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Electronic Transport in Single-Molecule Magnets on Metallic Surfaces

Department of Physics, Sejong University, Seoul 143-747, Republic of Korea School of Physics, Seoul National University, Seoul 151-742, Republic of Korea (Received February 2, 2008)

1

Gwang-Hee Kim1 ? and Tae-Suk Kim2 ?

2

arXiv:cond-mat/0402302v2 [cond-mat.mes-hall] 7 Apr 2004

An electron transport is studied in the system which consists of scanning tunneling microscopysingle molecule magnet-metal. Due to quantum tunneling of magnetization in single-molecule magnet, linear response conductance exhibits stepwise behavior with increasing longitudinal ?eld and each step is maximized at a certain value of ?eld sweeping speed. The conductance at each step oscillates as a function of the additional transverse magnetic ?eld along the hard axis. Rigorous theory is presented that combines the exchange model with the Landau-Zener model.

PACS numbers: 75.45.+j, 75.50.Xx, 75.50.Tt

Recently high-spin molecular nanomagnets such as Mn12 or Fe8 attracted lots of attention due to observation of quantum tunneling of magnetization and possible applications in information storage and quantum computing[1, 2, 3, 4, 5, 6]. These single-molecule magnets (SMMs) exhibit steps in the hysteresis loops at low temperature, which is attributed to resonant tunneling between degenerate quantum states or quantum tunneling of magnetization(QTM). These unique features of SMMs are the consequence of long-living metastable spin states due to the large spin and strong anisotropy of SMMs. QTM also made it possible to detect the interference e?ect of Berry’s phase on the magnetization at each step while the transverse ?eld along the hard axis is varied[5, 6]. Novel features of quantum tunneling are expected to manifest themselves in, if any, other observables. Especially the e?ects of QTM on the electronic transport remain to be explored in both experiments[7] and theories. In this paper we study theoretically the e?ects of QTM on the transport properties of SMMs which are deposited on a metallic surface with monolayer coverage. Placing the scanning tunneling microscopy(STM) tip right above one SMM, we compute the electric current which ?ows through a SMM when the bias voltage is applied between the STM tip and the metallic substrate(Fig. 1). We ?nd that the linear response conductance increases stepwise like the magnetization of a SMM as a longitudinal magnetic ?eld is increased. The stepwise behavior of conductance results from the QTM in SMM. The conductance at each step oscillates periodically as a function of additional transverse magnetic ?eld along the hard axis. Our theoretical predictions are not known in the literature as far as we know and can be tested experimentally. When a ?nite bias voltage is applied between the STM tip and the metallic substrate, the electrons will tunnel through a vacuum between the metal surface and the STM tip. Since the STM tip is placed right above the SMM in our model system, the tunneling electrons may well be scattered by the large spin of a SMM. Our model system can be considered as the conventional tunnel junc-

tion with a SMM sandwiched between two normal metallic electrodes. The metallic substrate and STM tip are conveniently called the left(p = L) and right(p = R) electrodes, respectively. Two electrodes are described by the featureless conduction bands with the energy dispersion ?pk , Hp = kα ?pk c? pkα cpkα . The Hamiltonian of the SMM will be introduced later. Tunneling electrons are modeled by the Hamiltonian[8, 9] H1 = TLR c? Lkα cRk′ α + H.c.

kk′ α

+

kα k′ β

JLR c? Lkα σαβ cRk′ β · S + H.c. , (1)

where α and β indicate the spin direction of electrons. The ?rst line represents the direct tunneling between two electrodes, while the second line describes the tunneling of electrons scattered by the spin S of SMM. Our theory is equally applicable to the molecular break junction geometry. The electric current can be computed using the Keldysh Green’s function method or equivalently the Fermi’s golden rule[8]. In this paper we study the very weak coupling limit so that the higher order process like

STM tip SMM Metallic substrate

FIG. 1: Schematic diagram of our model system. A singlemolecule magnet (SMM) is deposited on a metallic surface and the scanning tunneling microscopy (STM) tip is positioned right above the SMM. The easy axis of SMMs is directed normal to the metallic substrate.

