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the possibility of altering and controlling the spin - state of a single magnetic ion or of a small magnetic cluster with an external probe represents a unique opportunity towards the understanding and the exploitation of the magnetic interaction at the most microscopic level . possible areas of application for such ability may include spin - based quantum logic , where one necessitates to prepare , manipulate and read spin - qubits . it is then crucial to develop tools capable of addressing the single spin - limit . low - temperature scanning tunneling microscopy provides one of such tools . in general the method exploits a scanning tunneling microscope ( stm ) operated in spectroscopical mode , by which the inelastic electron tunneling spectroscopy ( iets ) at the spin - excitations of a given system is measured @xcite . this scheme is known as spin - flip iets ( sf - iets ) . the same stm can also be used to position and manipulate the magnetic atoms on a non - magnetic substrate @xcite , so that stm appears both as a fabrication and subtle characterization tool . transition metal magnetic atoms on insulating surfaces , in particular mn@xcite , co@xcite and fe@xcite , have been the focus of intensive research in the last few years . these have all partially filled @xmath0-shells , which are highly localized and responsible for the magnetic moment , and extended @xmath1-like electrons , which are responsible for the electron conduction . in general @xmath1 and @xmath0 electrons interact via exchange coupling so that the magnetic structure is coupled to the conducting electrons . the magnetic atoms are usually deposited on carefully prepared cun - decorated cu surfaces , where the typical electronic coupling is weak enough that the magnetism is preserved , but it is sufficiently strong to break the atomic central symmetry so that magnetic anisotropy develops . stm experiments are then conducted and the fingerprint of a magnetic excitation is a step in the differential conductance , @xmath2 , as a function of bias , @xmath3 ( @xmath4 is the stm current ) . these appear at the critical voltage necessary to open a new inelastic transport channel , i.e. at voltages corresponding to the given magnetic excitation energy . several methods aimed at modeling sf - iets have been recently developed . early theoretical work has focused on second order perturbation theory to describe the experimental conductance spectra of equilibrium spins by either using a master equation approach @xcite or a non - equilibrium green s function one @xcite . more recently this scheme has been extended to third order , which allows us to describe additional features in the @xmath5 line - shape that can not be accounted for at the second order level @xcite . these works have been very successful in describing conductance profiles , which appear symmetric with the external bias polarity , i.e. that they can not distinguish whether the current flows from the sample to the tip or in the opposite direction . however , recent experiments have shown that regardless on whether a non - spin - polarized @xcite or a spin - polarized @xcite tip is used the iets profiles exhibit an intrinsic asymmetry with respect to the applied voltage , i.e. @xmath6 .    in the case of a spin - polarized stm tip , where the tip density of states is spin split between majority ( spin up ) and minority ( spin down ) carriers , the asymmetry has been theoretically well explained @xcite . it has been shown that spin selection rules enforce a suppression of the inelastic scattering , which depends on the direction of the electrons flow . this results in a asymmetric conductance profile , where the magnitude of the asymmetry depends directly on the spin - polarization of the tip . it is also well understood that by driving spins out of equilibrium ( e.g. by decreasing the tip - sample distance ) the conductance line - shape changes @xcite . in this case we must assume that the tunneling electrons influence the spin - state of the atom as the time between inelastic events is small compared to the spin relaxation time . a tunneling electron can then encounter the local spin in an excited state far from the ground state . the non - equilibrium population of the various accessible spin - states then becomes bias - dependent and , for spin - polarized tips , this enhances the asymmetry of the @xmath5 line - shape . also in the case of a non - spin - polarized tip a bias asymmetry has been revealed experimentally @xcite . in particular this appears to be quite prominent for both single mn atoms and mn mono - atomic chains . this feature has been previously ascribed to a shift in the magnetic atom on - site energy , i.e. to an effect arising from the details of the density of states of the atom producing scattering . such a density of state effect produces a non - trivial slope in the conductance as a function of bias @xcite . the on - site energy shift however does not account for the asymmetry seen in the inelastic step heights , which also depends on bias . here we provide an alternative theoretical description , which allows us to better fit the experimentally found conductance line - shape .    