Synthetic helical liquid in a quantum wire
SSynthetic helical liquid in a quantum wire
George I. Japaridze , , Henrik Johannesson , and Mariana Malard Andronikashvili Institute of Physics, Tamarashvili 6, 0177 Tbilisi, Georgia Ilia State University, Cholokasvili Avenue 3-5, 0162 Tbilisi, Georgia Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden and Faculdade UnB Planaltina, University of Brasilia, 73300-000 Planaltina-DF, Brazil
We show that the combination of a Dresselhaus interaction and a spatially periodic Rashbainteraction leads to the formation of a helical liquid in a quantum wire when the electron-electroninteraction is weakly screened. The effect is sustained by a helicity-dependent effective band gapwhich depends on the size of the Dresselhaus and Rashba spin-orbit couplings. We propose a designfor a semiconductor device in which the helical liquid can be realized and probed experimentally.
PACS numbers: 71.30.+h, 71.70.Ej, 85.35.Be
The concept of a helical liquid − a phase of matterwhere spin and momentum directions of electrons arelocked to each other − underpins many of the fascinatingfeatures of the recently discovered topological insulators[1]. In the case of an ideal two-dimensional (2D) topo-logical insulator, electron states at its edges propagate inopposite directions with opposite spins, forming a one-dimensional (1D) helical liquid (HL) [2, 3]. Given theright conditions [4, 5], the spin-filtered modes of the HLmay serve as ballistic conduction channels [6], holdingpromise for novel electronics/spintronics applications.The HL is expected to exhibit several unusual proper-ties, such as charge fractionalization near a ferromagneticdomain wall [7], interaction-dependent response to pinch-ing the sample into a point contact [8], and enhancedsuperconducting correlations when two HLs are coupledtogether [9]. A particularly tantalizing scenario is the ap-pearance of Majorana zero modes in an HL in proximityto a superconductor and a ferromagnet [10]. However,testing these various predictions in experiments on theHgTe/CdTe quantum well structures in which the HLphase has been observed is a formidable challenge: Thesoftness and reactivity of HgTe/CdTe makes it difficultto handle [11], and moreover, charge puddles formed dueto fluctuations in the donor density may introduce a he-lical edge resistance [12]. Alternative realizations of theHL are therefore in high demand. The prospect of usingthe dissipationless current of an HL in future chip designsadds to the importance of this endeavor [13].One suggestion is to use a nanowire made of a “strongtopological insulator” material [1]. When pierced with amagnetic flux quantum, the electrons in the wire are pre-dicted to form an interacting HL [14]. In another scheme − appearing in attempts to engineer Kitaev’s toy model[15] for p-wave pairing [16] − electrons in a quantum wireform an HL when subject to a Rashba spin-orbit couplingcombined with a transverse magnetic field [17]. These,like most other proposals for HLs in quantum wires [18],specifically rely on the presence of a magnetic field.In this article we show that an HL can be produced andcontrolled in a quantum wire using electric fields only. FIG. 1: Device supporting a 1D synthetic helical liquid: Elec-trons in a single-channel quantum wire (blue) formed in aheterostructure supporting a Dresselhaus interaction are sub-ject to a modulated Rashba field from a periodic sequence ofcharged top gates (dark grey).
