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Development of electrostatically controlled quantum Hall ferromagnet, a new platform to realize high order non-Abelian excitations Aleksandr Kazakov,1 George Simion,1 Yuli Lyanda-Geller,1, 2 V. Kolkovsky,3 Z. Adamus,3 G. Karczewski,3 Tomasz Wojtowicz,3 and Leonid P. Rokhinson1, 2, 4 1 Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907 USA 2 Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907 USA 3 Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland 4 Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA (Dated: Submitted to Nature Materials on August 28, 2015) 1 Search for non-Abelian excitations is motivated by both scientific curiosity and a practical desire to alleviate decoherence problems of conventional qubits[1, 2]. While current efforts are primarily focused on the discovery of Majorana fermions, it is understood that braiding of Majoranas is not sufficient to perform universal quantum operations[3]. Proposals to realize higher order non-Abelian excitations with denser rotation group require properties not yet demonstrated in real systems. Here we report development of new heterostructure materials where quantum Hall ferromagnetic (QHFm) transitions can be induced locally by electrostatic gating. Helical domain walls (h-DW), formed at phase boundaries in the integer quantum Hall effect (QHE) regime, are similar to helical channels at the edges of two-dimensional topological isolators [4] and, coupled to an s-wave superconductor, should support Majorana fermions. In the fractional QHE regime h-DW are predicted to support parafermions[5] and, possibly, Fibonacci fermions[6], higher order non-Abelian excitations required for the realization of a fully protected topological quantum computer[3]. Beyond the study of topological excitation, new heterostructures can serve as model systems to study arbitrary configurations of Neél’s domain walls formed from excitations with fermionic or fractional anyonic statistics. In a quantum Hall effect (QHE) regime kinetic energy of electrons in a two-dimensional electron gas (2DEG) is quantized into Landau levels (LL), which are further split due to the presence of spin and electron-electron interactions[7]. Polarization of a 2DEG and, more importantly, of the top filled energy level, depends on the number of occupied energy levels ν = n/nφ (the filling factor is a ratio of electron n and magnetic flux nφ = eB/h densities), and changes as the system undergoes phase transitions between QHE states with different filling factors. If a 2D gas is separated into regions with different ν’s by, e.g., electrostatic gating, chiral current-caring states are formed at the boundary. These states are distinctly different from domain walls separating magnetic domains in ordinary ferromagnets. The actual order of spin-split energy levels is determined by an intricate balance between Zeeman, cyclotron and exchange energies. By shifting the balance it is possible to induce magnetic phase transitions between different QHE states with the same filling factor. One of the first studied examples was a transition between unpolarized and polarized ν = 2/3 states[8]. More quantum Hall ferromagnetic (QHFm) transitions have been studied by tilting 2 𝐸𝑠↓ < 0 𝐸𝑠↑ < 0 1.0 𝑄𝐻𝐸 0.5 𝐸𝑠↑↓ [K] h-DW 𝐸𝑠↑↓ 𝐸𝐹 0.0 𝑥 -0.5 𝐵2∗ 𝐵1∗ -1.0 0 3 6 9 12 magnetic field (T) (b) 60 𝜈=2 40 20 0 𝐸𝑛𝐶𝐹 + 𝐸𝑠↑↓ (meV) |𝑛, 𝑠 |3, ↑ |3, ↓ |2, ↑ |2, ↓ |1, ↑ |1, ↓ |0, ↑ |0, ↓ 80 1 𝑛 + 2 ℏ𝜔𝑐 + 𝐸𝑠↑↓ (meV) (a) |4, ↓ |3, ↓ |2, ↓ 0.2 |1, ↓ |4, ↑ |3, ↑ |2, ↑ |1, ↑ 0.0 1 5 , 3 3 -0.2 -20 0 (c) |𝑛, 𝑠 2 4 2 8 𝜈= , , , 3 3 5 5 0.4 5 7 10 magnetic field (T) (d) 8 9 10 magnetic field (T) FIG. 1. Formation of helical domain walls in a quantum Hall regime. (a) Schematic of edge states in a device where spin of the top level is changed under the gate (yellow). At the boundary counter-propagating edges form a helical domain wall (h-DW). Coupled to a superconducting contact (green), these h-DW should support non-Abelian excitations (magenta dots), which will be localized at the edges in the presence of sp in-orbit interactions. (b) Magnetic field dependence of spin subbands (Eq. 1) in a CdTe QW with xef f = 1.5% (solid lines) and 1.2% (dashed lines) of Mn paramagnetic impurities T = 100 mK. (c) Fan diagram of Landau levels for xef f = 1.5%). (d) Fan diagram for composite fermions Λ levels for xef f = 0.15% magnetic field[9, 10], introducing hydrostatic pressure[11], varying density[12], or inducing level crossing in dilute magnetic semiconductors [13–18]. If a system is developed where a QHFm transition can be locally controlled, helical domain walls (h-DW), which consist of two counter-propagating chiral channels with opposite polarization, will be formed at the phase boundary, see schematic in Fig. 1. In the presence of spin-orbit interactions these h-DW resemble Neel’s walls and allow investigation of electron transport in an arbitrary network of domain walls (unlike ordinary ferromagnets, the bulk of QHFm domains is not conducting). Even more excitingly, formation of h-DW from QHE states with integral and fractionalized excitations opens a new class of systems where Majorana and higher order non3 Abelian excitations can be realized. Recent demonstrations of induced superconductivity from superconducting contacts with high critical field in HgTe [19] and GaAs [20] makes demonstration of non-Abelian excitations in this system plausible. Unlike implementations bases on hybrid wire/superconductor structures [21, 22] QHE protection of h-DWs insure that exactly one helical 1D channel is present at the gate boundary. A network of these channels can be easily reconfigured in a multi-gate structure providing a path to the braiding of non-Abelian excitations. In the following we report development of new heterostructures where QHFm transitions can be controlled by electrostatic gating. High mobility two-dimensional electron gas is formed in CdTe quantum well (QW) heterostructures with engineered placement of paramagnetic impurities (Mn, spin S = 5/2). QWs with uniform Mn-doping have unusual spin splitting with spin levels crossing at high magnetic fields[23], and QHFm transition in both integer and fractional QHE regimes have been observed in tilted magnetic fields experiments [15, 16]. In the presence of magnetic field B spin-dependent energy in dilute magnetic semiconductors is [24]: g ∗ µB SB 1 ∗ ↑↓ , Es = ± g µB B + xef f Esd SBs 2 kB (T + TAF ) (1) where the first term is the Zeeman splitting and the second term is due to an s-d exchange between electrons in the QW and d-shell electrons localized on Mn. Here g ∗ ≈ −1.7 in CdTe, exchange Esd ≈ 220 meV [25, 26], xef f is an effective Mn concentration, and TAF is due to the Mn-Mn antiferromagnetic interaction. At low fields spin splitting is dominated by a large positive exchange term, while at high fields and low temperatures the Brillouin function Bs (B, T ) ≈ 1 and B-dependence is dominated by the negative Zeeman term. In Fig. 1 we plot spin splitting of energy levels (1) and spectrum of Landau levels (LL) for electrons (n + 1/2)~ωc + Es↑↓ and composite fermions (CF) EnCF + Es↑↓ , where energy gaps √ CF between CF levels En+1 − EnCF ≈ αC Ec /(2n + 1) ∝ B [27]. Here ~ is the reduced Plank’s constant, ωc is the cyclotron frequency, Ec = e2 /` is the charging energy, ` is the magnetic length, constant αC ≈ 0.01 − 0.03 depends on the confining potential [28], and n = 0, 1, 2, ... for electrons and n = 1, 2, 3... for CFs. The field of spin subbands crossing B ∗ for the same LL (|n, ↑i and |n, ↓i) or neighboring LLs (|n, ↑i and |n + 1, ↓i) depends on the strength of the s-d exchange interaction xef f Esd . Thus, engineering heterostructures with gate-tunable s-d exchange will allow local control of spin polarization in both integer and fractional QHE 4 Barrier QW FG, [Mn1] Back Gate 0.8 Front Gate Barrier 0.4 0.0 300 400 Barrier Mn2 0.4 0.0 FG [Mn2] BG [Mn2] 1.0 0.4 Quantum Well Barrier Ec [eV] 200 distance (nm) 1.2 0.8 100 0.3 0.2 (z)| 0 BG [Mn1] 1.1 (VG)(0) Ec [eV] 1.2 0.9 (b) 0.9 0.1 1.0 1.1 n(Vg) / n(0) Mn1 0.0 (a) 0.3 0.8 Mn2 0.2 0.4 0.0 0.1 Mn1 100 (z)| Ec [eV] Sample A (xeff=1.7%) 0.0 130 300 Samples B,C,D (xeff < 0.5%) (c) 0 100 110 120 130 distance (nm) distance (nm) FIG. 2. Modeling of a non-uniform dilute magnetic semiconductor CdMnTe quantum well heterostructure. (a) Band diagram of a 30nm CdTe QW heterostructure device is modeled using nextnano3 package [29] (top panel). Electron wavefunction is calculated for different voltages on the top (middle panel) and back (bottom panel) gates. In (b) an integral overlap χ(Vg ) between Mn-doped regions [M n1 ] and [M n2 ], normalized to the value at zero gate voltage χ(0), is plotted as a function of the 2D gas density change for front (FG) and back (BG) gates. (d) Mn doping distribution (red regions) in different wafers. regimes. The second term in (1) is a mean-field approximation to the exchange Hamiltonian hR i P ~ ~ ~ where interaction of an electron at a poJsd R~ i δ(~r − Ri )Si · ~σ ∝ [M n] |ϕ(z)|dz hSi, ~ i is approximated as an overlap of sition ~r with a large number of Mn ions at positions R the electron wavefunction ϕ(z) with a uniform Mn background within z ∈ [M n] and an ~ = hSz i = SBs (B, T ). For quantum wells with homogeneous average magnetization hSi R Mn distribution throughout the whole QW region an integral χ = [QW ] |ϕ(z)|dz has weak dependence on the shape of ϕ(z) and level crossing field B ∗ is almost independent of a gate voltage [16, 17]. We now consider non-uniform distribution of Mn inside a QW, e.g. Mn is confined 5 to regions [M n1 ] or [M n2 ] within the QW, see Fig. 2. In these regions ϕ(z) has strong dependence on the out-of-plane electric field and χ becomes gate dependent, χ = χ(Vg ). Application of positive (negative) voltage to the front gate shifts electron wavefunction closer to (away from) the surface, dχ/dVf g > 0 for [M n1 ] and dχ/dVf g < 0 for [M n2 ]. Gate voltage also changes electron density dn/dVf g > 0, thus dχ/dn > 0 for [M n1 ] and dχ/dn < 0 for [M n2 ] for the front gate. Application of a back gate voltage results in a density change dn/dVbg > 0 but electrical field shifts wavefunction in the opposite direction, thus dχ/dn < 0 for [M n1 ] and dχ/dn > 0 for [M n2 ] for the back gate. For the formation of well defined h-DWs we want to control B ∗ with a minimal change of n in order to remain at the same filling factor ν, or maximize |dχ/dn|. Quantum Hall effect in sample A is shown in Fig. 3a. A small peak at B ∗ = 7 Tesla in the middle of the ν = 2 state is a phase transition between |1 ↓i and |0 ↑i states which changes polarization of the top filled energy level. In the color plots magnetoresistance is