in our previous works @xcite we have combined the non - equilibrium green s function ( negf ) @xcite formalism with a perturbative expansion of the electron - spin interaction in order to describe sf - iets spectra in a manner , which is fully amenable to an implementation within density functional theory ( dft)@xcite . the scheme essentially consists in constructing an electron - spin interaction self - energy , which describes the inelastic tunneling events . the interacting self - energy was previously expanded first up to second order  @xcite and then to the third order  @xcite , with this latter describing the logarithmic decays of @xmath5 at each conductance step . although both non - equilibrium effects and spin - polarized iets have been well described up to the second order by the master equation approach , in this work we extend our formalism to include the extra line - shape features that can be ascribed to the third order self - energy . we also propose that the real part of the interacting self - energy , which has been well studied in the case of electron - phonon interactions  @xcite , is a necessary addition to the negf formalism in order to account for asymmetric features in non - spin polarized systems . the layout of the paper is as follows . in the next section we extend our negf formalism to account for spin polarized leads . in the same section we also derive a second - order electron - spin self - energy , which includes a second order expansion of the spin propagator . this provides the means to study non - equilibrium effects that result from high current densities . furthermore , in the case of non - spin polarized tips we derive an expression for the real part of the scattering self - energy up to second order . all of the above is combined with the third order contribution to the electron propagator . then we move to the results . first we study the non - equilibrium effects arising from an increase in current density for the case of a mn dimer . then we recreate the spin - polarized experiments of loth et al . on single mn and fe atoms @xcite . finally we test how the inclusion of the real part of the scattering self - energy modifies the sf - iets for a non - polarized tip probing mn monomers and trimers . we consider here the same single - orbital tight - binding model used in our previous works  @xcite to describe a magnetic system ( s ) coupled to two non - interacting electrodes representing respectively the stm tip ( tip ) and the substrate ( sub ) . the scattering region containing the magnetic nanostructure consists of a one - dimensional chain of @xmath7 magnetic atoms . each of the @xmath8-th atoms carries a quantum mechanical spin @xmath9 and it is characterized by an on - site energy @xmath10 . we assume that the tip and the substrate can only couple to one atom at a time in the scattering region thus to broaden the electronic level @xmath10 through the interaction with the electrode by @xmath11 . the hamiltonian in the scattering region is then described by @xmath12 , where @xmath13 is the tight - binding electronic part , @xmath14 is the spin part and @xmath15 describes the electron - spin interaction . the various terms can be written explicitly as @xmath16\big\}\:,\\ \label{eq:3 } & { h}_\mathrm{e - sp}=j_\mathrm{sd}\sum_{\lambda\:\alpha,\alpha'}(c_{\lambda\alpha}^{\dagger}[\boldsymbol{\sigma}]_{{\alpha}{\alpha'}}c_{\lambda\alpha'})\cdot\mathbf{s}_\lambda\ : \\ \nonumber & + j_\mathrm{0}\sum_{\lambda\:\alpha}c_{\lambda\alpha}^{\dagger}c_{\lambda\alpha}\:,\end{aligned}\ ] ] where the electron ladder operator @xmath17 ( @xmath18 ) creates ( annihilates ) an electron at site @xmath8 with spin @xmath19 ( @xmath20 ) and on - site energy @xmath10 . we model the spin - spin interaction between the localized @xmath21 spins by a nearest neighbour heisenberg hamiltonian with coupling strength @xmath22 . furthermore we include interaction with an external magnetic field @xmath23 ( @xmath24 is the bohr magneton and @xmath25 is the gyromagnetic ratio ) and both uni - axial and transverse anisotropy of magnitude @xmath26 and @xmath27 respectively  @xcite . the electron - spin interaction hamiltonian is constructed within the @xmath1-@xmath0 model  @xcite where the transport electron , @xmath1 , are locally exchanged coupled to quantum spins , @xmath21 ( @xmath0 indicates that the local moments originating from the atomic @xmath0 shell ) . in equation ( [ eq:3 ] ) the interaction strength is @xmath28 and @xmath29 is a vector of pauli matrices . the second term in eq .  ( [ eq:3 ] ) represents the potential scattering elastic contribution to the @xmath1-@xmath0 interaction given by the exchange parameter @xmath30 ( note that this enters as a shift of the on - site potential of a given atom ) . the ratio , @xmath31 , is typically in the range 1 - 2 @xcite . this term was not included in our previous works @xcite as it only becomes important for spin - polarized electrodes .    in order to construct an electron - spin interacting self - energy we must first consider the keldysh  @xcite contour - ordered single - body green s functions ( propagators ) for both the electronic ( @xmath32 ) and the spin ( @xmath26 ) sub - systems in the electron - spin many - body ground state @xmath33 @xmath34_{\sigma\sigma'}=-i{\langle}|t_c\{c_{\sigma}(\tau)c^{\dagger}_{\sigma'}(\tau')\}|{\rangle}\:,\ ] ] @xmath35_{nm}=-i{\langle}|t_c\{d_{n}(\tau)d^{\dagger}_{m}(\tau')\}|{\rangle}\:.