2 the Kondo e?ect may be safely neglected. In this case it is enough to compute the electric current up to the leading order term. Using the Fermi’s golden rule the electric current can be written as ILR = e

m

Pm

kα k′ β

WLkαm→Rk′ βm′

×f (?Lk )[1 ? f (?Rk′ )] ? (Lkαm ? Rk ′ βm′ ). (2) Here Wi→j is the transition rate from the state i to j , f (?) is the Fermi-Dirac distribution function and Pm is the probability for the SMM to be in the state Sz = m. The leading contribution to the transition rate is given by the expression Wi→j = 2π | < j |H1 |i > |2 δ (Ei ? Ej ), where i and j are the collective indices denoting the states {Lkαm} or {Rk ′ βm′ } and Epkαm = ? ?pk + Em with ? ?pk = ?pk + ?p (p = L, R). ?p is the chemical potential shift in the electrode p due to the source-drain bias voltage, and Em is the energy of the state Sz = m in the SMM. Up to the second order in TLR and JLR , we ?nd the electric current to be ILR = 2 e2 2 γT + Sz γJ V h e + γJ Pm [S (S + 1) ? m(m ± 1)] h m ×[ζ (Em ? Em±1 + eV ) ? ζ (Em ? Em±1 ? eV(3) )], where γT (γJ ) = 4π 2 NL NR |TLR |2 (|JLR |2 ) is a measure of the dimensionless direct (spin-scattered) tunneling rate, 2 2 Sz = m m Pm , V is the source-drain bias voltage given by eV = ?L ? ?R , and ζ (?) = ?/[1 ? exp(?β?)] with β ?1 = kB T . The linear response conductance is e2 2 γT + γJ gs (T ) , where gs (T ) = Sz + then G = 2h

m Pm [S (S + 1) ? m(m ± 1)]η (Em ? Em±1 ) with η (?) = dζ (?)/d?. We would like to emphasize that only the spinexchange tunneling re?ects the dynamics of the QTM inside the SMM. Due to the crystal electric ?eld arising from the structure of a magnetic molecule, the ground state spin multiplet cannot remain degenerate. The e?ective Hamiltonian for the ground state spin multiplet of independent SMMs such as Fe8 can be expanded as[6, 10] 2 2 2 4 4 HSMM = ?DSz + E (Sx ? Sy ) + C (S+ + S? ) ?g?B (Hz Sz + Hx Sx ), (4)

Hz = HM = M D/g?B . When Hz = HM , the two states Sz = ?S and Sz = S ? M are degenerate energetically. Turning on the transverse terms leads to mixing of two degenerate states, lifts the degeneracy at the resonant ?elds and results in the avoided level crossing. The scaled conductance gs can be simpli?ed as gs (M ) = S 2 + M n=0 nPS ?n at zero temperature by noting that ES < ES ?1 < ... and η (?) = θ(?), the step function. In deriving this expression of gs (M ) it is assumed that the weight transfers from Sz = ?S to Sz = S, S ? 1, · · · with increasing longitudinal magnetic ?elds. To compute the probability, we need to solve the time-dependent Schr¨ odinger equation for the Hamiltonian HSMM . The probability is de?ned as Pj ≡ limt→∞ |aj (t)|2 when the wave function is written S The time-dependent as |Ψ(t) = j =?S aj (t)|j . Schr¨ odinger equation for |Ψ(t) is reduced to the coupled 2S + 1 di?erential equations for the coe?cient aj (t). Recently it was numerically found[10, 11] that the two-level approximation can reproduce quite well the results of the full di?erential equations. In the ensuing discussion we adopt the two-level approximation to ?nd an analytic formula of the probability. The weight transfer is found to occur only between the states Sz = ?S and Sz = S ? M (0) at the resonant ?eld HM , for M = 0, 1, 2, .., until the complete depletion of the state Sz = ?S . The amount of such weight transfer depends on the magnitude of the tunnel splitting or mixing ?M between two states. At (0) the resonant ?eld HM the full Hamiltonian HSMM is approximated as the e?ective two-level model[10, 12] between the states Sz = ?S and Sz = S ? M , He? = ?(S ? M )g?B ct ?M /2 , ?M /2 Sg?B ct (5)

(0)

(0)

where c(= dHz /dt) is the ?eld sweeping speed. De?ning τ = g?B ct/?M and γM = g?B c/?2 M , we (0) ?nd the coe?cient[13] around the resonant ?eld HM , aS ?M (τ ) = 1 M 2 i τ + πλM 4 γ M j =0 √ ×D?iλM ?1 [?(1 + i) αM τ ] , (6) λM Fj exp ?

