Supplementary Information: Spin qubits in graphene quantum dots

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1 Supplementary Information: Spin qubits in graphene quantum dots jörn Trauzettel, Denis V. ulaev, Daniel Loss, and Guido urkard Department of Physics and stronomy, University of asel, Klingelbergstrasse 82, CH-456 asel, Switzerland Dated: December 26 I. GENERL MODEL We now present in detail how to derive solutions for bound states in a graphene quantum dot. The dot is assumed to be rectangular with width W and length L as illustrated in Fig. of the rticle. The basic idea of forming the dot is to take a strip of graphene with semiconducting armchair boundary conditions in x-direction and to electrically confine particles in y-direction. Transport properties of a similar system have been discussed in Ref. []. The low energy properties of electrons with energy ε in such a setup are described by the 4x4 Dirac equation σx i v x + σ y y Ψ + evyψ = εψ, σ x x + σ y y with the electric gate potential { Vgate, y L, Vy = V barrier, otherwise. In Eq., σ x and σ y are Pauli matrices, is Planck s constant devided by 2π, e is the electron charge, and v is the Fermi velocity. The four component envelope wave function Ψ = Ψ K, ΨK, ΨK, ΨK varies on scales large compared to the lattice spacing. Here, and refer to the two sublattices in the two-dimensional honeycomb lattice of carbon atoms, whereas K and K refer to the vectors in reciprocal space corresponding to the two valleys in the bandstructure of graphene. with 2 Plane wave solutions to Eq. take the form [2] Ψ + n,k r = χ+ n,k xeiky, χ + n,k x = a n,+ + b n,+ Ψ n,k r = χ n,k xe iky 3 e iqnx + a n,+ e iqnx + b n,+ e iq nx e iq nx 4 and χ n,k x = a n, + b n, e iqnx + a n, The complex number is given by e iqnx + b n, z n,k e iq nx e iqnx. 5 = ± q n + ik. 6 k2 + q 2 n The energy of the state in the barrier regions y < and y > L, where V = V barrier is given by ε = ev barrier ± v q 2 n + k 2. 7 In the dot y L, where V = V gate the wave vector k is replaced by k, satisfying ε = ev gate ± v q 2 n + k 2. 8 The ± sign in Eqs. 6 8 refers to conduction and valence bands. In the following, we concentrate on conduction band solutions of the problem keeping in mind that there is always a particle-hole conjugated partner solution. The transverse wave vector q n as well as the coefficients a n,±, a n,±, b n,±, b n,± of the n-th mode are determined up to a normalization constant by the boundary conditions at x = and x = W. We consider a class of boundary conditions for which the resulting parameters are independent of the longitudinal wave vectors k and k. We are particularly interested in semiconducting armchair boundary conditions defined by [3] Ψ x= = Ψ x=, 9 Ψ x=w = e i2πµ/3 e i2πµ/3 Ψ x=w, where µ = ± is defined by the width of the graphene strip W = a 3M + µ, with M a positive integer a =.246 nm is the graphene lattice constant, and is the 2x2 unit matrix. strip whose width is an integer multiple of three unit cells µ = is metallic and not suitable for spin qubit applications. The states of a semiconductor strip are non-degenerate in valley space: q n = π n + µ/3, n Z W

2 2 with a n,± = b n,± =, a n,± = b n,± for µ = or a n,± = b n,± =, a n,± = b n,± for µ =. Note that q n determines the size of the gap for each mode n that is due to the boundary conditions. The size of the gap of mode n is given by 2 vq n. For concreteness, we consider the case of µ = only. It can be shown that for the case of µ =, the bound states and normalized squared wave functions have exactly the same dependence on the parameters of a quantum dot, V barrier, V gate, W, and L. Our ansatz for a bound state solution at energy ε to Eq. then reads Ψ = α nχ n,k xe iky, if y <, β nχ + n, k xei ky + γ nχ n, k xe i ky, if y L, δ nχ + n,k xeiky L, if y > L. 2 For bound states, the wave function should decay for y ±, so we require that k = i q 2 n ε ev barrier 2 / v 2, 3 where vq n > ε ev barrier always has to hold. To find bound state solutions, we have to analyze the following set of equations coming from wave function matching at y = and y = L zn,k α n δ n = β n z n, k = β n z n, k + γ zn, k n e S + γ n zn, k, 4 e S, where α n = α na n,, β n = β na n,+, γ n = γ na n,, δ n = δ na n,+, and S i kl. We can write Eq. 4 as z n, k α n z n, k β n e S z n, ke S =. 