Supplemental Information

Transcription

1 Suppementa Information A Singe-eve Tunne Mode to Account for Eectrica Transport through Singe Moecue- Sef-Assembed Monoayer-based Junctions by A. R. Garrigues,. Yuan,. Wang, E. R. Muccioo, D. Thompson, E. de Barco, C. A. Nijhuis 1

2 2 Section 1 1 The rate equation formuation et M be the number of spin-spit orbitas in the moecue. The number of possibe eectronic configurations is equa to n=2 M i.e., each orbita can be empty or occupied. Each configuration can be described by a set of occupation numbers c α i, with α=1,,n i=1,,m, where c α i=0or1, 1 depending on whether the i-th orbita is occupied or not. Ca P α the probabiity of the α configuration, with P α 0 n α=1 P α =1. Under certain approximations, which negect memory effects discard correations between the eectrodes the moecue, the rate equation governing the change in the configuration probabiities over time can be expressed as 1,2,3] dp α dt = P α Γ α β + Γ β α P β, 2 βα βα where Γ α β is the rate of the α β transition. Equation 2 is the we-known Paui master equation 4]. We note here that there are severa ways to derive this equation from fundamenta theories, such as nonequiibrium quantum statistica mechanics 5,6,7], as we as from a semicassica Botzmann kinetic equation 1]. Here, rather than repeating those stard derivations, we focus on the use of the rate equation to compute transport proeperties in the context of the experiments reported in the main text. The main goa is to obtain expressions for the charge current across the moecue in different asymptotic regimes. We can express Eq. 2 in matrix form, where Λ αβ = dp α dt = β Λ αβ P β, 3 { α Γ α, ifα=β. 4 Γ β α, ifα β Notice that, consistent with the normaization condition, we have impying that n α=1 dp α dt n = n P α Γ α β + Γ β α P β α=1 βα α=1 βα 5 = n P α Γ α β + n P β Γ β α α=1 βα β=1 α β 6 = 0, 7 n Λ αβ =0, 8 α=1 as it can be easiy verified. On the other h, n Λ αβ = Γ α + Γ β α 9 β=1 α βα = Γ β α Γ α β. 10 βα

3 Transition rates 3 Thus, if Γ β α =Γ α β for a transitions, then the r.h.s. of this equation is equa to zero. This means that a trivia stationary soution exits where P α =1/n for a α=1,,n. However, when Γ β α Γ α β, other nontrivia stationary soution exist as we. 2 Stationary soutions In order to obtain a stationary soution to the rate equations set for a α=1,,n. This impies dp α dt =0 11 Λ αβ P β =0. 12 β Thus, to find the set of stationary probabiities {P α }, one needs to find the right eigenvector corresponding to the zero eigenvaue of the matrix Λ. 3 Transition rates et E α be the tota energy of the moecue N α be the tota number of eectrons in the α configuration. When an eectron hops from one of the eads into the moecue, energy conservation requires ε+e α =E β, 13 where αβ is the moecue s configuration beforeafter the hopping ε is the energy of the eectronic state in the ead. For this transition to take pace, the state with energy ε in the ead must have a finite occupation number, namey f >0, where µ is the ead s chemica potentia =R, T is the ε µ temperature. Here, fx denotes the Fermi-Dirac distribution, fx= 1 e x In the opposite case, when an eectron hops from the moecue into the ead, we have E α =ε+e β. 15 <1. Now, the occupation number of the state with energy ε must be such that f ε µ Ca R the eve widths due to the couping to the right eft eads, respectivey. We can spit the transition rate into two contributions, where 1,2,3] Γ α β / f = / 1 0, otherwise Eβ E α µ k BT Eα E β µ R Γ α β =Γ α β +Γ α β, 16, ifn β =N α +1d H c β,c α =1 ], ifn β =N α 1d H c β,c α =1,=R,, 17 where d H c β, c α is the Hamming distance between the binary sets c β c α. Namey, ony transitions where the number of eectrons in the moecue changes by one are aowed. Notice that the bias votage is equa to V =µ µ R /e, where e denotes the eectron charge. Equation 17 can be derived in a number of ways, with the most stard being Fermi s Goden Rue or time-dependent perturbation theory on the eve widths R.