plotted as a function of voltage on the front and back gates (Fig. 3b,c). For both gates density increases with gate voltage and the peak between ν and ν + 1 shifts to higher B. The QHFm transition B ∗ shifts in opposite directions as a function of Vf g and Vbg , consistent with the modelling in Fig. 2. Note that for B = 7 Tesla polarization of the top level can be tuned between |1 ↓i and |0 ↑i states by electrostatic gating, thus realizing the theoretical concept of Fig. 1. For small Mn concentration, relevant for the QHFm transition in the fractional QHE regime[15], gate-dependent xef f (Vg ) can be extracted from the modulation of Shubnikov-de Haas oscillations, where nodes are defined by the condition (n + 1/2)~ωc = |Es↑ − Es↓ | [30]. Positions of nodes are found to be gate dependent, as shown in Fig. 3d for the sample with xef f ≈ 0.34%. From the T -dependence of nodes positions we extract TAF ≈ 80 mK with no measurable gate dependence. Gate control of s-d exchange xef f (Vg )/xef f (0) = χ(Vg )/χ(0) is summarized in Fig. 4a for several wafers. Slopes dxef f /dn are in a good agreement with band simulations, the agreement is further improved when increase of Ec in the Mn-doped region is taken into account. Finally, we want to note that no peak is observed at ν = 1 (crossing of |0 ↑i and |0 ↓i levels), which is most likely due to a large exchange-induced anticrossing which we estimate to be ∼ 12 K. Exchange interactions are suppressed for the crossing of levels with different LL indexes. Nevertheless, we measured a small anticrossing of ≈ 1 K at ν = 2, see Fig. 4b, which 6 a) c) 0.75 5.0 0.50 2.5 0.25 Vfg (V) 7.5 0.3 5 6 7 8 1.0 b) 200 0.0 =3 d) 3.2 0 =2 |> -100 -200 5 6 0.0 6 8 B (T) VBG T = 30 mK 200 V 2.4 Rxx (k) 100 Vbg (V) 4 > |> =2 9 B (T) 2.0 0.1 0.00 4 (k) 0.2 |1 |0 0.0 Rxx > Rxy (h/e2) 1.00 Rxx (k) 10.0 1.6 0.8 7 8 0.0 0.1 9 B (T) -200 V 0.2 0.3 0.4 0.5 B (T) FIG. 3. Gate control of QHFm transition. (a) Longitudinal (Rxx ) and Hall (Rxy ) magnetoresistances in sample A measured at T = 400 mK. An additional maximum at ν = 2 is a QHFm transition between |1, ↑i and |0, ↓i spin-polarized levels. In (b) and (c) Rxx is plotted as a function of a voltage on front Vf g or back Vbg gates and magnetic field B. Position of the QHFm transition is highlighted by a white dotted line. For B = 7 T (green line) polarization of the top LL can be switched between ↑ and ↓ by the gate. Both plots have the same color scale. Measurements are performed at T = 300 mK. (d) Shift of SdH oscillations modulation in a sample with xef f = 0.34% with a back gate voltage measured at the base temperature T ≈ 30 mK. we attribute to the spin-orbit (SO) coupling between neighboring LLs. Energy spectrum in the presence of SO interactions is calculated by adding Dresselhaus γD κ · σ and Rashba γR E · (σ × k) spin-orbit terms to the single-particle Hamiltonian of a 2D gas in magnetic field in the presence of s-d coupling (1), square well confinement potential in z direction, and electric field potential eφ(z) ≈ eEz z, see Supplementary Material for details. Here γD and γR are the Dresselhaus and Rashba constants, and κ is defined as ({k̂x , k̂y2 − k̂z2 }, {k̂y , k̂z2 − k̂x2 }, {k̂z , k̂x2 − k̂y2 }). Energy spectrum near |0 ↑i and |1 ↓i levels crossing is plotted in the 7 Rxx (k) 800mK Rxx (k) xeff (VG) / xeff (0) 1.1 1.0 0.34% 0.2% 0.085% 1.71% 1.71% (FG) 0.9 (a) 0.9 200mK 2 1 0 6.5 7.0 7.5 8.0 B (T) 19.0 E (meV) Mn xeff 4 0.1 18.5 18.0 17.5 7.1 7.2 7.3 7.4 7.5 B (T) 1.0 (b) 200 1.1 n(VG) / n(0) 300 400 600 800 T (mK) FIG. 4. Gate control efficiency and effect