5 γ n z n, ke S e S δ n The allowed energy values ε are readily determined by finding the roots of the determinant of the matrix on the lhs of the latter equation e 2S z n, k 2 z n, k 2 =. 6 rather obvious solution is z n, k = which implies that S = corresponding to ε = ± vq n + ev gate. 7 However, the corresponding wave functions to the allowed energy solutions 7 vanish identically. So, in order to proceed, we have to find other less trivial solutions to the transcendental equation 6. Since the case = z n, k only has the trivial solution 7, we can assume that z n, k, which means that V barrier V gate. Then, we find that e S = ± z n, k. 8 z n, k To analyze the solutions of Eq. 8, we distinguish two cases, one where S = i kl 9 is purely imaginary i.e., k is real, and another, where S is real. The two cases are distinguished by the criterion ε ev gate vq n and ε ev gate < vq n, respectively. In the former case, since the lhs of Eq. 8 has modulus unity, also the rhs must be unimodular, which is satisfied if in addition ε ev barrier vq n. This case, where the equation for the argument of Eq. 8 remains to be solved, is discussed in Sec. II. The latter case where S is real has no solutions. Indeed, let us rewrite Eq. 8 as follows: e 2S = 2 kk q 2 n ε ev barrier ε ev gate / v 2 + kk. 2 Taking into account that q 2 n > ε ev barrier ε ev gate / v 2 and kk R, we find that the left and right sides of this equation have different signs, therefore, Eq. 8 has no roots for any purely imaginary k. II. OUND STTE SOLUTIONS We now restrict ourselves to the energy window ε ev gate vq n ε ev barrier. 2 Then k is real, therefore e S = and z n, k =. Furthermore, is real. We define z n, k e iθ n, where θ n = arctan k/q n. 22 It is easy to verify that in the energy window 2 z n, k =. 23 z n, k We can now rewrite Eq. 8 as sin θ n z 2 n,k tan kl = 2 + z 2 n,k cos θ n and further simplify this expression by using that sin θ n = cos θ n = fter some algebra, we obtain 24 k/q n + k/q n 2, 25 + k/q n i kk tan kl =. 27 ε ev barrier ε ev gate / v 2 q 2 n

3 3 The latter equation in combination with Eq. 3 yields Eq. 7 of the rticle. Graphical solutions to Eq. 27 are shown in Fig. 2 of the rticle. y applying different voltages to the gate and the barriers we shift the energy bands of the graphene ribbon under the barriers with respect to that of the quantum dot. bound state in the quantum dot is allowed once the energy of the state hits the band gap of the barriers. If the difference of the barrier and gate voltages V = V barrier V gate is less than the energy of the gap 2 v q n for n-th subband, k of a bound state lies in the interval [ k max, k max ], where k max is found from the condition ε k max = V barrier + v q n and, therefore, k max = e V + 2 v q n /e V. The number of bound states for a given subband index n is proportional to the length of the quantum dot L and is given by N = k max L/π x is the integer just larger than x. The number of the bound states of the n-th subband is maximal, when the barrier-gate voltage difference equals the energy band gap V = 2 v q n, and so N max = 8 q n L/π. In the case of V > 2 v q n, the top of the valence band of the graphene ribbon under the barriers becomes higher than the bottom of the conduction band of the quantum dot, therefore, there are no bound states with energies ε < ev barrier v q n and k of a bound state lies in the interval [ k min, k max ], where k min is found from the condition ε k min = V barrier v q n k min = e V 2 v q n /e V, therefore, bound states