4 4 Section 6 The rate equation approach is vaid when R, since it assumes a perfecty sharp energy ever in the moecue, when the tunneing through the moecue is sequentiay incoherent. Beow, we discuss the vaidity of the rate equation in more detai. Impicit in Eq. 17 is the assumption that the moecue s energy eves are sharp, such that R, are much smaer than other energy scaes of the probem, such as, ev, the separation of energy eves in the moecue. 4 Current The current coming from the eft ead is equa to where N α β =N β N α. I = e N α β Γ α β P α, 18 α,β 5 Tota energy The tota energy in the moecue can be broken down as foows constant charging energy mode: M E α = 1 2 N αn α 1E c ev g N α + c α iε i, 19 i=1 where E c is the charging energy, V g is the gate votage, {ε i } i=1,,m are the energies of the orbitas. 6 Singe-eve case et us appy this formuation compute the stationary current of a moecue with a singe orbita spiness case, in which case M =1 n=2. P 01 corresponds to the probabiity of the emptyfied state. The stationary probem is defined by the matrix Γ0 1 Γ Λ= 1 0, 20 where with Γ 1 0 R = + 21 R Γ 1 0 =Γ 1 0 +Γ 1 0, 22 = / f 0 µ Γ 1 0 = / 1 0 µ The eigenvector of the Λ matrix with zero eigenvaue corresponds to,=,r, 23 ],=,R. 24 P 0 = P 1 = Γ 1 0 Γ = Γ Γ =. 26

5 Singe-eve case 5 The current coming from the eft ead is equa to I = ep 0 P 1 Γ = e Γ 1 0 Γ R Γ = e R Γ R = e = e { R 1 R f R Γ E10 µ R E10 µ where E 10 =E 1 E 0 = ev g +ε 1. ] f E10 µ E10 µ R E10 µ R 1 ]} E10 µ 30 ], Finite eve width When the broadening of the energy eve is not negigibe, we have to modify the cacuations to account for the uncertainty in ǫ 1. et be the tota eve width D 1 ε the density of state profie associated to the singe-eve configuration; for instance, consider the orentzian profie D 1 ε= 1 π /2 ε ε 1 2 +/2 2, 32 with dεd 1 ε=1. The modified expressions for the transition rates are = / dεd 1 εf 0 µ,=,r, 33 ] Γ 1 0 = / dεd 1 ǫ 1 0 µ,=,r, 34 where Notice that Therefore, R E 1 E 0 = ev g +ε. 35 R +Γ 1 0 = R / dεd 1 ε= R / 36 +Γ 1 0 = / dεd 1 ε= /. 37 +Γ 1 0 = /, 38 1 P 0 = dεd 1 ε ] 1 0 µ 39 1 ] ε µ = dǫεd 1 ε+ev g 1 40 P 1 = = 1 dεd 1 ε f 1 dεd 1 ε+ev g f 0 µ ε µ

6 6 Section 6 Going back to the expression defining the current through the eft ead, we find I = ep 0 P 1 Γ = e 1 Γ 1 0 Γ R Γ = e 1 R Γ R = e 1 R Γ R Γ 1 0 R + Γ 1 0 ] 45 R Γ 1 0 ] 46 = e { R dεd 1 ε dε D 1 ε evg +ε µ 1 f R ]} evg +ε µ f R evg +ε 1 µ ] f = e { } R dεd 1 ε dε D 1 ε evg +ε f µ evg +ε µ R = e = e = e h R dεd 1 ε f R dεd 1 ε+ev g f R dε evg +ε µ ε µ ε ε 1 +ev g 2 +/2 2 f ] evg +ε µ R ] ε µr ε µ ε µr evg +ε µ ]. 51 This expression generaizes the previous one, Eq. 31, to incude a finite eve width. It is straightforward to check that Eq. 51 recoves Eq. 31 when 0. It is natura to assume that the tota eve width can be broken into three components, = + 0, 52 where 0 represents the broadening caused by effects other than the eakage of charge through the eads. 6.2 Adding spin To add spin, we spit the configuration where the moecue eve is occupied into two,, resuting in a tota of three configurations: i=0,, we forbid doube occupancy by assuming that the charging energy E c is a very arge energy scae, namey, E c,ev, ε 1. et us assume that the moecuar eve is spin degenerate. Then, the tota current through the eft ead is given by the expression The rate equations are I = ep 0 Γ 0 +Γ 0 P Γ 0 P Γ 0 ]. 53 dp 0 dt dp dt dp dt = P 0 Γ 0 +Γ 0 +P Γ 0 +P Γ 0, 54 = P Γ 0 +P 0 Γ 0, 55 = P Γ 0 +P 0 Γ 0. 56

