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Berry-phase blockade in single-molecule magnets

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We formulate the problem of electron transport through a single-molecule magnet (SMM) in the Coulomb blockade regime taking into account topological interference effects for the tunneling of the large spin of a SMM. The interference originates from
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    a  r   X   i  v  :  c  o  n   d  -  m  a   t   /   0   6   1   0   6   5   3  v   1   [  c  o  n   d  -  m  a   t .  m  e  s  -   h  a   l   l   ]   2   4   O  c   t   2   0   0   6 Berry-phase blockade in single-molecule magnets Gabriel Gonz´alez 1 , 2 and Michael N. Leuenberger 1 , 2 ∗ 1 NanoScience Technology Center, University of Central Florida, Orlando, FL 32826 and  2 Department of Physics, University of Central Florida, Orlando, FL 32816  (Dated: February 6, 2008)We formulate the problem of electron transport through a single-molecule magnet (SMM) in theCoulomb blockade regime taking into account topological interference effects for the tunneling of thelarge spin of a SMM. The interference srcinates from spin Berry phases associated with differenttunneling paths. We show that in the case of incoherent spin states it is essential to place the SMMbetween oppositely spin-polarized source and drain leads in order to detect the spin tunneling in thestationary current, which exhibits topological zeros as a function of the transverse magnetic field. PACS numbers: 73.23.Hk, 03.65.Vf, 75.45.+j, 75.50.Xx Single-molecule magnets (SMMs), such as Mn 12  [1, 2] and Fe 8  [3, 4], have become the focus of intense research in the last decade since experiments on bulk samplesdemonstrated the quantum tunneling of a single mag-netic moment on a macroscopic scale. These moleculesare characterized by a large total spin, a large magneticanisotropy barrier, and anisotropy terms which allow thespin to tunnel through the barrier. Transport throughSMM offers several unique features with potentially largeimpact in applications for magnetic devices based onSMM such as high-density magnetic storage as well asquantum computing applications [5]. Recently experi-ments have pointed out the importance of the interfer-ence between spin tunneling paths in molecules and itseffects on electron transport scenarios involving SMMs.For instance, measurements of the magnetization in bulkF 8  samples (see Ref. [6]) have observed oscillations in thetunnel splitting ∆ E  s, − s  between states  S  z  =  s  and − s  asa function of a transverse magnetic field at temperaturesbetween 0 . 05 K and 0 . 7 K. This effect can be explained bythe interference between Berry phases acquired by spintunneling paths of opposite windings using a coherentspin-state path integral approach [7, 8, 9, 10]. The ability to manufacture molecular structures withfixed magnetic properties has given rise to a field knownas molecular magnetism. To date, several experimentshave shown the possibility to work with an individualSMM preserving the magnetic properties [11], therebydemonstrating the Coulomb blockade effect at low tem-perature in a single SMM transistor geometry [11]. Thetheoretically predicted Kondo effect in SMMs [12, 13] has not been observed