ f /κ ∼ √ r D .[27] The factor (2 κ/κ ) is of order unity if κ ≃ κ are not quitedifferent.[28] D. Culcer and R. Winkler, Phys. Rev. B , 235417 (2008).[29] A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B ,235126 (2011).[30] Here d c is obtained from the criterion d = λ where λ = p r /q s = ( d/ k F α ) / .[31] D. G. Polyakov and B. I. Shklovskii, Phys. Rev. Lett.74