\end{aligned}\ ] ] these propagators describe a non - equilibrium system at zero - temperature . here @xmath36 is a quasi - particle creation operator defined by the relation @xmath37 where @xmath38 and @xmath39 are eigenstates of @xmath40 of energy @xmath41 . the matrix elements @xmath42 determine the transition rates from the initial state @xmath43 to the final state @xmath44 . the quasi - particle operators are assumed fermionic in nature@xcite . therefore the equilibrium population of a given state @xmath44 is given by @xmath45 where @xmath27 is the energy , @xmath46 the temperature and @xmath47 the boltzmann constant . in the non - interacting case ( @xmath48 ) the electronic system is not in equilibrium as the interaction with the electrodes establishes a steady state current . in contrast the spin system is in thermal equilibrium at the temperature @xmath46 . in this case the energy resolved lesser and greater green s functions take the form @xmath49_{\sigma\sigma'}=\frac{[\sigma^{\lessgtr}_{\mathrm{tip - s}}(e)]_{\sigma\sigma'}+[\sigma^{\lessgtr}_{\mathrm{sub - s}}(e)]_{\sigma\sigma'}}{(e-\varepsilon_0)^2+\gamma^2}\:,\\ \label{eq:72 } & [ d_0^{\lessgtr}(e)]_{mn}=\frac{[\pi^{\lessgtr}_{0}(e)]_{mn}+[\pi_0^{\lessgtr}(e)]_{mn}}{(e-\varepsilon_m)^2+(k_\mathrm{b}t)^2}\:.\end{aligned}\ ] ] in the electronic non - interacting green s function [ see eq . ( [ eq:71 ] ) ] the coupling of the sample to the tip and the substrate causes a broadening of the bare on - site level @xmath10 of magnitude @xmath50_{\sigma\sigma'}$ ] , where @xmath51 . the non - interacting self - energies take the form @xmath52_{\sigma\sigma'}=[1-f_{\eta}(e , v)][\gamma_{{\eta}-s}]_{\sigma\sigma'}$ ] and @xmath53_{\sigma\sigma'}=f_{\eta}(e , v)[\gamma_{{\eta}-s}]_{\sigma\sigma'}$ ] , where @xmath54 is the fermi function in each of the @xmath55-th leads at a bias @xmath3 . in contrast the local spin green s function of equation ( [ eq:72 ] ) describes a system , which is adiabatically coupled to a heat - bath of temperature @xmath46 . this provides a very weak broadening of the single spin states @xmath56 of magnitude @xmath57 . such a heat bath keeps the spin - system in equilibrium and in the non - interacting case the population then resides mostly in the ground state [ see eq . ( [ eq:7 ] ) ] . for a ground state population of the spin system @xmath58 we have @xmath59_{mn}=\delta_{mn}(1-p^0_m)k_\mathrm{b}t$ ] and @xmath60_{mn}=\delta_{mn}p^0_mk_\mathrm{b}t$ ] .      in order to evaluate the effects that the interaction has on the electronic motion we must calculate the electron - spin self - energy . here we take a perturbative approach and formally expand equation  ( [ eq:4 ] ) up to the @xmath43-th order in the interaction hamiltonian , @xmath61 , as @xmath62_{\sigma\sigma'}=\sum_n\frac{(-i)^{n+1}}{n!}\int\limits_c{d}\tau_1\dots\int\limits_c{d}\tau_n\ \times \nonumber \\ & \frac{{\langle}0|t_c\{{h}_\mathrm{e - sp}(\tau_1)\dots{h}_\mathrm{e - sp}(\tau_n)c_{\sigma}(\tau)c_{\sigma'}^{\dagger}(\tau')\}|0{\rangle}}{u(-\infty,-\infty)},\end{aligned}\ ] ] where @xmath63 is the time - evolution unitary operator and the time - averages are performed over the known non - interacting @xmath64 ground state @xmath65 . the time integration over @xmath66 is ordered on the contour @xmath67 going from @xmath68 to @xmath69 and then returning from @xmath69 to @xmath68 , since the ground state of the non - equilibrium system can only be defined at @xmath68  @xcite .    in the following we consider the tip to have a spin polarization @xmath55 ( @xmath70 ) . this is defined as the spin asymmetry in the electronic coupling between the tip and the sample . an such the spin resolved electronic broadening is given by @xmath71_{\uparrow\uparrow}=\frac{(1+\eta)}{2}\gamma_\mathrm{tip - s}$ ] and @xmath71_{\downarrow\downarrow}=\frac{(1-\eta)}{2}\gamma_\mathrm{tip - s}$ ] , where @xmath72 is the non - spin - polarized broadening . the substrate is assumed to remain non - magnetic . as a result @xmath73_{\uparrow\uparrow}\neq[g^{\lessgtr}_0(e)]_{\downarrow\downarrow}$ ] so that we now must retain the spin indexes when constructing the self - energy . the self - energy for the majority ( @xmath74 ) and minority ( @xmath75 ) spins writes respectively as @xmath76_{\uparrow\uparrow}^{(2)}=-j^2_\mathrm{sd}\sum_{mn}[g^{\lessgtr}_0(e\pm\omega_{mn})]_{\uparrow\uparrow } \times\nonumber \\ & \big(\delta_{nm}\chi p_ns_{mn}^z+ p_n(1-p_m)|s_{mn}^z|^2\big)\nonumber \\ & -j^2_\mathrm{sd}\sum_{mn}[g^{\lessgtr}_0(e\pm\omega_{mn})]_{\downarrow\downarrow}p_n(1-p_m)|s^+_{mn}|^2\end{aligned}\ ] ] and @xmath77_{\downarrow\downarrow}^{(2)}=-j^2_\mathrm{sd}\sum_{mn}[g^{\lessgtr}_0(e\pm\omega_{mn})]_{\downarrow\downarrow } \times\nonumber \\ & \big(-\delta_{nm}\chi p_ns_{mn}^z+p_n(1-p_m)|s_{mn}^z|^2\big)\nonumber \\ & -j^2_\mathrm{sd}\sum_{mn}[g^{\lessgtr}_0(e\pm\omega_{mn})]_{\uparrow\uparrow}p_n(1-p_m)|s^-_{mn}|^2\:.\end{aligned}\ ] ] the lesser ( greater ) self - energy describe an incoming ( outgoing ) electron that excites ( relaxes ) the spin system by @xmath78 , with a probability that depends on the occupation of the spin levels @xmath79 and @xmath80 and on the spin selection rules @xmath81 ( note @xmath82 and @xmath83 ) . the first term in both the equations ( [ eq:10a ] ) and ( [ eq:10b ] ) , proportional to @xmath84 , corresponds to the magnetoresistive elastic term of the @xmath1-@xmath0 hamiltonian of equation ( [ eq:3 ] ) . the remaining contributions are inelastic in nature and depend on the spin orientation of the electron transferred from the tip .      