Introduction

Quantum interference of electrons in mesoscopic systems leads to the striking transport phenomena, the so-called universal conductance fluctuations (UCF)1. It predicts that disordered mesoscopic systems with sizes smaller than the phase coherent length exhibit sample-to-sample conductance fluctuations in the order of e2/h, independent of the details of the system such as material properties, disorder strength and sample size. The UCF only depends the dimensionality and universality class of the system, as can be understood in the framework of the random matrix theory2,3. Time reversal symmetry (TRS) and spin rotational symmetry (SRS) play an important role in determining the universality class. There exist three types of universality classes: circular orthogonal ensemble (β = 1), where TRS and SRS are present; circular unitary ensemble (β = 2), where TRS is broken; circular symplectic ensemble (β = 4), where SRS is broken and TRS is preserved.

Recent discoveries of novel two-dimensional (2D) materials4 have raised several interesting issues in the quantum interference effects such as weak (anti)localization5,6 and UCF7,8,9,10,11,12,13. In particular, graphene exhibits unusual UCF behavior. When disorder is governed by long-range potentials, the inter-valley scattering of Dirac fermions is suppressed in graphene, hence the amplitude of the UCF is increased by a factor of 2 compared with conventional metals7,8,9. Moreover, it was suggested that the finite size effects in graphene can lead to a systematic deviation from the universal behavior10. More importantly, in the presence of strong spin-orbit interaction (SOI) in graphene, Qiao and coworkers11 have shown that the universal spin Hall conductance fluctuation, an analogue of the UCF in spin Hall conductance12,13, does not follow the conventional value for the circular symplectic ensemble of β = 4. Although it was argued that a new type of universality class exists, its origin is not clearly resolved.

Strong SOIs in 2D systems, on the other hand, give rise to a wide range of intriguing physical phenomena such as spin-valley coupling40,41, such as tij = 2.6 between the C atoms in pristine graphene, tij = 3.2 between the edge C atoms and γad = 5.72 between the host C atom and adatom in the case of H adatoms with concentrations nad. Other type of adatoms should give qualitatively similar results except for the breaking of electron-hole symmetry42. The adatom-induced SOI26,28 is not included.

For a device model in which a disordered graphene nanoribbon (GNR) of 100 nm length is sandwiched between two semi-infinite GNR electrodes, the two-terminal conductance is calculated by using the Landauer-Büttiker formula, G = (e2/h)Tr(tt†), where t is the transmission matrix. To examine the sample-to-sample fluctuation behavior in the conductance of the system, we analyze the data statistically. Individual data points are obtained by taking the ensemble average over more than 2,000 different configurations.

Results and Discussion

We first discuss the effect of the intrinsic SOI (HKM) on the electronic structure of pristine systems in Fig. 1. To identify the one-dimensional edge state, we consider two armchair graphene nanoribbons (N-aGNRs) of about 10 nm width, where N is the number of C-C dimer lines across the ribbon. For 83-aGNR and 82-aGNR, the bulk band gaps are 0.04 and 0.15 eV, respectively. In the strong coupling regime (λKM = 0.1, λR = 0), the band gaps of both aGNRs increase to 0.95 eV due to the SOI and the topological edge states appear in the gaps [Fig. 1(a,b)]. When periodic boundary conditions (PBC) are imposed along the transverse direction in 82-aGNR, the edge states are removed whereas the bulk gap remains unchanged [Fig. 1(c)].

Figure 1
figure 1

Band structures of graphene TIs.

Band structures of (a) 83-aGNR, (b) 82-aGNR and (c) 82-aGNR with periodic boundary conditions in the presence of intrinsic spin-orbit coupling (λKM = 0.1). The bulk and edge states are illustrated as black regions and blue lines, respectively. For comparison, the bulks states of pristine aGNRs for λKM = 0 are illustrated as gray shaded regions.

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To understand the influence of disorder on the transport properties of graphene TI, we plot the averaged conductance of disordered GNRs as a function of channel energy for λKM = 0.1 in Fig. 2(a). For moderate strengths of Anderson disorder (W = 1.5, 2.0), the edge states (E > − 0.5 eV) are unaffected by the disorder and hence the electron transport is ballistic, indicating the robustness of the topological edge states22. In the case of adatom disorder, however, the edge states become slightly damaged close to the neutrality point due to the formation of strongly localized adatom defect states40. Despite this, most of the states are still in the (quasi-)ballistic transport regime. If we remove the edge states by imposing PBC, the conductance value drops to zero [dotted lines in Fig. 2(a)]. On the other hand, for the bulk states (E < −0.5 eV) in the diffusive regime, conductance is suppressed and exhibits large sample-to-sample fluctuations depending on the disorder strength.