The advantages of employing electric fields rather thanmagnetic fields are manifold. Most importantly, an elec-tric field does not corrupt the feature that counterprop-agating helical modes carry antiparallel spins. Also, anelectric field can easily be generated and applied locally,and eliminates many of the design complexities that comewith the use of magnetic fields [19]. Our proposed device(see FIG. 1) exploits an unexpected effect that appearswhen interacting electrons are subject to a Dresselhausspin-orbit interaction combined with a spatially periodic
Rashba interaction: When the electron density is tunedto a certain value, determined by the wavelength of theRashba modulation, a band gap tied to the helicity of theelectrons opens. This gives rise to an HL. Notably, the re-quired setup for realizing this HL is built around standardnanoscale semiconductor technology, and is very differentfrom the recently proposed all-electric setup in Ref. 20using carbon nanotubes. In what follows we derive aneffective model that captures the surprising effect fromthe interplay between the Dresselhaus and the modulatedRashba interaction. We analyze the model and explainhow the HL materializes, and also discuss the practicalityand robustness of this novel type of a synthetic HL.We consider a setup with a single-channel quantumwire formed in a gated 2D quantum well supported bya semiconductor heterostructure. The electrons in the a r X i v : . [ c ond - m a t . s t r- e l ] J un well are subject to two types of spin-orbit interactions,the Dresselhaus and
Rashba interactions [21]. For a het-erostructure grown along [001], with the electrons con-fined to the xy -plane, the leading term in the Dresselhausinteraction takes the form H D = β ( k x σ x − k y σ y ) with β a material-specific parameter. The Rashba interactionis given by H R = α ( k x σ y − k y σ x ) , where α depends onseveral distinct features of the heterostructure [22, 23],including the applied gate electric field. The latter fea-ture allows for a gate control of the Rashba coupling α [24]. It is important to mention that large fluctuations of α [22] may drive the HL to an insulating state throughan Anderson-type transition [25]. We shall return to thisissue below.Taking the x -axis along the wire, adding to H D and H R the kinetic energy of the electrons as well as the chemicalpotential, one obtains − using a tight-binding formula-tion − the Hamiltonian H + H DR , where H = − t (cid:88) n,α c † n,α c n +1 ,α + µ (cid:88) n,α c † n,α c n,α + h.c. , (1) H DR = − i (cid:88) n,α,β c † n,α (cid:104) γ D σ xαβ + γ R σ yαβ (cid:105) c n +1 ,β +h.c. , (2)with H DR the second-quantized projection of H D + H R along the wire. Here c † n,α ( c n,α ) is the creation (anni-hilation) operator for an electron with spin α = ↑ , ↓ onsite n (with spin along the growth direction ˆ z ), t is theelectron hopping amplitude, and µ a chemical potentialcontrollable by a back gate. The signs and magnitudesof γ D ≡ βa − and γ R ≡ αa − ( a being the lattice spac-ing) depend on the material as well as on the particulardesign of the heterostructure.We now envision that we place a sequence of equallycharged nanoscale electrodes on top of the heterostruc-ture (cf. FIG. 1). As a result, the Rashba coupling willpick up a modulated contribution due to the modula-tion of the electric field from the electrodes. Taking theirseparation to be the same as their extension along thewire (cf. FIG. 1), we model the Rashba modulation by asimple harmonic, H modR = − iγ (cid:48) R (cid:88) n,α,β cos( Qna ) c † n,α σ yαβ c n +1 ,β + h.c. , (3)with γ (cid:48) R the amplitude and Q its wave number. Besidesthe modulation of the Rashba interaction, also the chem-ical potential gets modulated by the external