lie in the energy window ev barrier v q n ε ev barrier + v q n as shown in Fig. 2 of the rticle and the number of the bound states is given by N = k max L/π k min L/π. With increasing the barrier-gate voltage difference, a m-th bound state appears at V = v q n + v q 2 n + π/l 2 m 2 with the energy ε m = V gate + V + v q n and ends up at V = v q n + v q 2 n + π/l 2 m 2 with the energy ε m = V gate + V v q n see Fig. 2 in the rticle. III. WVE FUNCTION Following rey and Fertig [3], we write the wave function as Ψx, y = Ψ K Ψ K Ψ K Ψ K x, y x, y x, y x, y 28 and give solutions for each component separately. s mentioned above, the subscripts and refer to the two sublattices in the two-dimensional honeycomb lattice of carbon atoms and the superscripts K and K refer to the two valleys in graphene. Note that the normalization condition [3] ddx dy [ Ψ K µ x, y + Ψ K µ x, y] = 2 29 for µ =, each finally determines the normalization constant of the wave function. With the ansatz 2 we explicitly Ψ /q n Ψ Ψ -5 5 y q n FIG. : Ground-state wave function. Normalized squared wave function Ψ = Ψ K 2 = Ψ K and Ψ = Ψ K 2 = Ψ K for the ground state solution of a dot of length q n L = 5 with corresponding energy ε =. vq n. Here, ev barrier V gate =.5 vq n. The dotted lines indicate the dot region yq n 5. obtain the following components of the wave function Ψ K x, y = Ψ K x, y = Ψ K x, y = Ψ K x, y = α n e iqnx e iky, β n e iqnx e i ky + γ n z n, ke iqnx e i ky, δ n e iqnx e iky L, α n e iqnx e iky, β n z n, ke iqnx e i ky + γ n e iqnx e i ky, δ n e iqnx e iky L, α n e iqnx e iky, β n e iqnx e i ky + γ n z n, ke iqnx e i ky, δ n e iqnx e iky L, α n e iqnx e iky, β n z n, ke iqnx e i ky + γ n e iqnx e i ky, δ n e iqnx e iky L In the latter equations, the first line corresponds to the region in space, where y <, the second line to y L, and the third line to y > L. Thus, we obtain that Ψ K x, y 2 = Ψ K x, y 2, 34 Ψ K x, y 2 = Ψ K x, y We now plot the normalized squared wave function of a ground-state solution and a excited-state solution of a dot with length q n L = 5 in Figs. and 2, respectively. These are obtained from Fig. 2 of the rticle under the choice that ev barrier V gate =.5 vq n. Evidently, the ground-state solution has no nodes in the dot region, whereas the excited-state solution has nodes.

4 4 IV. LONG-DISTNCE COUPLING Ψ /q n Ψ Ψ -5 5 y q n FIG. 2: Excited-state wave function. Normalized squared wave function Ψ = Ψ K 2 = Ψ K and Ψ = Ψ K 2 = Ψ K for the first excited state solution of a dot of length q n L = 5 with corresponding energy ε =.34 vq n. Here, ev barrier V gate =.5 vq n. The dotted lines indicate the dot region yq n 5.. Long-distance coupling of two qubits Here, we discuss a particular example of long-distance coupling of two qubits separated by a distance d. The coupling is achieved via a continuum of states in the valence band of the barrier region as shown in Fig.3b of the rticle. Therefore, the long-range coupling is enabled by the Klein paradox. In the weak tunneling regime, the hopping matrix element is given by t ε Ψ L x, yψ Rx, ydx dy, 36 where Ψ L,R x, y = Ψx, y ± d + L/2 are the spinor wave functions of the left and right dots and ε is the single-particle energy of the coupled levels. The integration in transverse x- direction is trivial and just gives a factor W. The integration in longitudinal y-direction can be restricted to the integration window y [ d/2, d/2] if the wave functions are predominantly localized in the dot regions. Then, the hopping matrix element can be estimated for d L as t 4εα δ Wdz,k exp d k, 37 Vbarrier L Pin/Pout=7.2 V gate L V barrier2 t=.6 ε d=l Pin/Pout=2.7 R V gate2 FIG. 3: Long-distance coupling of a ground state and an excited state. The normalized squared wave functions Ψ = Ψ K 2 + Ψ K + Ψ K 2 + Ψ K of two qubits separated by a distance d = L, where L is the length of each quantum dot, are plotted next to each other. ground state