M ?1

where Sx , Sy , Sz are three components of the spin operator, S± = Sx ± iSy , D and E are the second-order and C the fourth-order anisotropy constants, and the last term is the Zeeman energy. In the absence of transverse terms, the energy level of the state Sz = m is Em = ?Dm2 ? g?B Hz m. When we start with a ground state Sz = ?S corresponding to a large negative longitudinal ?eld, the level crossing with states Sz = S ? M (M = 0, 1, 2, · · · ) occurs at resonant ?elds,

2 where αM = (2S ? M )/(2γM ), λM = 1/(8αM γM ), Fj = exp(?2πλj ) and D is the parabolic cylinder function[14]. The desired probabilities are then PS ?M = M M ?1 (1 ? FM )( j =0 Fj ) and P?S = j =0 Fj . Note that Fj and 1 ? Fj denote the probability for an SMM not to transfer and to transfer from Sz = ?S to Sz = S ? j at the j -th resonant ?eld, respectively. To illustrate the above analytical results with concrete number, we compute the scaled conductance, g ?s (≡ gs ? S 2 ) at zero temperature for an octanuclear iron(II) oxo-hydroxo cluster of formula [Fe8 O2 (OH)12 (tacn)6 ]8+

3

6

M=4 3 c=0.014 c=0.08 c=0.26 M=2 M=3

10

M=4

5

<S >

z

5

M=3

0

M=2

g (M)

4

2

g (M)

-5

M=1

M=4

_

_

3

s

s

-10 1E-5

1E-4

1E-3

0.01

0.1

1

M=3

1 M=1

2

c (T/sec)

1

0

M=1

M=2

0

0.0 0.2 0.4 0.6 0.8 1.0

1E-5

1E-4

1E-3

0.01

0.1

1

Hz(T)

c (T/sec)

FIG. 2: The scaled conductance g ?s (M ) vs. the longitudinal ?eld Hz at zero temperature for three typical sweeping speed (T/sec). M = 1, 2, 3, 4 indicate the positions of the resonant (0) ?elds, HM = 0.215, 0.429, 0.643, 0.858. in units of Tesla.

FIG. 3: Dependence of g ?s (M ) on the ?eld sweeping speed c at each resonant ?eld. Inset: the magnetization Sz vs. c at each resonant ?eld.

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where tacn is a macrocyclic ligand[2]. We adopt the model parameters from Refs. 6 and 10: D = 0.292K, E = 0.046K, C = ?3.2 × 10?5 K. The tunnel splitting ?M is calculated for Hx = 0.1Hz at the resonant ?eld by employing the numerical diagonalization[10] or the perturbation method[15]. We obtain qualitatively the same results when Hx has the ?xed value at all resonant ?elds[16]. M i?1 The scaled conductance, g ?s (M ) = i=1 j =0 Fj ? is dis≤ Hz < Fj which is valid for M played in Fig. 2 for three typical ?eld sweeping speeds. Similar to the magnetization curve, the scaled conductance is featured with the stepwise increase as a function of magnetic ?elds. The jumps in g ?s (M ) occur at the resonant ?elds and are caused by the QTM in SMMs. The (0) step height is very tiny (? 0.318 × 10?4) at H1 = 0.215 T for all three sweeping speeds. At the second and third resonant ?elds the step heights are more pronounced and their magnitude depends sensitively on the value of c. Some steps are missing depending on both the sweeping speed and the resonant ?elds. To study in more detail the structure of the steps in the conductance we plot in Fig. 3 the scaled conductance g ?s (M ) at each resonant ?eld as a function of the sweeping speed c. In comparison the magnetization, M M i?1 Sz = S ? i=1 j =0 Fj ? (2S ? M ) j =0 Fj , is displayed in the inset. The magnetization is a monotonically decreasing function of c while the conductance is nonmonotonic and maximized at the speci?c value of c. (0) Since the weight transfer, 1 ? Fj at Hj , from Sz = ?S to Sz = S ? j is monotonically decreasing with increasing c, the magnetization is expected to decrease with c. Unlike the magnetization, the conductance has contributions only from the transferred states but not from Sz = ?S . Since Fj is increasing with c, P?S is an inM j =0 (0) HM (0) HM +1 ,

M=3, 4

2

g (M)

M=2 1

_

s

M=1

0 0.00 0.25 0.50 0.75 1.00 1.25 1.50

Hx(T)

FIG. 4: Oscillation of g ?M as a function of transverse ?eld for c = 0.014T/sec. For this sweeping speed the M = 3, 4 curves are almost identical except Hx ? 0, 0.4, and 0.8 T.