yet. A theoretical description of theobserved Coulomb blockade effect has recently been givenby means of   coherent   spin states in Ref. [14].In this letter we propose the Berry-phase blockade effectby coupling an individual SMM to spin-polarized leads.We analyze the transport properties of the system in theCoulomb blockade regime for the ground state of a SMMin the presence of a longitudinal and transverse mag-netic field. Since the decoherence time between degen-erate spin states can be as low as  T  2  = 10 − 9 s  [15, 16], we work with  incoherent   spin states. We show that inthe case of   incoherent   spin tunneling it is essential to useoppositely spin-polarized source and drain leads in orderto be able to observe variations of the stationary currentas a function of longitudinal or transverse magnetic field.In particular, the current can be suppressed due to theBerry-phase interference of the spin tunneling paths. Inthe case of fully polarized leads, complete current sup-pression coincides precisely with the topological zeros of the spin tunneling. Even in the case of partially polar-ized leads there are still fingerprints of the Berry-phaseblockade which lead to significant changes in the station-ary current through the SMM.In the following we present our calculations that are validfor any SMM in the Coulomb blockade regime coupled topolarized leads and in the presence of a longitudinal andtransverse magnetic field. We derive the (generalized)master equation for the low energy states and calculatethe stationary current through the SMM for the cases of unpolarized, fully, and partially polarized leads. We ap-ply our results to the newly synthesized SMM Ni 4 , whichhas a spin of   s  = 4 and a ground state tunnel splitting of ∆ E  s, − s  ≈ 0 . 01 K at zero magnetic field [18], correspond-ing to an angular frequency of   ω  = 10 9 s − 1 .Consider a SMM in the Coulomb blockade regime whichis tunnel-coupled to two polarized leads at the chemicalpotentials  µ l , where  l  =  L,R  denote the left and rightlead, respectively. The total Hamiltonian is given by H   =  H  l  + H s  +  H  m ,  (1)where  H  l  =   lkσ  ǫ lk c † lkσ c lkσ  represents the energy of theleads.  c † lkσ  creates an electron in lead  l  with orbital state k , spin  σ ,and energy  ǫ lk . The coupling of the leads tothe molecule is described by  H  m  =   lpkσ  t σlp c † lkσ d  pσ  + H.c. , where  t σlp  denotes the tunneling amplitude and  d †  pσ creates an electron on the molecule in orbital state  p . Theterm  H spin  is typically given by the spin Hamiltonian of a SMM in an external transverse magnetic field  H  ⊥  anda longitudinal magnetic field  H  z , i.e. H s  =  − A q S  2 q,z  +  B q 2  S  2 q, +  +  S  2 q, −  +  B 4 ,q 3  S  4 q, +  +  S  4 q, −   2+ gµ B H  z S  q,z  + 12( h ∗⊥ S  q, +  +  h ⊥ S  q, − ) ,  (2)where the easy axis is taken along  z ,  S  q, ±  =  S  q,x ± iS  q,y ,and the integer index  q   denotes the charging state of theSMM, where  q   =  − 1 adds one electron to the moleculeand  q   = 0 when the molecule is neutral. The transversemagnetic field  H  ⊥  =  H  x  +  iH  y  =  | H  ⊥ | e iϕ lies in the xy  plane. We use the abbreviation  h ⊥  =  gµ B H  ⊥ . Inthis Hamiltonian, the dominant longitudinal anisotropyterm creates a ladder