of the series with the transverse quantum number n = in the left dot is coupled to an excited state of the same series with n = in the right dot. The coupling is as large as t =.6ε, where ε is the ground-state energy of the left dot. Furthermore, the qubits are still highly localized, which can be seen from the ratio Pin/Pout. Here, Pin is the probability of the electron to be inside the corresponding dot and Pout is probability to be outside the dot in the barrier regions. The parameters chosen for the potential in units of vq /e are V barrier = V barrier3 =, V barrier2 =.65, V gate =.5, and V gate2 =.9. R Vbarrier3 where α and δ are wave function amplitudes specified in Eqs In Eq. 37, we assumed that only levels of the series corresponding to the n = transverse mode are coupled. It is easy to relax this assumption because, if higher transverse modes form bound states, then only modes with n L = n R contribute to t, where n L/R is the transverse quantum number in the left/right dot. In Fig. 3, we demonstrate that a rather large coupling of t =.6ε can be achieved over a distance as large as ten times the size of the quantum dots see also Fig. 5 of the rticle for comparison. Note that the qubits in this example are well localized in the corresponding dot regions: The probability of the electron in the left dot to be in the dot region Pin is 7.2 times larger than to be in the barrier regions Pout. For the right dot, the ratio of Pin/Pout = 2.7 is a bit smaller but the electron is still predominantly localized in the dot region.. Long-distance coupling in multiple quantum dot setup In Fig.6 of the rticle, we propose a triple quantum dot setup in which dot and dot 3 are strongly coupled and the center dot 2 is decoupled by detuning. It is important that dot and dot 3 are coupled via the valence band states of dot 2 and not via the detuned qubit level of dot 2. Otherwise, the spin of the decoupled qubit level would be affected by the coupling of the other qubits which is unwanted in the proposed longdistance coupling scheme. We assume that the gates that put the three dots in the Coulomb blockade regime are set in such a way that cotunnelling processes from dot via dot 2 to dot 3 happen in the following order: First, an electron tunnels from dot 2 to dot 3 and then an electron tunnels from dot to dot 2.

5 5 The system is described by the Hamiltonian H = H + H T, 38 where the kinetic term describes three qubit levels α =, 2, 3 and the continuum of states in the valence band of dot 2 H = E α,σ a α,σa α,σ + ε k,σ b k,σ b k,σ 39 α= 3 σ=, k σ=, and the tunnelling Hamiltonian reads H T = t a,σ a 2,σ + a 2,σ a 3,σ + H.c. σ + t a,σ b k,σ + b k,σ a 3,σ + H.c.. 4 k,σ In Eq. 39, a α,σ and b k,σ annihilate electrons with spin σ in the qubit level of dot α and in the valence band of dot 2, respectively. We assume that E,σ = E 3,σ i.e. qubit and qubit 3 are on resonance and ε = E 2,σ E,σ E gap = 2 vq see Fig. 6 of the rticle. In Eq. 4, we make the approximation that all tunnelling matrix elements t depend only very weakly on energy and are real. The transmission rate from an initial state i to a final state f can be calculated using Fermi s golden rule W f i = 2π f Tε i i δε f ε i 4 with the transition matrix given by up to second order in H T Tε = H T + H T ε + iη H H T We can put η = in the latter equation because we are only interested in off-resonant cotunnelling processes. This means that ε 2 see Fig. 6 of the rticle should be finite because we want to have well-localized qubit states. The corresponding matrix elements of the T-matrix 42 may be written as Tε k,k = H T k,k + H T k,k k H T k ε ε,k k Now, we want to calculate T 3 3 TE,σ, where and 3 are the ground states of qubit and 3, respectively. The lowest non-vanishing contribution of T 3 is of cotunnelling type, i.e., of fourth order in t. It is possible to separate the different contributions to T 3 into three terms, namely T 3 = T Q 3 + T V Re [T Q 3 T V 3 ]. 