creasing function of c while PS ?M has the maximum value as a function of c. The conductance g ?s (M ) has the contribution δ g ?s = M PS ?M from the M -th resonance and is expected to have the maximum value at some value of c. Such a sweeping speed can be com(max) puted approximately as cM ? [π/(2 g?B )][?2 M /(2S ? j M ?1 M ?1 where νi = M )][log[M i=0 νi /(1 + j =1 i=0 νi )]]

2 (T/sec) are (2S )?2 i /[(2S ? i)?0 ]. The values of cM ?5 5.1 × 10 , 1.08 × 10?2, 0.182 at M = 1, 2, 3, respectively. Even though there exists a maximum in the scaled conductance at M = 4, the value of c = 5.16 (T/sec) lies beyond experimentally meaningful range. In order to observe the steps in conductance at M = 3 or M = 4 resonance, the sweeping speed should be larger than about 0.01 or 0.1 (T/sec), respectively. The conductances at the resonant ?elds are displayed in Fig. 4 as the transverse ?eld is varied along the hard (max)

4 axis. Similar to the magnetization the conductance at each resonant ?eld oscillates with almost the same period of ? 0.4 T. Such oscillatory conductance faithfully re?ects the structure of the tunnel splittings as a function of the transverse ?eld[16]. The periodic modulation of tunnel splittings by the transverse ?eld results from the interference between two spin paths of opposite windings around the hard axis[5, 6, 17, 18]. The tunneling splitting is known to vanish at the lattice of the diabolic ?elds[18]. At such ?elds the tunneling probability is zero so that the jump in the conductance vanishes. Depending on the parity of M , the oscillations of the conductance have the di?erent phase. The M = 2 curve is out of phase compared to the M = 1, 3 curves. For example, the conductance for M = 2 takes on the minimum value at the transverse ?eld where the conductance for M = 1 is maximized. This parity behavior originates from the impossibility of matching an evenvalued wave function with an odd-valued one which gives rise to diabolic ?elds. Weak structures around Hx = 0, 0.4, and 0.8 T for M = 3, 4 curves can be made conspicuous with varying the ?eld sweeping speed. Though the overall structure of oscillatory conductance persists, the amplitude of oscillations depends sensitively on the sweeping speed[16]. We brie?y address the e?ect of experimentally relevant issues on our theoretical results. It may be important to consider the e?ect of environmental degrees of freedom such as phonons, nuclear spin and dipolar interaction [19] on the magnetization process of SMMs. Such interactions make the SMM relax to the true ground state Sz = S and the relaxation process helps the magnetization to recover its full stretched value. Since all the transferred states Sz = S ? M (M = 1, 2, · · · ) lose the weight to the ground state, we expect that the value of g ?s will rise stepwise with increasing ?eld and might vanish in the end due to the relaxation process. Since the elapsed time between steps, which is of the order of 10 sec or less for the typical sweeping speeds (see Fig. 2), is much smaller than the relaxation time of magnetization (? 104 sec)[2, 19], we believe that the stepwise behavior of the conductance can be observed experimentally in the typical ?eld sweeping speeds. The e?ect of anisotropy in SMMs on the conductance was clari?ed in our work. In the absence of anisotropy gs = S (S + 1) so that the anisotropy in SMMs modi?es the conductance by the amount S out of S (S + 1). In the case of Fe8 or Mn12 S = 10 so that the modi?ed conductance is estimated to about 10% which lies in the experimentally detectable range. Possible exchange anisotropy in spinscattered tunneling can be addressed[16] by considering y x z 2 the ratio, a = (JLR + JLR )2 /4[JLR ] . When a > 1, the conductnce steps are more enhanced than the isotropic case(a = 1). For the case of a < 1, the steps are reduced or can be negative depending on the value of a. In summary we studied the current-voltage characteristics of the STM-SMM-metal system at low temperature. We found that the quantum tunneling of magnetization (QTM) in SMMs has a substantial e?ect on the electronic transport. The QTM in SMMs leads to the stepwise behavior in the conductance (just like the magnetization) when the magnetic ?eld is applied along the easy axis. Unlike the magnetization the conductance at each resonance is nonmonotonic with the sweeping speed and reaches the maximum at some sweeping speed. In addition, the conductance at the resonant ?elds is oscillating as a function of the transverse ?eld applied along the hard axis. G.-H.K. was supported by Korea Research Foundation Grant (KRF-2003-070-C00020). T.-S.K. was supported by Korea Research Foundation Grant (KRF2003-C-00038) and grant No. 1999-2-114-005-5 from the KOSEF.

? ?

[1] [2] [3]

[4]

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[18] [19]

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