gates: H modcp = µ (cid:48) (cid:88) n,α cos( Qna ) c † n,α c n,α + h.c. (4)As follows from the analysis in Ref. 26, this termhas no effect at low energies unless the electron den-sity is tuned to satisfy the commensurability condition | Q − k F | << O (1 /a ) mod 2 π , with k F the Fermi wavenumber: At all other densities, including those for which an HL emerges, H modcp in Eq. (4) is rapidly oscillatingand gives no contribution in the low-energy continuumlimit. Hence, we shall neglect it here.Given the full Hamiltonian H = H + H DR + H modR , wepass to a basis which diagonalizes H + H DR in spin space, (cid:18) d n, + d n, − (cid:19) ≡ √ (cid:0) − ie − iθ c n, ↑ + e iθ c n, ↓ e − iθ c n, ↑ − ie iθ c n, ↓ (cid:1) , (5)with 2 θ = arctan γ D /γ R . The index τ = ± of the oper-ator d n,τ label the new quantized spin projections alongthe direction of the combined Dresselhaus ( ∝ γ D ˆ x ) anduniform Rashba ( ∝ γ R ˆ y ) fields. Putting γ (cid:48) R = 0 in Eq.(3) and using (5), the system is found to exhibit fourFermi points ± k F + τ q , where q a = arctan (cid:112) (˜ t/t ) − t = (cid:112) t + γ + γ , and where k F = πν/a with ν = N e / N , N e [ N ] being the number of electrons [lat-tice sites]. The corresponding Fermi energy (cid:15) F is givenby (cid:15) F = − t cos( k F a ) + µ .To analyze what happens when γ (cid:48) R is switched on, wefocus on the physically relevant limit of low energies, lin-earize the spectrum around the Fermi points and takethe continuum limit na → x . By decomposing d n,τ intoright- and left-moving fields R τ ( x ) and L τ ( x ), d n,τ → √ a (cid:0) e i ( k F + τq ) x R τ ( x ) + e i ( − k F + τq ) x L τ ( x ) (cid:1) , and choosing | Q − k F + τ q ) | << O (1 /a ) mod 2 π onethus obtains an effective theory with two independentbranches, H → (cid:80) i =1 , (cid:82) dx H i , where H applies to theFermi points ± k F ∓ q , and H to ± k F ± q . We herechoose Q = 2( k F + q ), and come back to the generalcase below. Omitting all rapidly oscillating terms thatvanish upon integration, one finds H = − iv F (: R †− ∂ x R − : − : L † + ∂ x L + :) (6) H = − iv F (: R † + ∂ x R + : − : L †− ∂ x L − :)+ iλ ( R † + ∂ x L − + L †− ∂ x R + ) , (7)where v F = 2 a ˜ t sin( πν ), λ = aγ (cid:48) R γ D ( γ R + γ D ) − / , : ... :denotes normal ordering, and where we have absorbedthe constant phase e i ( k F + q ) a into R + ( x ).While the nondiagonal term in Eq. (7) isrenormalization-group (RG) irrelevant in the absence of e-e interactions it may turn relevant and open a gap atthe Fermi points ± k F ± q when the e-e interaction H e - e = (cid:88) n,n (cid:48) ; α,β V ( n − n (cid:48) ) c † n,α c † n (cid:48) ,β c n (cid:48) ,β c n,α , (8)is included. Its low-energy limit can be extracted by fol-lowing the procedure from above, and we obtain H e - e → (cid:82) dx H e - e , where H e - e = g : R † τ L τ L † τ (cid:48) R τ (cid:48) : + g : R † τ R τ L † τ (cid:48) L τ (cid:48) :+ g L † τ L τ L † τ (cid:48) L τ (cid:48) : + L → R ) , (9)with τ, τ (cid:48) = ± summed over, and where g ∼ ˜ V ( k ∼ k F )and g ∼ ˜ V ( k ∼ V ( k ) being the Fourier transform ofthe screened Coulomb potential V ( n − n (cid:48) ) in Eq. (8). Thebackscattering ∼ g is weak in a semiconductor structureand renormalizes to zero at low energies also in the pres-ence of spin-orbit interactions [27]. In effect we are thusleft with only the dispersive and forward scattering chan-nels ∼ g in Eq. (9), to be added to H and H fromEqs. (6) and (7). Passing to a bosonized formalism [28],the resulting full Hamiltonian density can be written as H = H (1) + H (2) + H (12) with H ( i ) = H ( i )0 + λ δ i √ πKa cos( √ πKφ ) ∂ x θ , i = 1 , H (12) = g Kπ ∂ x φ ∂ x φ , (11)where K ≈ (1 + g /πv F ) − / . Here H ( i )0 = u [( ∂ x θ i ) +( ∂ x φ i ) ] is a free boson theory with u ≈ v F / K , and with θ i the dual field to φ i . The indices “1” and “2” taggedto the fields label the two branches originating from Eqs.(6) and (7).We should point out that our fields φ i ( i = 1 ,