when is magnetic and the tunneling current tip carries a finite spin - polarization the spin system can be dragged out of equilibrium , in particular if the current density is intense . this means that the equilibrium conditions employed previously  @xcite , namely @xmath85 and @xmath86 is no longer valid . as a consequence we must now derive also an expression for the propagator and thus for the self - energy associated to the local spins . the derivation , up to second order in the electron - spin interaction is described in details in the appendix . in particular the total spin - self - energy also includes a zeroth - order contribution , which accounts for the non - interacting ( @xmath48 ) case . this is approximated by @xmath87_{kk}=p_k^0k_\mathrm{b}t$ ] and @xmath88_{kk}=(1-p_k^0)k_\mathrm{b}t$ ] where @xmath89 is the ground state population . therefore , in absence of inelastic scattering , the spin system will remain in thermal equilibrium with the heat bath and only the ground state will be occupied .    by combining the zeroth and second order contributions to self - energy we can write down a master equation , describing the non - equilibrium spin - population , in terms of the total self energy @xmath90@xcite @xmath91_{nm}[d_0^<(e)]_{mn } \nonumber \\ & -\pi^{<}(e)]_{nm}[d_0^>(e)]_{mn}\big\}\:.\end{aligned}\ ] ] after some rearrangement this can be written in more compact form as @xmath92\nonumber \\ & + ( p_n^0-p_n)k_\mathrm{b}t\:,\end{aligned}\ ] ] where the bias dependent transition rate from an initial state @xmath93 to a final state @xmath94 is calculated after evaluating the integral described in equation ( [ eq:14 ] ) of the appendix . this finally writes @xmath95_{\uparrow\uparrow}[\gamma_{\eta'}]_{\uparrow\uparrow}-[\gamma_{\eta}]_{\downarrow\downarrow}[\gamma_{\eta'}]_{\downarrow\downarrow}\big ) + \nonumber \\ &   + |s^z_{nl}|^2[\gamma_{\eta}]_{\uparrow\uparrow}[\gamma_{\eta'}]_{\uparrow\uparrow}+|s^z_{nl}|^2[\gamma_{\eta}]_{\downarrow\downarrow}[\gamma_{\eta'}]_{\downarrow\downarrow } + \nonumber \\ &   + |s^+_{nl}|^2[\gamma_{\eta}]_{\downarrow\downarrow}[\gamma_{\eta'}]_{\uparrow\uparrow}+|s^-_{nl}|^2[\gamma_{\eta}]_{\uparrow\uparrow}[\gamma_{\eta'}]_{\downarrow\downarrow}\big\}\:,\end{aligned}\ ] ] where @xmath96 and @xmath97 is the chemical potential in lead @xmath51 . note that @xmath98 is such that for @xmath99 the resulting transition rates @xmath100 are bias independent and do not contribute to the current . however , they do contribute to the spin relaxation time i.e. to the time taken by the localized spin system to relax back to its equilibrium state . such a relaxation time is reduced if the coupling between the sample and the leads is increased . furthermore , the smaller is the inelastic energy transition @xmath78 , the longer the spins will remain in the excited state before relaxing back to equilibrium . finally , we note that the above expression is based on the assumption that the on - site energy is large enough for the density of states of the spin system to remain constant in the small energy window of interest . therefore @xmath101 .    returning to equation ( [ eq:17 ] ) note that we are only interested in the steady state non - equilibrium population of the spin states at a given bias . therefore we can set @xmath102 and reduce eq . ( [ eq:17 ] ) to system of linear equations , which can be solved self - consistently . for an initial guess of the populations ( @xmath103 ) we can iterate eq . ( [ eq:17 ] ) , which define @xmath80 , with equation ( [ eq:14 ] ) of the appendix , which define the self - energy @xmath104 , until self - consistency is reached . we can then combine the resulting non - equilibrium population with the second order electronic self - energy calculated in eq . ( [ eq:4 ] ) to obtain the current . in order to provide an explanation to the inherent asymmetry that has been observed in most of the stm experiments on magnetic atoms using a non - magnetic tip we return to the expression for the full retarded self - energy . this is defined by the hilbert transform ( note we will only consider the non - spin polarized case for simplicity ) @xmath105 by using the expressions derived in section  [ spse ] for the 2@xmath106 order lesser and greater self - energies we can find an analytic expression for the real contribution to the retarded self energy @xmath107=2\rho j^2_\mathrm{sd}\sum_{i , m , n}|s^i_{mn}|^2p_n(1-p_m)\times \\ \nonumber &   \frac{1}{\gamma}\big\{2\pi\varepsilon_0+\sum_{\eta}\gamma_{\mathrm{\eta - s}}\:\mathrm{ln}\big[\frac{(e+\omega_{mn}-\mu_{\eta})^2+(k_bt)^2}{(e-\omega_{mn}-\mu_{\eta})^2+(k_bt)^2}\big]\big\}\:.