structure in the molecule spectrumwhere the eigenstates  |± m q   of   S  z  are degenerate. Theweaker transverse anisotropy terms couple these states.The coupling parameters depend on the charging stateof the molecule. For example, it is known that Mn 12 changes its easy-axis anisotropy constant (and its totalspin) from  A 0  = 56  µ eV ( S  0  = 10) to  A − 1  = 43  µ eV( S  − 1  = 19 / 2) and  A − 2  = 32  µ eV ( S  − 2  = 10) when singlyand doubly charged, respectively [19]. Experiments withNi 4  show that  B 4 ,q =0  =  − 0 . 003 K, i.e.  B 4 , 0  is negative[18]. In this case, in order to see the Berry phase oscilla-tion, a magnetic field  H  ⊥  must be applied in the  xy  plane[10] along specific angles  ϕ ( B q ,B 4 ,q ). It is also possibleto tune the tunnel splitting by means of the longitudinalmagnetic field  H  z .For weak coupling between the leads and the SMM weuse the standard formalism suitable to describe a system(SMM) coupled to a reservoir (polarized leads)[20]. Themaster equation describing the electronic spin states of the SMM is given in Born and Markov approximation by˙ ρ m,n  =  i   [ ρ,H  ] m,n  +  δ  m,n  l  = m ρ n W  m,l − γ  m,n ρ m,n ,  (3)where  γ  m,n  =  12  l ( W  l,n  + W  l,m ) + 1 /T  2  is the total de-coherence rate which contains the spin decoherence time T  2  due to e.g. nuclear spins and the rates  W  m,n  of tran-sition between the states of the SMM. Figs. (1) and (2) show the ground states  s  and  − s  of a SMM placed be-tween unpolarized and polarized leads, respectively.  w ( l ) ↓↑ represents the spin-dependent transition rate from the l  =  L,R  lead to the SMM and are defined in Fermi’sgolden rule approximation by  w ( l ) ↓  = 2 π D  ν  ( l ) ↓  | t ( l ) ↓  | 2 /    and w ( l ) ↑  = 2 π D  ν  ( l ) ↑  | t ( l ) ↑  | 2 /   , respectively, where  D   is the den-sity of states and  ν  ( l ) ↑  and  ν  ( l ) ↓  are fractions of the num-ber of spins polarized up and down of lead  l  such that ν  ( l ) ↓  +  ν  ( l ) ↑  = 1.  t ( l ) ↑  and  t ( l ) ↓  are the tunneling amplitudesof lead  l , respectively. Typical values for the tunnelingrate of the electron range from around  w  = 10 6 s − 1 to w  = 10 10 s − 1 (see Refs. [11, 21]). In order to see coherent spin tunneling, the de-coherence time must be increased for example bydeuteration[22] such that 1 /T  2  ≪  ∆ E  s, − s /    and at thesame time the contact to the leads must be so weak that W  m,n  ≪ ∆ E  s, − s /   . Another possibility is to increase thetransverse magnetic field  | H  ⊥ |  beyond the Berry-phaseoscillations. In this case unpolarized leads can be usedto measure the tunnel splitting between the coherent spinstates ( | s ′  + |− s ′  ) / √  2 and ( | s ′ −|− s ′  ) / √  2 by varyingthe gate or bias voltage. However, only partially or fullypolarized leads allow us to probe the incoherent tunnel-ing rate Γ s, − s  between the ground states  s  and  − s  for q   = 0 and also between  s ′ and  − s ′ for  q   =  − 1. As weprove below, both incoherent tunneling rates Γ s, − s  andΓ s ′ , − s ′  contribute to the total polarized current throughthe SMM. The sequential tunneling rates for absorptionof an electron in Eq. (3) for ground states with spin  s and  s ′ and energy differences ∆ ± s ′ , ± s  in the case of lowtemperatures are