of spin-orbit interaction. (a) Gate dependence of the measured effective Mn concentration for different wafers. Efficiency of s-d exchange control depend on the |dxef f /dn| slope: for QHFm transition in the integer QHE regime (B ∗ (Vg )−B ∗ (0)) ∝ (xef f (Vg ) − xef f (0)). (b) Arrhenius plot of the Rxx T-dependence at the QHFm transition, the activation energy is 0.096 meV. Top inset: temperature dependence of Rxx near ν = 2. Bottom plot: anticrossing of |0, ↑i and |1, ↓i levels calculated using spin-orbit Hamiltonian, see text. insert in Fig. 4b. The value of the anticrossing gap is found to depend only on the Rashba spin-orbit coupling ∆SO √ 2 2|γR hEz i| = . ` (2) 2 For an average electric field of hEz i = 35000 V/cm, B = 7 T and γR = 6.9 eÅ the gap ∆SO = 70 µeV, in a good agreement with the experimentally measured activation gap of 96 µeV. We note that an ability to open a topologically trivial (spin-orbit) gap is required for the localization of non-Abelian excitations [5]. Methods In order to demonstrate electrostatic control of QHFm transition several Cd1−x Mnx Te/ Cd0.8 Mg0.2 Te QW heterostructures were grown by molecular beam epitaxy (MBE), see [15, 16] for details. Mn was introduced into the QD region either as a digital δ-doping (for xef f > 1%, wafer A) or as a continuous doping (for xef f < 0.5%, wafers B-D), as shown schematically in Fig. 2c. Samples were patterned into Hall bar geometry and semitransparent 10 nm Ti front gate was thermally evaporated. Copper foil glued to the back of samples 8 served as a back gate. Devices were illuminated with a red LED at 4 K, low temperature electron density and mobility were in the range of 3.2 − 3.5 · 1011 cm−2 and 2 − 3 · 105 cm2 /V·s in different samples. Electron transport was measured in a dilution refrigerator using standard ac technique with 10 nA excitation. Acknowledgements Authors acknowledge support by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Awards DE-SC0008630 (A.K and L.P.R.), by the Department of Defence Office of Naval research Award N000141410339 (A.K, T.W., G.S. and Y. L-G.), by the National Science Centre (Poland) grant DEC2012/06/A/ST3/00247 (V.K., Z.A., G.K., and T.W.), and by the Foundation for Polish Science (T.W.). Authors contribution L.P.R. and T.W. conceived and designed the experiments, V.K, Z.A., G.K. and T.W. have grown heterostructures, A.K. and T.W. performed measurements and data analysis, G.S. and Y.L-G. provide theoretical analysis, A.K. and L.P.R written the manuscript with input from all authors. [1] A. Y. Kitaev, Ann. Phys. 303, 2 (2003). [2] M. H. Freedman, Proceedings of the National Academy of Sciences 95, 98 (1998). [3] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Rev. Mod. Phys. 80, 1083 (2008). [4] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [5] D. J. Clarke, J. Alicea, and K. Shtengel, Nat. Commun. 4, 1348 (2012). [6] R. S. K. Mong, D. J. Clarke, J. Alicea, N. H. Lindner, P. Fendley, C. Nayak, Y. Oreg, A. Stern, E. Berg, K. Shtengel, and M. P. A. Fisher, Phys. Rev. X 4, 011036 (2014). [7] R. E. Prange and S. M. Girvin, eds., The quantum Hall effect (Springer-Verlag, New York, 1990). [8] J. P. Eisenstein, H. L. Stormer, L. N. 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Jaroszynski, and G. Karczewski, Physical Review Letters 88, 186803 (2002). 