\end{aligned}\ ] ] such a final expression is heavily dependent on the on - site energy @xmath10 but is also an odd function of the energy and the bias via its logarithmic dependence on the spin level eigenvalues with opposite polarity for @xmath108 and @xmath109 . we will show in the results section that this is at the origin of the conductance asymmetry found in experiments . we can finally unveil the effects that a non - equilibrium spin - population bares on the conductance profile of a magnetic nanostructure by taking the derivative of the current with respect to the bias voltage @xmath3 . the current , @xmath110 , flowing at the electrode @xmath111 can be written as @xmath112-[\sigma_{\eta}^{<}(e)g^{>}(e)]\}\:,\end{aligned}\ ] ] where @xmath113 are the full many - body lesser / greater electronic green s functions . these are finally defined as @xmath114=\frac{[\sigma^{\lessgtr}_{\mathrm{tip - s}}(e)]+[\sigma^{\lessgtr}_{\mathrm{sub - s}}(e)]+[\sigma^{\lessgtr}_{\mathrm{int}}(e)]}{(e-\varepsilon_0-\mathrm{re}[\sigma_{\mathrm{int}}(e)])^2+(\gamma-\mathrm{im}[\sigma_{\mathrm{int}}(e)])^2},\end{aligned}\ ] ] and @xmath115=[\sigma^{\lessgtr}_{\mathrm{int}}(e)]^{(2)}+[\sigma^{\lessgtr}_{\mathrm{int}}(e)]^{(3)}$ ] where the third order self - energies are calculated following ref . here for simplicity we take the expression for the third order contribution to @xmath116 obtained by neglecting any explicit spin - polarization . such approximation is justified by the fact that the effects due to spin - polarization are small at the third order and that in doing so we avoid a rather cumbersome formulation . we do however consider the combination of non - spin polarized 3rd order effects with 2nd order spin - polarized self energies to highlight subtle differences in the spectra . we start our analysis by first looking at the effects originating from driving the spin system out of equilibrium with an electronic current . this attempts at explaining the experiments reported in ref . [ ] , in which a stm tip ( non - magnetic ) is positioned above a mn dimer deposited onto a cun substrate . the conductance spectra are measured for different tip to sample distances . varying the stm tip height is equivalent to changing both the current density and the electronic coupling between the tip and the sample . non - equilibrium effects then appear as variations of the conductance profiles as a function of the stm tip height . many of the parameters needed by our model can be extrapolated from a similar experiment carried on over mn linear atomic chains deposited on cun  @xcite . the five unpaired electrons in the mn 3@xmath0 shell suggest a @xmath117 ground state , as confirmed both by experiments and theory  @xcite . the spin - spin exchange interaction between two mn atoms is antiferromagnetic and has an estimated value of @xmath118  mev . as a result the ground state of the dimer is a singlet ( total spin @xmath119 ) . the first excited state is a triplet with total spin @xmath120 and the energy splitting between the ground state and such first excited state is exactly @xmath22 . the next excited level is the quintuplet with total spin @xmath121 and it is separated from the first excited state by @xmath122 . this pattern continues throughout the spin manifold ( see figure  [ fig1 ] ) . the axial and transverse anisotropies are found to be @xmath123  mev and @xmath124  mev respectively . these cause the lifting of the spin multiplets degeneracy . the temperature is set at @xmath125  k. the value of @xmath28 is estimated from density functional theory ( dft ) to be of the order of 500  mev  @xcite , while @xmath126 is also found from dft to be approximately 100  mev . in contrast @xmath72 remains an adjustable parameter with the chosen values ranging from 0.125  mev to 200  mev . finally , in order to ensure a nearly constant density of states around the fermi energy ( @xmath127 ) we set the on - site energy of the atom under the stm tip to be @xmath128  ev . figure  [ fig2 ] shows the conductance spectra obtained by simply taking the numerical derivative of the current [ eq .  ( [ eq:23 ] ) ] with respect to the bias . we consider three different tip to sample distance , corresponding respectively to weak ( @xmath129  mev ) , intermediate ( @xmath130  mev ) and strong ( @xmath131  mev ) electronic coupling . the evolution of the conductance lineshape as a function of @xmath72 is a direct consequence of the spin system being driven out of equilibrium . for @xmath129  mev the stm tip is far enough from the sample to ensure that the spin system is always in its ground state between two subsequent electron tunneling events . therefore the only transition detected in the @xmath5 profile is that between the @xmath119 ground state and the first excited state with @xmath120 . this has an excitation energy equal to @xmath22 and it does manifest itself as a conductance step at a voltage @xmath132 , with @xmath133 being the electron charge .    as the tip is brought closer to the sample ( @xmath130  mev ) the first excited triplet level ( @xmath120 ) starts to populate . now an incoming electron with sufficiently large energy ( @xmath122 ) can induce a second transition from from the first to the second excited state . note that the @xmath121 state is not accessible with a single electron tunneling process from the ground state and it can be reached only if the spin system does not have enough time between tunneling events to relax back to the ground state . for this case the transition appears as a reduction of the conductance at the critical voltage @xmath134 . the same spectroscopical feature is further enhanced at an even larger current density ( @xmath135  200  mev ) , when a third conductance step appears at @xmath136 . this is associated to a transition from the @xmath121 to the @xmath137 spin state and it becomes possible only if the occupation of the @xmath121 level is not zero , i.e. if the system is driven to this highly excited state . these results are in almost perfect quantitative agreement with the experimental data ( see fig .  