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Magnetic Quantum Coherence Effect in Ni4 Molecular Transistors

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Magnetic Quantum Coherence Effect in Ni4 Molecular Transistors Magnetic quantum coherence effect in Ni4 molecular transistors Gabriel Gonzalez´ 1, ∗ and Michael N. Leuenberger2, 3, y 1Departamento de Matematicas´ y F´ısica, Instituto Tecnologico´ y de Estudios Superiores de Occidente, Periferico´ Sur Manuel Gomez´ Mor´ın 8585 C.P. 45604, Tlaquepaque, Jal., MEXICO 2NanoScience Technology Center, University of Central Florida, Orlando, FL 32826, USA, 3Department of Physics, University of Central Florida, P.O. Box 162385, Orlando, FL 32816-2385, USA (Dated: November 1, 2019) We present a theoretical study of electron transport in Ni4 molecular transistors in the presence of Zeeman spin splitting and magnetic quantum coherence (MQC). The Zeeman interaction is extended along the leads which produces gaps in the energy spectrum which allow electron transport with spin polarized along a certain direction. We show that the coherent states in resonance with the spin up or down states in the leads induces an effective coupling between localized spin states and continuum spin states in the single molecule magnet and leads, respectively. We investigate the conductance at zero temperature as a function of the applied bias and magnetic field by means of the Landauer formula, and show that the MQC is responsible for the appearence of resonances. Accordingly, we name them MQC resonances. PACS numbers: 73.63.Nm, 75.50.Xx, 75.45.+j, 05.60.Gg Keywords: Single molecule magnets, Magnetic quantum tunneling, Landauer formula INTRODUCTION spin ground state of Ni4 is S = 4. The article is organized as follows. First we will start with the model Hamiltonian of the Ni4 molecular transistor taking into Single-molecule magnets (SMMs), such as Mn12 (see Refs. account the magnetic quantum coherence of the two lowest 1, 2) and Fe8 (see Refs. 3, 4), have become the focus of ground spin states of the SMM. Then we will solve this model intense research since experiments on bulk samples demon- Hamiltonian and use the solution to calculate the conductance strated the magnetic quantum tunneling of a single magnetic through the molecular transistor as a function of the applied moment on a macroscopic scale. These molecules are char- bias by means of the Landauer formalism at zero temperature acterized by a large total spin, a large magnetic anisotropy for the SMM Ni4. The conclusions are summarized in the last barrier, and anisotropy terms which allow the spin to tunnel section. through the barrier. It is well known that magnetic quantum coherence (MQC) is realized when the SMM tunnels several times between degenerate spin states with opposite spin pro- MODEL HAMILTONIAN jections on the magnetic easy axis before the coherence is destroyed by the environment. Evidence of MQC has been The total Anderson-type Hamiltonian of a system formed reported for various superconducting systems and for antifer- by a single-molecule magnet (SMM) attached to two metallic romagnetic nanoclusters. [5–10] leads can be separated into three terms Electronic transport through SMMs offers several unique features with potentially large impact on applications such Htot = Hlead + HSMM + HSMM−lead: (1) as high-density magnetic storage as well as quantum computing.[11, 12] Recent experiments have pointed out the We will consider the leads as a one dimensional linear chain importance of the interference between spin tunneling paths in of sites. Thus, the first term on the right-hand side of Eq. (1) molecules. For instance, measurements of the magnetization is given by, in bulk Fe8 have observed oscillations in the tunnel splitting X X y X X y y ∆s;−s between states Sz = s and −s as a function of a trans- Hlead = σciσciσ −v ciσcjσ + cjσciσ ; verse magnetic field at temperatures between 0:05 K and 0:7 i σ=";# hiji σ=";# K (see Ref. 13, 14). This effect can be explained by the in- (2) arXiv:1208.0963v2 [cond-mat.mes-hall] 1 Jul 2014 y terference between Berry phases associated to spin tunneling where the operator ciσ (ciσ) creates (annihilates) electronic path of opposite windings.[15–19] states in the leads with spin orientation σ ="; #, and energy In this article we investigate coherent magnetic quantum tun- σ. The symbol hiji implies the sum over nearest neighbors. neling in the SMM Ni4 in which the tunneling rate is faster The potential of the wire is taken to be zero and the hopping in than the rate of decoherence and at a temperature at which the wire is v. The on site energies in the leads and in the SMM ∆z tunneling occurs only between the lowest spin states. We have are given by σ = 2 [σz]σσ and "0σ, respectively. ∆z = chosen to study the SMM Ni4 because of its high symmetry gµBHz is the Zeeman energy splitting in the leads where σz (S4) and large tunnel splittings (∼ 0.01K) at zero magnetic is the Pauli matrix.