11 Supplementary Materials Development of electrostatically controlled quantum Hall ferromagnet, a new platform to realize high order non-Abelian excitations A. Determination of TAF Antiferromagnetic coupling TAF in Eq. 1 of the main text can be obtained from the temperature dependence of nodes positions in the modulation of Shubnikov-de Haas oscillations: gµB B 1 ∗ ~ωc = g µB B + xef f SEs−d Bs n+ 2 kB [T + TAF ] Temperature dependence of magnetoresistance for a wafer with xef f = 0.34% is shown in Fig. S1a. From the fits to the temperature dependence of nodes positions we obtain TAF = 84 mK for this sample, we found that TAF ≈ 80 mK in all samples with xef f < 0.5% (in samples with larger Mn concentrations there is no easy way to determine TAF ). b) #081214 33 mK offset - 0,25 k Rxx [k] 3 2 1 625 mK 0 0,0 0,2 0,4 0,6 # 081214 node 1 node 2 node 3 node 4 node 5 node 6 node 7 node 8 0,6 node position [T] a) 0,4 0,2 xeff = 0.34% 0,0 0,0 0,8 B [T] TAF = 84 mK 0,2 0,4 0,6 0,8 1,0 T [K] FIG. S1. a) Magnetoresistance in a sample with xef f = 0.34% as a function of temperature. b) Temperature dependence of nodes extracted from the beating of Shubnikov-de Haas oscillations. Solid lines are two-parameter fittings (xef f and TAF ) to the nodes positions. sup-1 B. Calculation of spin-orbit–induced anticrossing of LLs in the presence of s-d exchange A general single-particle Hamiltonian can be written as: H0 = 2 −i~∇ + ec A + 21 gµB B · σ − eφ(r) + Vb (z) P − Jσ · i Si δ (Ri − r) + γD κ · σ + γR E · (σ × k) , 1 2m∗ (S1) where σ is a vector containing Pauli matrices, φ is an electric potential, and Vb is a confinement potential in z direction, the fifth term is s-d exchange with Mn impurities and the last two terms are Dresselhaus and Rashba spin-orbit coupling. κ is defined as ({k̂x , k̂y2 − k̂z2 }, {k̂y , k̂z2 − k̂x2 }, {k̂z , k̂x2 − k̂y2 }), where {A, B} = (AB + BA)/2 and k = −i~∇+eA/c. Magnetic field B = (0, 0, B) corresponds to a vector potential A = (0, Bx, 0). For exchange interaction we use a mean-field model described in the main text −Jρi hSi · σ = −Jρi BS (gµB S|B|/kB T )σ ·b , where ρi is the density of ions, B is the Brillouin function, kB is Boltzmann’s constant, and b is a unit vector in the direction of magnetic field. We consider high-field limit BS (x) = 1. We also assume uniform Mn doping in the range zmin < z < zmax . Electric potential is φ(x, z) ≈ −Ez z, we consider Ez > 0. The Hamiltonian describing motion in z direction is Hz = − ~2 ∂ 2 + Vb (z) − e|Ez |z − Jθ(z − zmin )θ(zmax − z)σz . 2m ∂z 2 (S2) Its eigenvalues for the lowest subband are: " 31 # ~2 Λs = e|Ez | −w − ai1 (S3) 2me|Ez | 0 sJ 2 2 0 2 2 + 0 2 Ai [Ξ (zmin )] − Ai [Ξ(zmax )] − Ai [Ξ (zmin )] Ξ (zmin ) + Ai [Ξ (zmax )] Ξ (zmax ) Ai (ai1 ) where Ai is the Airy function, ai1 is its first zero, Ξ(z) = (w + z)(2me|Ez |/~2 )1/3 + ai1 , and s = 1 for spin up states and s = −1 for spin down. In the presence of perpendicular magnetic field an effective Hamiltonian is √ γR |Ez | † † 3 2 2 † D ~ωC a† a† + 21 + 21 (gµB B + δΛ) i √γ2` â ââ − a − 2` k â + 2 ` â 3 z , H0 = √ γR |Ez | † γD 1 1 † † 3 2 2 † † √ −i 2`3 ââ â − (â ) − 2` kz â + 2 ` â ~ωC a a + 2 − 2 (gµB B + δΛ) (S4) √ √ where lowering and rising operators are defined as a† = (k̂y − ik̂x )/ 2, a = (k̂y + ik̂x )/ 2, p ` = eB/~ is the magnetic length, and ~ωC = ~eB/m is the cyclotron energy. sup-2 We treat spin-orbit couplings as perturbations and found that only Rashba term has a non-zero matrix element between |0 ↑i and |1 ↓i energy levels. In the vicinity of crossing the energy spectrum is 1 E± = ~ωC ± 2 r (~ωC − gµB B − δΛ)2 + and the anticrossing gap is ∆SO 8γR2 Ez2 , `2 (S5) √ 2 2|γR hEz i| . = ` (S6) 2 For an average electric field of hEz i = 35000 V/cm, B = 7 T and γR = 6.9 eÅ , ∆SO ≈ 70 µeV. A similar value is obtained from a numerical solution of Eq. S4. sup-3