given by W  s ′ ,s  =  l  W  ( l ) s ′ ,s  , W  ( l ) s ′ ,s  =  w ( l ) ↓  f  l (∆ s ′ ,s ) ,W  − s ′ , − s  =  l  W  ( l ) − s ′ , − s  ,W  ( l ) − s ′ , − s  =  w ( l ) ↑  f  l (∆ − s ′ , − s ) , (4)and the tunneling rates for the emission of an electronare given by W  s,s ′  =   l  W  ( l ) s,s ′  , W  ( l ) s,s ′  =  w ( l ) ↓  [1 − f  l (∆ s,s ′ )] ,W  − s, − s ′  =   l  W  ( l ) − s, − s ′  ,W  ( l ) − s, − s ′  =  w ( l ) ↑  [1 − f  l (∆ − s, − s ′ )] , (5)where  f  l (∆ s ′ ,s ) = [1+ e (∆ s ′ ,s − µ l ) /kT  ] − 1 is the Fermi func-tion. The diagonal elements of (3) yield˙ ρ s  =  i   [ ρ,H  ] s,s  +  n  = s ρ n W  s,n − ρ s  n  = s W  n,s ,  (6)and the off-diagonal elements of (3) yield˙ ρ s,s ′  =  i   [ ρ,H  ] s,s ′  − γ  s,s ′ ρ s,s ′ .  (7)Since we are interested in the long time behavior  t  ≫ 1 /γ  m,n , we can set ˙ ρ s,s ′  = 0 in equation (7) to obtain thefollowing coupled differential equations for the diagonalelements of the density matrix˙ ρ s  =  ∆ E  s, − s 2   22 γ  s, − s ( H  s − H  − s ) 2 /  2 + γ  2 s, − s ( ρ − s − ρ s )+ W  s,s ′ ρ s ′  − W  s ′ ,s ρ s , (8)˙ ρ − s  =  ∆ E  s, − s 2   22 γ  s, − s ( H  − s − H  s ) 2 /  2 + γ  2 s, − s ( ρ s − ρ − s )+ W  − s, − s ′ ρ − s ′  − W  − s ′ , − s ρ − s . (9)The other two differential equations are obtained by justreplacing  s ↔ s ′ in the above equations. Solving the setof differential equations for  ρ s ,  ρ − s ,  ρ s ′  and  ρ − s ′  in thestationary case ( t ≫ 1 /W  m,n ) we obtain ρ s  = ( W  s,s ′ ( W  − s ′ , − s  + Γ s, − s )Γ s ′ , − s ′ + W  − s, − s ′ ( W  s ′ ,s  + Γ s ′ , − s ′ )Γ s, − s ) /η,ρ − s  = ( W  s,s ′ ( W  − s, − s ′  + Γ s ′ , − s ′ )Γ s, − s + W  − s, − s ′ ( W  s ′ ,s  + Γ s, − s )Γ s ′ , − s ′ ) /η,ρ s ′  = ( W  s ′ ,s ( W  − s, − s ′  + Γ s ′ , − s ′ )Γ s, − s + W  − s ′ , − s ( W  s,s ′  + Γ s, − s )Γ s ′ , − s ′ ) /η,ρ − s ′  = ( W  s ′ ,s ( W  − s ′ , − s  + Γ s, − s )Γ s ′ , − s ′ + W  − s ′ , − s ( W  s,s ′  + Γ s ′ , − s ′ )Γ s, − s ) /η, (10)  3where  η  is a normalization factor given by η  = (Γ 4 , − 4  + Γ 7 / 2 , − 7 / 2 )( W  − 4 , − 7 / 2 W  7 / 2 , 4 + W  4 , 7 / 2 W  − 7 / 2 , − 4 ) + 2[Γ 4 , − 4 W  − 4 , − 7 / 2 W  4 , 7 / 2 +Γ 7 / 2 , − 7 / 2 W  − 7 / 2 , − 4 W  7 / 2 , 4  + Γ 4 , − 4 Γ 7 / 2 , − 7 / 2 ( W  7 / 2 , 4  +  W  − 7 / 2 , − 4  +  W  4 , 7 / 2  +  W  − 4 , − 7 / 2 )] , (11)such that   n  ρ n  = 1. The incoherent tunneling rate isΓ s, − s  =  ∆ E  s, − s 2    2 2 γ  s, − s ( H  s − H  − s ) 2 /   2 +  γ  2 s, − s ,  (12)which is a Lorentzian as a function of the longitudinalZeeman splitting  H  s − H  − s  =  gµ B H  z [ s − ( − s )]. We nowproceed to define the current through the SMM in termsof the density matrix elements. In the case of Ni 4  wehave  s  = 4 and  s ′ = 7 / 2, therefore the current reads I   =  e  W  4 , 7 / 2 ρ 7 / 2  +  W  − 4 , − 7 / 2 ρ − 7 / 2  .  (13)Taking into consideration the asymmetry of the leads ,i.e. w ( L ) ↓↑   =  w ( R ) ↓↑  , and restricting ourselves to the case of un-polarized leads, i.e.  ν  ( L ) ↑  =  ν  ( L ) ↓  =  ν  ( R ) ↓  =  ν  ( R ) ↑  = 1 / 2, weobtain the following conditions for the transition rates W  7 / 2 , 4  =  W  − 7 / 2 , − 4 , W  4 , 7 / 2  =  W  − 4 , − 7 / 2 .  (14)Substituting the values of   ρ 7 / 2  and  ρ − 7 / 2  into Eq. (13)and using Eq. (14) we obtain eI  unp = 1 W  7 / 2 , 4 + 1 W  − 4 , − 7 / 2 ,  (15)which does not depend on the tunnel splitting energy of the SMM. Thus it is impossible to observe Berry-phaseoscillations for the case of unpolarized leads and inco-herent spin states. Eq. (15) can be interpreted as tworesistances in series [17], where the only transitions thatcontribute to the current through the SMM are 4 ↔ 7 / 2and  − 4 ↔− 7 / 2 (see Fig. 1).In the case of leads that are fully polarized in oppositedirections, i.e.  ν  ( L ) ↑  =  ν  ( R ) ↓  = 1 or  ν  ( L ) ↓  =  ν  ( R ) ↑  = 1, weget the following conditions for the transition rates: W  − 4 , − 7 / 2  =  W  7 / 2 , 4  = 0 ,W  4 , 7 / 2  =  W  − 7 / 2 , − 4  = 0 .  (16)Choosing the case  ν  ( L ) ↑  =  ν  ( R ) ↓  = 1 and using Eq. (16)we can then substitute the values of   ρ 7 / 2  and  ρ − 7 / 2  intoEq. (13) to obtain eI   p = 2 W  − 7 / 2 , − 4 + 1Γ 4 , − 4 + 2 W  4 , 7 / 2 + 1Γ 7 / 2 , − 7 / 2 ,  (17)which reflects the fact that the current through the SMMdepends on the tunnel splittings and can be interpretedas four resistances coupled in series in a loop (see Fig. 2).Notice the clockwise direction of the transition rates be-tween the different states  s ,  s ′ , − s ′ and  − s  of the SMM. LRL w  R w s’ -s’ E c s -s FIG. 1: Diagram for the transitions between the ground states4 ↔ 7 / 2 and − 4 ↔− 7 / 2 in the case of unpolarized leads. Thetransitions arise from the sequential tunneling of unpolarizedelectrons in and out of the SMM. E c  is the charging energy. -s LRL w  R w s’-s’s E c FIG. 2: Diagram for the transitions between the ground states4 → 7 / 2 →− 7 / 2 →− 4 in the case of fully polarized leads inopposite directions  ν  ( L ) ↑  =  ν  ( R ) ↓  = 1. If we chose to work with fully polarized leads of the form ν  ( L ) ↓  =  ν  ( R ) ↑  = 1, then the direction of the transition ratesbetween states would be opposite, i.e. anticlockwise.Fig. (3) shows the current as a function of the trans-verse magnetic field  H  ⊥  for fully polarized leads. If thetunnel splitting ∆ E  4 , − 4  or ∆ E  7 / 2 , − 7 / 2  is topologicallyquenched, then Γ 4 , − 4  or Γ 7 / 2 , − 7 / 2  vanishes [see Eq. (12)],which leads to complete current suppression according toEq. (17). Since this current blockade is a consequenceof the topologically quenched tunnel splitting, we call itBerry-phase blockade. Note that the current can also besuppressed by applying the longitudinal magnetic field H  z  which follow immediately from Eqs. (12) and (17). If we consider now partially polarized leads (i.e.  ν  ( R ) ↑  >ν  ( R ) ↓  , ν  ( L ) ↓  > ν  ( L ) ↑  ) and calculate the current through the  4 FIG. 3: The graph shows the logarithm base 10 of the sta-tionary current versus the transverse magnetic field  H  ⊥  for ν  ( R ) ↑  =  ν  ( L ) ↓  = 1,  ν  ( R ) ↓  =  ν  ( L ) ↑  = 0,  w ( L ) ↓  = 1 × 10 9 s − 1 and w ( R ) ↓  = 10 w ( L ) ↓  .The polarizations of the left and right lead aregiven by  P  R = − P  L =  ν  R ↑  − ν  R ↓  = 100%. At the zeros of thetunnel splitting ∆ E  s, − s  or ∆ E  s ′ , − s ′  the current is completelysuppressed. The scale varies from I p  = 0 . 