[22] field, which have been confirmed by high frequency EPR and The second term on the right-hand side of Eq. (1) denotes the magnetic relaxation experiments (see Ref. 20, 21). The total SMM part, which can be broken into spin, charging, and gate 2 contributions, SMM Ni4 in order to switch the total spin ground state from S = 4 to S = 9=2 or S = 7=2, the final ground state will be (q) HSMM = Hspin + Ec − q eVg; (3) the result of the exchange interaction between the total spins in the SMM. When the nanomagnet is singly charged and with where Ec denotes the charging energy, q is the number of ex- the application of the longitudinal magnetic field Hz the SMM cess electrons (the charge state of the molecule), and Vg is the will only allow electrons with spin down (up) polarization in electric potential due to an external gate voltage. In the pres- single electron tunneling transport due to spin blockade. If the ence of an external magnetic field, the spin Hamiltonian of the SMM has a total spin ground state of S = 9=2 or S = 7=2, SMM Ni4 reads then there will be transitions from j − 9=2i to |±i and from (q) j − 7=2i to |±i, respectively. For S = 9=2 spin up electrons H = −D S2 + C (S4 + S4 ) − µ gS~ · H~ spin q q;z q q;+ q;− B are transmitted through the SMM, whereas for S = 7=2 spin 2 4 4 1 ∗ down electrons are transmitted through the SMM. The energy = −DqSz + Cq(Sq;+ + Sq;−) − (h?Sq;+ + h?Sq;−) 2 levels as a function of the orientation of the magnetic mo- +hkSq;z; (4)ment is pictorially shown in Figure (1), where the gap between j+ > and |− > has been exaggerated to help visualization of z S = where the easy axis is taken along the direction and q;± the electronSequential transmission tunneling through thethrough SMM transistor. |-4> Sq;x ± iSq;y. The magnetic field components were rescaled We will be using two pairs of spin ground states j ± siq=0 to h? = gµB(Hx + iHy) and hk = gµBHz for the transver- sal and longitudinal parts, respectively, where g = 2:3 denotes j<0 j=0 j>0 the effective gyromagnetic ratio for the giant spin of the SMM −4 s 4 and Dq=0 = 0:75K and Cq=0 = 2:9 × 10 K. The total spin as well as the anisotropy constants D and C depend on t Z q q the charging state of the molecule, i.e. if the SMM is singly (a) 7/2 Sequential7/2 tunneling through |-4> s' 7/2 and doubly charged.[23] The longitudinal magnetic field Hz H z tilts the double potential well favoring those spin projections aligned with the field. At zero magnetic field the spin projec- The red regionj<0 is given by the bias voltage V.j=0 j>0 tions j4i and j − 4i have nearly the same energy and magnetic s 4 Gate voltage Vg is applied in such a way that spin down electrons flow through the SMM. quantum tunneling is possible. Importantly, for an individual Z t Ni4 nanomagnet quantum magnetic tunneling is possible only (b) between states that differ by 4 spin units, i.e. if the selection 0 9/2 rule sq − sq = 4k is satisfied, where k is an integer.[24] The 9/2 s' 9/2 H transverse magnetic field in the xy plane lifts the degeneracy z S ∆ 0 of the eigenstates of q;z by an energy sq ;sq , the so-called The red region is given by the bias voltage V. tunnel splitting, and leads to states that are coherent superpo- FIG. 1: Schematic illustration of tunneling through the SMM tran- Gate voltage Vg is applied in such a way that spin up electrons flow through the SMM. sitions of the eigenstates of Sq;z. Denoting λs;−s as the cou- sistor for s = 7=2 and s = 9=2. The dotted line corresponds to the pling matrix element between the states jsiq=0 and j − siq=0, direct electron path across the SMM transistor. the magnetic quantum coherence in the single molecule mag- net is given by the following effective Hamiltonian and j ± siq=1 for the uncharged and charged SMM, respec- tively. For the sake of clarity, we will denote the charged 0 HMQC = λs;−s (|−si00hsj + jsi00h−sj) : (5) ground states by j ± s i and the uncharged ground states by j ± si. With this notation and by restricting the Hilbert space where λs;−s represents the source of spin flipping. The to the lowest spin doublet of the SMM we can then write down most general coherent superpositions for the two lowest levels the Hamiltonian which represents the interaction of the SMM jsiq=0 and j − siq=0 is given by with the leads q 2 2 0 y 0 ∓∆s;−s + ∆s;−s + λs;−s |−si0 ± λs;−s jsi0 HSMM−lead = −t |±i h−s j c1# + c1# |−s i h±| + |±i = ; 0 y 0 N± |±i h−s j c1" + c |−s i h±| + (6) 1" 0 y 0 where |±i h−s j c−1# + c−1# |−s i h±| + r 0 y 0 q 2 |±i h−s j c−1" + c−1" |−s i h±| (8) 2 2 2 N± = λs;−s + ∓∆s;−s + ∆s;−s + λs;−s : (7) where t is the lead-molecule tunneling amplitude.
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