44 In the latter equation, T Q 3 determines the transition rate via the qubit level of dot 2 the unwanted process, T V 3 determines the transition rate via the continuum of states in the valence band of dot 2 the wanted process, and 2 Re [T Q 3 T V 3 ] is the interference term of the two paths. It is straightforward to derive that for a given spin σ of qubit 2 T Q 3 E,σ = t 2 E,σ E 2,σ. 45 This has to be compared with where T V 3 E,σ = t 2 ν E = E gap /2 de ν E E,σ E Lt2 π v ln4 /E gap, 46 L vπ E E 2 E gap / is the density of states of the mode n = with E gap = 2 vq. In Eq. 46, we integrate over the whole band width of the valence band bounded by 6 ev. The approximate result in the second line of Eq. 46 holds for the hierarchy of energies E gap E,σ. In a more general case, the integral in Eq. 46 can still be evaluated analytically but yields a less compact expression. The contribution coming from T Q 3 is evidently the smallest term of the three terms on the rhs of Eq. 44. If we want to compare the rate that does not affect the spin of the qubit level in dot 2 which we call Γ V with the largest of the rates that does affect the spin of the qubit level in dot 2 which we call Γ Q, we can estimate Γ V T V 3 = Γ Q 2 Re [T Q 3 T V 3 ] L W ln4 /E gap. 48 The latter result is Eq. 9 of the rticle. It shows that by increasing the aspect ratio L/W we can increase the weight of the coupling via the valence band states of dot 2 which is wanted as compared to the weight of the coupling via the qubit state of dot 2 which is unwanted. V. DECOHERENCE Finally, we give some arguments and rough estimates for the spin decoherence in graphene. It is generally believed that spin-orbit effects are weaker in carbon than in Gas due to the lower atomic weight. Therefore, the dominating mechanism for decoherence will be the hyperfine coupling to the nuclear spins that are present in the material. The coherence time given by the hyperfine coupling can be estimated as [4] t N/c, where N is the number of atoms in the dot for a typical graphene dot, N 4, cn is the number of atoms per dot with a nuclear spin, and the coupling constant of the hyperfine interaction. Since we mainly deal with π orbitals in graphene, the contact hyperfine interaction is strongly reduced and the hyperfine interaction is dominated by its dipolar part. From the best available calculations for the dipolar hyperfine matrix elements [5, 6], one obtains dip = 8 µ 5 4π µ µ.38 µev, 49 r 3

6 6 where µ is the nuclear magneton of 3 C, µ is the ohr magneton, and µ the vacuum dielectric constant. This estimated value for dip is smaller than Gas 9 µev by more than two orders of magnitude. The natural abundance of 3 C is about c % which yields, with the values quoted above, a coherence time of approximately t µs, about a thousand times longer than in Gas. Unlike in Gas, this value can be improved by isotopic purification. Reducing the 3 C content by a factor of about already decreases the average number of nuclear spins per dot to about one. This allows for a preselection of the dots without any nuclear spin to be used as qubits. [] Silvestrov, P.G. & Efetov, K.. Quantum dots in graphene. Preprint at [2] Tworzydlo, J., Trauzettel,., Titov, M., Rycerz,. & eenakker, C.W.J. Quantum-limited shot noise in graphene. Phys. Rev. Lett. 96, [3] rey, L. & Fertig, H.. Electronic states of graphene nanoribbons studied with the Dirac equation. Phys. Rev. 73, [4] W.. Coish and D. Loss, Hyperfine interaction in a quantum dot: Non-Markovian electron spin dynamics. Phys. Rev. 7, [5] Tang, X.-P. et al., Electronic structures of single-walled carbon nanotubes determined by NMR. Science 288, [6] ntropov, V. P., Mazin, I. I., ndersen, O. K., Liechtenstein,. I. & Jepsen, O., Dominance of the spin-dipolar NMR relaxation mechanism in fullerene superconductors. Phys. Rev. 47,

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