2) arerotated with respect to the conventional bosonic fields φ R,Lτ ( τ = ± ) [29] representing the original fermion fields R τ and L τ , φ i = φ R ± + φ L ∓ , with upper (lower) sign at-tached to i = 1 ( i = 2). This nonstandard spin-mixingbasis { φ i } is suitable for revealing how the non-diagonalterm in Eq. (7) combines with the e-e interaction in Eq.(9) to gap out the states near ± k F ± q : The term inEq. (7) transforms into the sine-Gordon-like potentialin Eq. (10) [30], controlled by e-e interactions throughthe Luttinger liquid K -parameter. As we shall see, thetheory brought on the form of Eqs. (10) and (11) can beefficiently handled by using an adiabaticity argument.To make progress we pass to a Lagrangian formalismby Legendre transforming Eqs. (10) and (11). Using thatΠ i = √ K∂ x θ i serves as conjugate momentum to φ i / √ K ,Π i can be integrated out from the partition function Z ,with the result Z ∼ (cid:90) D φ D φ e − ( S (1) + S (2) + S (12) ) , (12)with Euclidean actions S ( i ) = S ( i )0 − δ i m πa (cid:90) dτ dx cos( √ πKφ ) , i = 1 , S (12) = g Kπ (cid:90) dτ dx∂ x φ ∂ x φ . (14)Here S ( i )0 = (1 / (cid:82) dτ dx [(1 /v )( ∂ τ φ i ) + v ( ∂ x φ i ) ] is afree action with v = 2 u , and m = λ / Kva .Having brought the theory on the form of Eqs. (13)and (14), valid for a Rashba modulation with Q =2( k F + q ), we first consider the auxiliary problem wherethe amplitude g in Eq. (14) is replaced by a tunable FIG. 2: (Color online) Schematic plot of the dispersion re-lations for the two types of helical liquid phases, with (a) Q = 2( k F + q ) and (b) Q = 2( k F − q ). parameter, g (cid:48) call it. Putting g (cid:48) = 0 and refermioniz-ing S (1) we then obtain a helical Dirac action for thefirst branch (with Fermi points ± k F ∓ q ), with the sec-ond branch (with Fermi points ± k F ± q ) described bya sine-Gordon action, S (2) . The cosine term in S (2) be-comes RG relevant for K < /
2, driving this branch toa stable fixed point with massive soliton-antisoliton ex-citations [30]. The energy to create a soliton-antisolitonpair defines an insulating gap ∆, and one finds from theexact solution of the sine-Gordon model [31] that∆ = c ( K )Λ( m Λ ) / (2 − K ) , K < , (15)where Λ = v/a is an energy cutoff, and c ( K ) is ex-pressible in terms of products of Gamma functions. Theopening of a gap implies that the field φ gets pinnedat one of the minima of the cosine term. Thus, in theneighborhood of the fixed point its gradient is suppressedwith the effect that the action S (12) remains vanishinglysmall also after g (cid:48) has been restored to its true value, g (cid:48) → g . In particular, it follows that S (12) cannotclose the gap. Note that this “argument by adiabaticity”is perfectly well controlled as the approach to a stablefixed point rules out any non-analyticities in the spec-trum. In summary, when K < /
2, a Rashba modulation Q = 2( k F + q ) opens a gap in the second branch whichbecomes insulating, leaving behind a conducting helicalelectron liquid in the first branch (see FIG 2(a)).The analysis above is readily adapted to the case with Q = 2( k F − q ), and one finds that the gap now opensin the first branch. Note that our results remain validin the presence of the weakened commensurability con-dition | Q − k F + τ q ) | (cid:28) O (1 /a ) mod 2 π, τ = ± , asthis condition still allows us to throw away the rapidlyoscillating terms in the low-energy limit of H modR .Our interpretation of the dynamically generated gap∆ as an effective band gap − as in FIG. 2 − draws on aresult by Schulz [32] where a bosonized theory similar tothat defined by our Eqs. (10) and (11) is refermionizedinto a non-interacting two-band model, with the bandsseparated by a gap corresponding to the dynamic gap ofthe bosonized theory. This picture − while heuristic only − helps to conceptualize the role of the commensurabilityconditions for the emergence of the synthetic HL.The fact that e-e interactions can open a gap in an HLis well-known from the literature [4, 5, 16]. In particular,Xu and Moore [5] noted that if a dynamically generatedgap opens in one of two coexisting Kramers’ pairs ( alias ‘branches’ 1 and 2 in our model), this gives rise to a stableHL in the other pair. Their observation pertains to thecase where the scattering within each branch is governedby distinct strengths of the e-e interaction: a gap maythen open in the branch with the stronger interaction.For this reason the Xu-Moore observation does not applyto the realistic case of of a single quantum wire with thesame interaction strength in the two spin-split branches.This is where our proposal injects a novel element intothe picture: By properly combining a modulated Rashbaspin-orbit interaction with a Dresselhaus interaction wefind that the gap-opening mechanism from e-e interac-tions can indeed be triggered in such a way as to open agap in one of the branches only, leaving behind a stableHL in the other. This HL is of a new type compared tothe ones hitherto probed experimentally: It owes its ex-istence neither to being ‘holographic’ [33] (like the edgestates of an HgTe QW [6]) nor to being ‘quasi-helical’ [18](as is the case for magnetic-field assisted HLs [34]). Thetime-reversal analogue of the notorious fermion-doublingproblem [35] is instead circumvented by the fact that thegapped branch breaks time-reversal symmetry sponta-neously by developing a spin-density wave [36]. As thereis no need to apply a magnetic field to realize the syn-thetic HL, it escapes the complications from time-reversalsymmetry breaking that mar a quasi-helical liquid [18].By this, it becomes an attractive candidate for renewedMajorana fermion searches [37].Having established a proof of concept that a syntheticHL can be sustained in a quantum wire by applicationof electric fields only, is our proposal also a ‘deliverable’in the lab? The query can be broken down into threespecific questions: (i) Is it feasible to realize a regimewith sufficiently strong e-e interactions (as required bythe condition K < / V ( k ∼ ≈ e π(cid:15) (cid:15) r ln( 2 dη ) + O ( η d ) (16)with η the half width of the wire, and where (cid:15) r is theaveraged relative permittivity of the dopant and cap-ping layers between the QW and a metallic back gateat a distance d from the wire [39]. The commonly usedIn − x Al x As capping layer has (cid:15) r ≈
12 when x = 0 . η ≈ v F ≈ × m/s [40], taking d > µ mand using that g = 4 ˜ V ( k ∼ /π (cid:126) [28], one verifies that K ≈ (1+ g /πv F ) − / < /
2. Thus, the desired “strong-coupling” regime is attainable without difficulty.Turning to (ii), we need to attach a number to the gap∆ in Eq. (15). Reading off data from Ref. 24, applicablewhen the InAs QW is separated from the top gates by asolid PEO/LiClO electrolyte, the Rashba coupling (cid:126) α isfound to change from 0 . × − eVm to 2 . × − eVmwhen tuning a top gate from 0 . . a ≈ (cid:126) γ R = 8 meV and (cid:126) γ (cid:48) R = 60 meV,assuming that [the spacers between] the top gates in Fig.1 are biased at [0.3 V] 0.8 V. As for the Dresselhauscoupling, experimental data for InAs QWs come withlarge uncertainties. We here take (cid:126) γ D = 5 meV, guidedby the prediction that 1 . < α/β < . λ = aγ (cid:48) R γ D ( γ + γ ) − / into Eq. (15), and choosing, say, K = 1 / c (1 /