Figure 2
figure 2

Electronic transport characteristics.

(a) Averaged conductance and (b) its deviation values are plotted as a function of channel energy for 83-aGNRs of 100 nm length in the presence of Anderson and adatom disorders. For comparison, the results are also given for 82-aGNRs with periodic boundary conditions (empty symbols). The lines in (b) represent the deviation values predicted by the UCF theory for circular orthogonal ensembles (β = 1), circular unitary ensembles (β = 2) and circular symplectic ensembles (β = 4) in quasi-one-dimensional systems1,2.

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Graphene-based TI, like any metallic or semiconducting systems, exhibits universal conductance fluctuations when the electron transport is diffusive. However, the amplitude of these fluctuations is distinguished from that of previously studied spin-orbit coupled 2D systems. Figure 2(b) shows the deviation of conductance as a function of channel energy in 83-aGNR and 82-aGNR. Since the edge states are robust against the disorder, conductance fluctuations are mostly zero for energies above –0.5 eV. The conductance fluctuation in the case of adatom disorder is associated with the hop** between defect states40. In the diffusive transport regime below –0.5 eV, regardless of the channel energy, disorder type and disorder strength, the conductance fluctuation exhibits the universal behavior with the UCF value of 0.52 e2/h. Note that this UCF value corresponds to that estimated from the circular unitary ensemble (β = 2) and differs from the estimated value of 0.365 e2/h for the Rashba SOI (β = 4)1,, where λH is the coupling constant. Therefore, including sz and substituting λH for λKM in HHaldane exactly result in the KM Hamiltonian [Eq. (1)]. Due to the sz term, HKM preserves the TRS and its system belongs to the circular symplectic ensemble (β = 4). That is to say, the combination of two β = 2 ensembles results in the β = 4 ensemble. Since HKM is just a direct sum of spin-up and spin-down Haldane terms with each component having the opposite sign, , the conductance values obtained by HKM are identical to those of HHaldane. As a result, the intrinsic SOI in graphene TI (HKM) leads to the exactly the same UCF value as that derived from HHaldane, which is larger by a factor of compared with the extrinsic SOI case (HR) [Fig. 3(b)].

Figure 3
figure 3

The UCF in the presence of various interactions.

Black dotted, solid and dashed lines denote the predicted UCF values for β = 1, β = 2 and β = 4, respectively, as in Fig. 2(b). Red, green and blue symbols represents the calculated deviation values for disorder strengths of W = 1.5, W = 2.0 and nad = 0.5%, respectively. Channel energy is set to –1.2 eV. (a) Deviations as a function of an external magnetic field (B) for λKM = λH = λR = 0, (b)-(c) deviations as a function of λH, λKM and λR for B = 0 and 50 T and (d) deviations as a function of λKM in the presence of Rashba coupling (λR = 0.15) for B = 0 and 50 T. In (b), two dashed lines denote the results for 82-aGNRs with periodic boundary conditions.

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Here, we emphasize that our analysis for the graphene KM model can be extended to generic 2D systems. The microscopic SOI is described by the Hamiltonian of HSO~s ⋅ (∇V × p). When (1) electrons are confined in 2D systems () and (2) the mirror symmetry with respect to the 2D plane is present (∂V/∂z = 0), the allowed interaction is given by sz(∂V/∂xpy − ∂V/∂ypx). Due to the sz-related term, the spin-flip scattering does not take place, indicating that spin-up and spin-down states are well separated. Furthermore, spin-up and spin-down components are identical except for the sign as in the KM model [Eq. (1)]. These results imply that, when 2D materials have both the intrinsic SOI and the mirror symmetry about the plane, the same UCF behavior as that of the KM model should be observed. A more fundamental origin of such a distinct UCF value in graphene-based TI, therefore, is the perfect 2D nature of graphene and the intrinsic SOI that preserves the mirror symmetry.

We compare the effects of an external magnetic field B, HHaldane, HKM and HR on the UCF behavior in Fig. 3. When only an external magnetic field is applied perpendicular to the 2D plane, the universality class changes from a circular orthogonal ensemble (β = 1) to a circular unitary ensemble (β = 2). The gradual evolution of the UCF value from 0.72 to 0.52 e2/h with increasing of B is independent of the type and strength of disorders [Fig. 3(a)]. The reduced UCF value by a factor of is attributed to the elimination of the particle-particle channels (so-called Cooperons), according to the diagrammatic perturbation theory1.