2 of ref .  [ ] ) . the evolution of the population of the various spin states ( up to @xmath121 ) as a function of bias is presented in figure  [ fig3 ] . this is calculated in the case of strong tip to sample electronic coupling @xmath135  200  mev . in the figure one can note the strong spin - pumping from the ground state into both the first and the second excited state . the excitation to the @xmath138 excited state occurs at approximately 18  mev but is too weak to be observed on this scale . we now move on to consider the situation where the tip is magnetic , i.e. when the injected current is spin - polarized . again we use as guide the experimental work of loth _ et al._@xcite . the stm tip is now spin - polarized by placing an additional mn atom at its apex , while also applying a strong magnetic field perpendicular to the substrate of 3  t. in this case the spectrum is collected from a single mn or fe ion on the surface ( not from a dimer ) . in the case of mn , the atom exhibits a weak anisotropy on cun ( see previous section ) and the strong magnetic field effectively produces a zeeman split of the six levels of the @xmath117 mn spin manifold . the direction of the magnetic field in these experiments is chosen so that the ground state of the mn spin corresponds to the magnetic quantum number @xmath139 . since the same magnetic field is applied to the mn atom on the tip s apex , the tip and atom are both spin - polarized and collinear . figure  [ fig4 ] shows the calculated spectra for the system described above . in particular we consider magnetic field strength of 3  t and either weak ( @xmath140  mev ) , intermediate ( @xmath141  mev ) or strong ( @xmath142  mev ) tip to sample couplings . the on - site energy is fixed at @xmath143ev and the value of @xmath144mev is infered from the work of lucignano  _ _ et al.__@xcite . the tip spin - polarization constant and the inelastic ratio that best fit the experimental data are respectively @xmath145 and @xmath146 . in the weak coupling regime ( when the local spin remains always close to equilibrium ) the local spin resides almost entirely in its @xmath139 ground state . due to the spin - exchange selection rules and to the collinearity of the tip and the sample , only the minority carriers can excite the local spin out of the ground state . for a tip spin - polarization of @xmath145 , there are more minority electrons coming from the tip than those coming from the substrate . as a result , the intensity of the inelastic interaction will change depending on the direction of the current . this creates an asymmetry in the conductance spectrum with respect to the applied bias . the additional lineshape features appearing in the weak coupling case ( the conductance decay following a conductance step ) are due to the third order kondo - like self - energy , which produces a logarithmic decay at the conductance steps . this result is in good agreement with experiments ( see fig .  4 of ref .  ) . when the spin of the mn ion is driven further out of equilibrium , in particular in the strong coupling case , the bias asymmetry becomes more pronounced . such spin - pumping phenomenon can be appreciated by looking at figure  [ fig5](a ) , where we show the populations of the six spin states of the mn atom as a function of bias for strong tip to sample coupling ( @xmath142  mev ) . from the figure one can see that as the bias increases the @xmath139 ground state gets depleted in favour of populating the other five excited states . in particular already at @xmath147  mv the population of the @xmath148 level is larger than that of the ground state . [ fig5 ]    in th figure we also plot the average magnetization , which is defined as @xmath149 [ see panel ( b ) ] . intriguingly we find that for negative biases the spin is effectively flipped from @xmath139 to @xmath150 over 25mev range . such spin flipping results in a large dip in the conductance for negative biases as the tip is no longer collinear to the sample .    in fig [ fig6 ] we present the calculated spectra for the fe atom in the spin polarized case . we choose parameters in this case that conform with the experimental data of loth _ the fe atom is assumed to carry a quantum mechanical spin of @xmath121 and it also exhibits a transverse easy axis anisotopy of @xmath151mev and an axial anisotropy of @xmath152mev . we again assume a large value of the onsite energy , @xmath153ev , and we examine the spectra in the strong coupling case of @xmath142mev with a tip polarisation of @xmath154 and magnetic field strength 3 t as used in experiments . in fig 6(a ) we present the conductance spectra for the two cases when the magnetic field is parallel and perpendicular to the easy axis of the atom ( the @xmath155axis in this model ) . as previously , we present this for both second and third order calculations . firstly , we notice that the spin polarized tip affects the spectra only in the case of parallel magnetic field where a clear bias assymetry is produced . no significant assymetry is found in the perpendicular case . this conforms with the experimental findings and is due to the fact that electron spins in the tip are no longer colinear with localised spin of the fe atom . as found in previous works @xcite the inclusion of third order effects is vital in reproducing the corresct