1 nA to 1 fA. SMM, we obtain I   pp  =  e [Γ 4 , − 4 W  − 4 , − 7 / 2 W  4 , 7 / 2 ( W  − 7 / 4 , − 4  +  W  7 / 2 , 4 )+Γ 4 , − 4 Γ 7 / 2 , − 7 / 2 ( W  − 4 , − 7 / 2  +  W  4 , 7 / 2 )( W  − 7 / 4 , − 4  +  W  7 / 2 , 4 )+Γ 7 / 2 , − 7 / 2 W  − 7 / 4 , − 4 W  7 / 2 , 4 ( W  − 4 , − 7 / 2  +  W  4 , 7 / 2 )] /η. (18)Fig. 4 shows the current as a function of the transversemagnetic field  H  ⊥  for partially polarized leads. TheBerry-phase blockade is still visible in the stationary cur-rent even at a spin polarization of 60 %, which makes itexperimentally accessible since recent experiments haveachieved near 100 % spin polarization [23].In summary, we have shown the Berry-phase blockade fora SMM placed between polarized leads. This behavior isdue to Berry-phase interference of the SMM spin betweendifferent tunneling paths. We have shown that in the caseof incoherent spin states it is essential to use polarizedleads in order to observe the Berry-phase blockade. Acknowledgment  . We thank E. del Barco, S. Khon-daker and E. Mucciolo for useful discussions. ∗ Electronic address: mleuenbe@mail.ucf.edu[1] C. Paulsen  et al. , J. Magn. Magn. Mater.  140-144 , 379(1995); J. R. Friedman  et al. , Phys. Rev. Lett.  76 , 3830(1996); L. Thomas  et al. , Nature (London)  383 , 145(1996).FIG. 4: The graph shows the logarithm of the stationarycurrent versus the transverse magnetic field  H  ⊥  for  ν  ( R ) ↑  = ν  ( L ) ↓  = 0 . 8,  ν  ( R ) ↓  =  ν  ( L ) ↑  = 0 . 2,  w ( L ) ↓  = 1 × 10 9 s − 1 , and  w ( R ) ↓  =10 w ( L ) ↓  . The polarizations of the left and right lead are givenby  P  R =  − P  L =  ν  R ↑  − ν  R ↓  = 60%. In this case at the zerosof the tunnel splitting ∆ E  s, − s  and ∆ E  s ′ , − s ′  the current issuppressed approximately by a factor of 3.[2] E. del Barco  et al. , J. Low Temp. Phys.  140 , 119 (2005).[3] C. Sangregorio  et al. , Phys. Rev. Lett.  78 , 4645 (1997).[4] W. Wernsdorfer and R. Sessoli, Science  284 , 133 (1999).[5] E. Chudnovsky and L. Gunther, Phys. Rev. Lett.  60 , 661(1988); J. Tejada  et al. , Nanotechnology  12 , 181 (2001);M. N. Leuenberger and D. Loss, Nature  410 , 789 (2001).[6] W. Wernsdorfer  et al. , Europhys. Lett.  50 , 552 (2000);[7] D. Loss, D. P. DiVincenzo, and G. Grinstein, Phys. Rev.Lett.  69 , 3232 (1992).[8] J. von Delft and C. L. Henley, Phys. Rev. Lett.  69 , 3236(1992).[9] A. Garg, Europhys. Lett.  22 , 205 (1993).[10] M. N. Leuenberger and D. Loss, Phys. Rev. B  63 , 054414(2001).[11] H. B. Heersche  et al. , Phys. Rev. Lett.  96 , 206801 (2006);M.-H. Jo  et al. , Nano Lett.  6 , 2014 (2006).[12] C. Romeike  et al. , Phys. Rev. Lett.  96 , 196601 (2006).[13] M. N. Leuenberger and E. R. Mucciolo, Phys. Rev. Lett. 97 , 126601 (2006).[14] C. Romeike, M. R. Wegewijs, and H. Schoeller,Phys. Rev. Lett.  96 , 196805 (2006).[15] E. del Barco  et al. , Polyhedron  24 , 2695 (2005).[16] S. Hill  et al. , Science  302 , 1015 (2003).[17] M. N. Leuenberger and D. Loss, Phys. Rev. B  61 , 1286(2000).[18] Sieber  et al. , Inorg. Chem.  44 , 4315 (2005).[19] R. Basler  et al. , Inorg. Chem.  44 , 649 (2005);N. E. Chakov  et al. ,  ibid.  44 , 5304 (2005).[20] K. Blum,  Density Matrix Theory and Applications  (Plenum, New York, 1996), Chap. 8.[21] J.Park,  PhD. Thesis  , (1996).[22] W. Wernsdorfer  et al. , Phys. Rev. Lett.  84 , 2965 (2000).[23] Z. H. Xiong  et al. , Nature (London)  427 , 821 (2004).
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