7 Singe-eve case 7 Soving for the steady state yieds P 0 = Γ 0 Γ 0 Γ 0 Γ 0 +Γ 0 Γ 0 +Γ 0 Γ 0, 57 P = P = Γ 0 Γ 0 Γ 0 Γ 0 +Γ 0 Γ 0 +Γ 0 Γ 0, 58 Γ 0 Γ 0 Γ 0 Γ 0 +Γ 0 Γ 0 +Γ 0 Γ Assuming spin degeneracy in the eads, we find Γ 0 =Γ 0 = / dεd 1 εf 0 µ Γ 0 =Γ 0 = / dεd 1 ε 1 0 µ Therefore,,=,R, 60 ] Γ 1 0,=,R. 61 Γ 0 =Γ 0 = dεd 1 ε R f R ε+ f ε] 62 Γ 0 =Γ 0 = dεd 1 ε R f R ε f ε]/ 63 = / dεd 1 ε R f R ε+ f ε]/ 64 = / Γ 1 0, 65 where we introduced f ε=f 0 µ. 66 Notice that rates probabiities do not depend on spin. Thus, we can recast the probem in terms of P 0 P 1 =P +P. Then, we find Γ P 0 = Γ Γ P 1 = Γ Pugging them into the expression for the current, we get I = e2p 0 P 1 Γ = 2e Γ 1 0 Γ Γ where = 2e Γ R 1 0 = 2 e h = Γ R 0 1 Γ 1 0 / + 71 R + dε ε ε 1 +ev g 2 +/2 2 f dε ε ε 1 +ev g 2 +/2 2 R f ε µ ε µ + f ε µr ], 72 ] ε µr /. 73

8 8 Section 7 Notice that because of the term in the denominator of the prefactor in Eq. 72, the expression for the current in the presence of spin is not exacty equa to twice that for the spiness case. However, if we are ony interested in inear response, we can set µ = µ R = µ in Eq. 73, in which case we obtain /, provided that ε 1 ev g <µ namey, when the energy eve is brought beow the Fermi energy in the eads. Then, the factor of 2 is approximatey canceed we recover the expression for the spiness current. The current for the spinfu case is ony exacty equa to twice that for the spiness case when the charging energy in the moecue is zero non-interacting imit, in which case conductance through the moecue is spin degenerate. 7 Exact soution of the singe-eve case spiness It is possibe to sove exacty the fuy coherent singe-eve case by using the Kedysh non-equiibrium technique 7], or even scattering theory, since no many-body interactions are present 8]. The resut is the foowing: the probabiity of the eve to be occupied is equa to P 1 = dε 2π f ε µ ε ε 1 +ev g 2 +/2 2, 74 where = R + absence of any eve broadening other than eakage of charge through the eads. The probabiity of the empty eve configuration is P 0 =1 P 1. The expressions for the probabiities are identica to those obtained with the rate equations after the broadening of the energy eve is incorporated. The fact that the coherent incoherent formuations yied the same resuts for the probabiities is not surprising. For singe channe eads a singe eve in the moecue, interference pays no roe since there is ony one conduction path. When the moecue has mutipe independent paths for eectrons to hop in out, then the coherent incoherent predictions depart, since interference between paths can resut in enhancement or depetion of certain configuration occupations. An expression for the current was derived by Jauho, Wingreen, Meir 9] using the Kedysh Green s function technique. Their resut is I = e ] dε R ε µ ε µr h ε ε 1 +ev g 2 +/2 2 f. 75 Contrary to our previous derivation using rate equations, this expression fuy takes into account coherence. Yet, Eq. 75 Eq. 51 are identica, provided that we set = namey, no eve broadening other than that due eakage through the eads. To some extend this shoud come as a surprise, as the coherent transport formuation contains incoherent, sequentia regime as a imit. Notice that for the spinfu case, one simpy need to insert a factor of 2 on the right-h-side of Eq. 75. Asymptotic imits for the current Notice that ε µ f ε µr = cosh evb sinh ε+ef k BT 2k BT + cosh evb, 76 2k BT where ev b =µ µ R E F =µ +µ R /2. Defining ε =ε ε 1 +ev g, we can then rewrite Eq. 75 as I = e evb R dε Dε sinh 2k BT ε cosh +ε 1 ev g+e F evb, + cosh k BT 2k BT where 77 Dε = 1 /2 π ε 2 +/