logarithmic decay at each of the conductance steps , which is particularly noticable for the perpendicular magnetic field . more significantly , experimental spectra for the parallel case exhibit a zero bias conductance dip which is absent in the 2nd order spectra but appears strongly when third order effects are included . this can also be seen the calculation of the second derivative of the current in fig 6(b ) where a clear zero bias anomaly is evident in the third order case . we finally move to discuss the inherent asymmetry measured in the conductance profile , which is usually observed even if the tip is not spin - polarized  @xcite . we model this lineshape feature by including the real part of the full interacting electron - spin self - energy in the description [ see equation ( [ eq:22 ] ) ] . the structure of this contribution to the self - energy shows an explicit dependence on the on - site energy , @xmath10 , and also a logarithmic peak of width @xmath57 at the onset of an inelastic transition ( @xmath156 ) . the asymmetry arises from the difference in polarity of the logarithmic peak for @xmath157 . the self - energy is thus an odd function of both energy and bias . this results in the conductance profile having a decrease of the step heights for @xmath158 and a increase of them for @xmath159 . we test this approach by considering the case of a non - spin - polarized tip and a single mn atom . we use the same anisotropy parameters as for the mn dimer but , for the sale of simplicity , we keep the spin always its equilibrium state and choose @xmath140mev . figure  [ fig7 ] shows the resulting conductance spectra for three different choices of the on - site energy @xmath160 . it is cear that the closer @xmath160 is to the fermi energy ( 0  ev ) , the greater is the bias asymmetry , while as @xmath160 is increased , the conductance profile becomes more symmetric . in this respect , the formalism outlined here is in agreement with the fano lineshape argument  @xcite where the degree of asymmetry for electrons tunneling through a single impurity is given by a ratio of the real to the imaginary contributions to the interacting green s function@xcite .    as an additional test we consider the case of a mn trimer , whose spectrum was shown first by hijibehedin _ _ et al.__@xcite to exhibit a large bias asymmetry when measured with a non - magnetic tip . we model this system by choosing an antiferromagnetic nearest neighbour exchange coupling @xmath1612.3  mev . furthermore , in order to accurately describe the position of the principle conductance steps in the conductance profile , we also include a ferromagnetic second - nearest - neighbour interaction between the local spins of magnitude @xmath162 - 1.0  mev  @xcite . we again choose to keep the spin system in equilibrium and therefore consider weak coupling between the stm tip and the second atom in the trimer chain ( @xmath140  mev ) . the best fit to the experimental data is found with @xmath163ev . figure  [ fig8 ] shows the model fit to the experimental data ( from ref .  [ ] ) ) . whereas previous calculations did not predict any conductance asymmetry @xcite it is clear from the figure that the inclusion of the real part of the self - energy in the description produces a significant conductance asymmetry this is most prominent at the principle step height @xmath164 for each bias polarity . although the step height for the negative bias is not as small as that found experimentally , the qualitative trends are similar . in particular we notice the logarithmic conductance increase ( reduction ) that occurs before ( after ) the onset of the step at @xmath165mev , which also originates from the third order contribution to the self - energy .    in this work , based on a perturbative approach of the s - d model , we have shown that the entire lineshape description can be re - conciliated with experiments by considering an expansion of the self - energy to the third order , which also includes its real part . as such we have shown that the conductance asymmetry can be described also if the electronic orbitals forming the sample s spin are not explicitly taken into account as also suggested by delgado and fernandez - rossier @xcite . in conclusion we have studied the lineshape details of the conductance profile of mn atoms deposited on cun and probed with a stm tip either or not carrying spin - polarization . in particular we have looked closely at the asymmetry of the conductance with the bias polarity . firstly , we have extended our perturbative approach to spin - scattering to the spin - polarized case and considered an expansion of the complex part of the electronic propagator up to the third order . this allows us to reproduce the logarithmic decay of the conductance subsequent a conductance step , which is observed in experiments but could not be explained by a second order theory .    