For B = 0, all the UCF values converge to 0.52 e2/h for the KM and Haldane interactions [Fig. 3(b)], following the trend of the B field only. The calculated conductance and its deviation values are exactly the same for the KM and Haldane interactions, due to the underlying symmetries of the Hamiltonians as discussed earlier. It is clear again that the existence of the topological edge states does not affect the UCF behavior [dashed lines in Fig. 3(b)]. On the other hand, when the Rashba SOI involving spin-flip scattering is considered, where the SOI breaks both SRS and the mirror symmetry, the UCF value is reduced by a factor of 2 as expected1,2,43,44.

To illustrate the difference between the Haldane and KM interactions, we now turn on the magnetic field in such systems. For a magnetic field of B = 50 T, the UCF values are plotted as a function of interaction parameters (λH, λKM and λR) in Fig. 3(c). In the Haldane interaction, since TRS is already broken, increasing the magnetic field does not change the universality class, thus, the UCF value remains unchanged. In the KM interaction, the magnetic field reduces the UCF value by a factor of , in contrast to the case of B = 0 [Fig. 3(b)]. The reduced UCF value is attributed to different symmetries in the two interactions: the magnetic field breaks the symmetry between up and down spins in HKM, while HHaldane maintains the spin symmetry even for B ≠ 0. In the Rashba interaction, the magnetic field further decreases the UCF value from 0.365 to 0.258 e2/h (by a factor of )47.

We examine the combined effect of KM and Rashba interactions on the deviation of conductance. We choose the Rashba coupling of λR = 0.15 to ensure that the system is initially in the circular symplectic ensemble (β = 4). For B = 0, as the KM coupling λKM increases from 0 to 0.2, the UCF does not change with the value of 0.365 e2/h, implying that the system maintains the universality class of β = 4. When a magnetic field of 50 T is additionally applied, the UCF value is reduced to 0.258 e2/h for all the values of λKM, similar to the Rashba case in Fig. 3(c). The UCF behavior, therefore, is not influenced by the intrinsic SOI as long as the extrinsic SOI involving spin-flip scattering is present in the system.

In graphene, it may be difficult to realize a spin-orbit coupled system without spin-flip scattering, because enhancing the SOI generally involves the breaking of mirror symmetry which leads to the Rashba-type interaction26,27,28,29,30,31,32,33. However, graphene TIs in sandwiched structures35,36 and monolayer transition metal dichalcogenides in trigonal prismatic (1H) structure14,15 can provide a promising platform for testing our prediction, because they have both strong SOI and mirror symmetry. In such systems, increasing the Rashba coupling by applying an external electric field would lead to the 1/ reduction of the UCF value, as illustrated in Fig. 3(d), instead of the 1/2 reduction in Fig. 3(b). On the other hand, if spin-flip scattering initially exists in the system, an electric field will not affect the UCF behavior. Therefore, the 1/ reduction of the UCF value by applying an electric field could serve as a signature reflecting the absence of spin-flip scattering in 2D materials. In graphene devices17,26, the electric-field-induced Rashba coupling was predicted to be extremely small (about 10−8 ~ 10−5 eV) presumably due to neglecting the effect of orbital angular momentum48. However, a substantial Rashba splitting up to 0.1 eV was observed49 when Au atoms were intercalated between graphene and substrate.

As a final remark, our understanding of the UCF behavior in spin-orbit coupled systems would enable us to reveal the exotic spin Hall conductance fluctuations predicted in 2D TIs. It was reported that, in 2D materials such as graphene and HgTe quantum well, the intrinsic SOI leads to an amplitude of the universal spin Hall conductance fluctuation (= 0.285 e/4π)11,50, which is larger than that of the extrinsic (Rashba-type) SOI (= 0.18 e/4π)11,12,13, similar to the UCF case. The reason for this discrepancy remains to be clarified. Although the arguments in our study for charge transport cannot be directly used to explain the spin Hall conductance behavior, our results provide important insights to understanding the role of intrinsic and extrinsic SOIs in 2D systems.

In conclusion, we have shown that the underlying symmetries of the system, rather than the topological edge states, play a key role in characterizing the UCF in 2D TIs. In 2D materials with both strong SOI and mirror symmetry, we find that the UCF value can be reduced by a factor of by applying an external electric field, while it is not affected when mirror symmetry is initially broken. Thus, our findings can be experimentally confirmed by observing the effect of an electric field on the UCF, which reflects the existence of spin-flip scattering in 2D materials.

Additional Information

How to cite this article: Choe, D.-H. and Chang, K. J. Universal Conductance Fluctuation In Two-Dimensional Topological Insulators. Sci. Rep. 5, 10997; doi: 10.1038/srep10997 (2015).