9 Exact soution of the singe-eve case spiness 9 It is easy to show that dε Dε =1. 79 Without oss of generaity, we can set ev g =E F. Thus, ε 1 becomes the position of the energy eve with respect the Fermi energy in the eads at zero bias. Then, I = e et us ook at some asymptotic imits. R evb sinh dε Dε 2 cosh ε +ε cosh evb e V b ε 1 : Weak broadening, finite bias, arge temperature. I e R evb dε Dε 1 2 ε cosh +ε 1 k BT e R evb e ε1/kbt. 82 The current in this case shows an activation behavior, with the activation energy being the offset between the energy eve in the moecue the Fermi energy in the eads. inear bias regime. On resonance, we find I e R evb. 83 ǫ 1 e V b : Weak broadening, intermediate temperature, arge bias, neary on resonance. I e R evb sinh dε Dε evb 1+cosh 2 e R. 85 The current is approximatey temperature bias independent non-inear bias regime. ǫ 1 e V b : Simiar to the previous case, more off-resonance. I e R e e Vb /2kBT dε Dε 1 2 cosh e ε +ε 1 k BT +e e Vb /2kBT /2 86 R. 87 The current is again approximatey temperature bias independent. e V b < ǫ 1 : Simiar to the previous case, but even more off-resonance. I e R e e Vb /2kBT dε Dε 1 2 cosh +e e Vb /2kBT /2 e ε +ε 1 k BT 88 R e e Vb /2 ε1 /kbt. 89 The current shows activation behavior is highy non-inear.,e V b, ǫ 1 : High-temperature regime. I e R evb dε Dε e R evb. 91 4

10 10 Section 9 The current decreases with the inverse of the temperature inear bias regime. : ow-temperature regime; strong broadening. ε 1+eV b/2 I e R dε ε Dε 92 1 ev b /2 2e ] R ε1 +ev arctan b /2 ε1 ev arctan b /2. 93 h /2 /2 Notice that Eq. 93 is the starting point of a we-known theoretica description of eectronic transport in soft moecuar eectronics 10,11]. The current is temperature independent becomes inear with the bias votage when e V b : 8 Concusions I e2 h 4 R V b. 94 Given that for singe-channe, singe-eve conductance both fuy coherent sequentiay incoheren approaches ead to the same expression for the current, we can concude that the most genera expression at ow bias is given by ε µ ε µr I = e h R dε ε ε 1 +ev g 2 +/2 2 f ], 95 where we aow the tota eve width to incude some broadening due to energy reaxation mechanisms other than eakage through the eads, namey, = References 1. D. V. Averin A. N. Korotkov, Zh. Eksp. Teor. Fiz. 97, Infuence of discrete energy spectrum on correated eectron-eectron tunneing via a mezoscopic sma meta granue, Sov. Phys. JETP 70, ]. 2. J. H. Davis, S. Hershfied, P. Hydgaard, J. W. Wikins, Current rate equations for resonant tunneing, Phys. Rev. B 47, S. A. Gurvitz Ya. S. Prager, Microscopic derivation of rate equations for quantum transport, Phys. Rev. B 53, H.-P. Breuer F. Petruccione, The Theory of Open Quantum Systems Oxford Press, New York, P. Kadanoff G. Baym, Quantum Statistica Mechanics W. A. Benjamin, Reading, MA, V. Kedysh, Zh. Eksp. Teor. Fiz Diagram technique for nonequiibrium processes, Sov. Phys. JETP 20, ]. 7. H. J. W. Haug A.-P. Jauho, Quantum Kinetics in Transport Optics of Semiconductors, 2nd edition Springer, J. Koch, F. von Oppen, A. V. Andreev, Theory of the Frank-Condon bockade regime, Phys. Rev. B 74, A.-P. Jauho, N. S. Wingreen, Y. Meir, Time-dependent transport in interacting noninteracting resonant-tunneing systems, Phys. Rev. B 50, I. Badea H. Koppe, Sources of negative differentia resistance in eectrica nanotransport, Phys. Rev. B 81, I. Badea, Ambipoar transition votage spectroscopy: Anaytica resuts experimenta agreement, Phys. Rev. B 85,