when the current density is increased and the tip is spin - polarized the conductance profile starts to develop a significant asymmetry with respect to the bias polarity . these are indicative of the spin system being driven out of equilibrium . we have then derived a second order expansion of the spin - propagator capable of evaluating the non - equilibrium population of the various spin energy levels . this was put favorably to the test against a series of experiments probing a single mn and fe ions with a spin - polarized stm tip in an intense magnetic field . furthermore the same formalism was capable of describing excitations occurring away from the ground state for a mn dimer probed by a non - magnetic tip . also in this case the agreement with experiments is very satisfactory . finally , in an attempt to describe the bias asymmetry in the case of non - spin - polarized stm tips we have derived an analytic expression for the real part of the electron - spin interacting self - energy . this contains logarithmic peaks at the excitation energies that are odd with respect to energy and voltage . such parity results in an asymmetry in the conductance profiles . such a scheme was tested for the case of a mn monomer and a mn trimer and compares reasonably well with experiments . this work is sponsored by the irish research council for science , engineering & technology ( ircset ) . nb and ss thank science foundation of ireland ( grant no . 08/era / i1759 ) and crann for financial support . computational resources have been provided by the trinity centre for high performance computing ( tchpc ) . we wish to thank cyrus hirjibehedin for making the experimental data shown in figures [ fig7 ] available to us . here we wish to derive a method for calculating the steady - state non - equilibrium distribution of the spin energy levels populations , @xmath166 , due to the coupling with the electrodes . in order to do so we expand equation ( [ eq:5 ] ) up to the @xmath43-th order in the interaction hamiltonian @xmath167_{nm}=\sum_n\frac{(-i)^{n+1}}{n!}\int\limits_c{d}\tau_1\dots\int\limits_c{d}\tau_n\ \times \nonumber \\ & \frac{{\langle}0| t_c\{{h}_\mathrm{e - sp}(\tau_1)\dots{h}_\mathrm{e - sp}(\tau_n)d_{n}(\tau)d_{m}^{\dagger}(\tau')\}|0{\rangle}}{u(-\infty,-\infty)}\:,\end{aligned}\ ] ] where @xmath63 is the time - evolution unitary operator and the time - averages are over the known non - interacting @xmath64 ground state @xmath65 . as in equation ( [ eq:8 ] ) the time integration over @xmath66 is ordered on the contour @xmath67 going from @xmath68 to @xmath69 and then returning from @xmath69 to @xmath68 @xcite .    by inserting the expression for @xmath61 from equation ( [ eq:3 ] ) into the equation above and by expanding up to the second order we obtain [ note for the ease of the description we omit the elastic contribution of @xmath168 , which is then included in the final expression in equation ( [ eq:12 ] ) ] @xmath169^{(2)}_{nm}=\frac{(-i)^{3}}{2!}j^2_\mathrm{sd}\sum_{\alpha,\alpha',\beta,\beta'}\int\limits_c{d}\tau_1\int\limits_c{d}\tau_2\ \times \nonumber \\ & { \langle}0|t_c\{c_{\alpha}^{\dagger}(\tau_1)c_{\alpha'}(\tau_1)c_{\beta}^{\dagger}(\tau_2)c_{\beta'}(\tau_2)d_{n}(\tau)d_{m}^{\dagger}(\tau')\}|0{\rangle } \nonumber \\ & \times \sum_{i , j}{\langle}0|t_c\{{s^{i}}(\tau_1){s^{j}}(\tau_2)\}|0{\rangle}[{\sigma}^i]_{{\alpha}{\alpha'}}[{\sigma}^j]_{{\beta}{\beta'}}\:,\end{aligned}\ ] ] where the indices @xmath8 and @xmath170 run over the cartesian coordinates @xmath171 , @xmath172 and @xmath173 for the given spin coupled to the tip ( the tip make electronic contact with one spin only ) . we now substitute into equation ( [ eq:9 ] ) the operator breakdown of the spin from equation ( [ eq:6 ] ) @xmath174^{(2)}_{nm}=\frac{(-i)^{3}}{2!}j^2_\mathrm{sd}\sum_{k , k',l , l'}\int\limits_c{d}\tau_1\int\limits_c{d}\tau_2 \nonumber \\ & \times { \langle}0|t_c\{d_{n}(\tau)d_{k}^{\dagger}(\tau_1)d_{k'}(\tau_1)d_{l}^{\dagger}(\tau_2)d_{l'}(\tau_2)d_{m}^{\dagger}(\tau')\}|0{\rangle } \nonumber \\ & \times \sum_{\alpha,\alpha',\beta,\beta'}{\langle}0|t_c\{c_{\alpha'}(\tau_1)c_{\beta}^{\dagger}(\tau_2)c_{\beta'}(\tau_2)c_{\alpha}^{\dagger}(\tau_1)\}|0{\rangle } \nonumber \\ & \times \sum_{i , j}s_{kk'}^is_{ll'}^j[{\sigma}^i]_{{\alpha}{\alpha'}}[{\sigma}^j]_{{\beta}{\beta'}}\:.\end{aligned}\ ] ] the time - ordered contractions of the two brackets in equation ( [ eq:10 ] ) can be re - written in terms of their respective non - interacting green s functions , @xmath175 and @xmath176 as follows @xmath177^{(2)}_{nm}=-j^2_\mathrm{sd}\sum_{k , k',l , l'}\int\limits_c{d}\tau_1\int\limits_c{d}\tau_2 \nonumber \\ & \times \delta_{nk}\delta_{lk'}\delta_{ml'}[d_0(\tau,\tau_1)]_{nn}[d_0(\tau_1,\tau_2)]_{ll}[d_0(\tau_2,\tau')]_{mm } \nonumber \\ & \times \sum_{\alpha,\alpha',\beta,\beta'}\delta_{\alpha'\beta}\delta_{\alpha\beta'}[g_0(\tau_1,\tau_2)]_{\beta\beta}[g_0(\tau_2,\tau_1)]_{\alpha\alpha } \nonumber \\ & \times \sum_{i , j}s_{kk'}^is_{ll'}^j[{\sigma}^i]_{{\alpha}{\alpha'}}[{\sigma}^j]_{{\beta}{\beta'}},\end{aligned}\ ] ] where the extra factor of 2 emerges from the fact that a second contraction of the time - ordered bracket merely exchanges @xmath178 and @xmath179 . then , by using dyson s equation  @xcite , one can write the second order contribution to the interacting spin self - energy ( @xmath180 ) . this reads @xmath181^{(2)}_{nm}=-2j^2_\mathrm{sd}\sum_{\alpha,\beta}[g_0(\tau_1,\tau_2)]_{\beta\beta}[g_0(\tau_2,\tau_1)]_{\alpha\alpha } \nonumber \\ & \times \sum_{l}[d_0(\tau_1,\tau_2)]_{l , l}\sum_{i , j}s^i_{nl}s^j_{lm}[{\sigma}^i]_{{\alpha}{\beta}}[{\sigma}^j]_{{\beta}{\alpha}}\:,\end{aligned}\ ] ] where we have evoked the assumption that the electrons are spin degenerate thus omitting the spin index on @xmath182 and including a factor of 2 . we now calculate the real - time quantities , such as the lesser ( greater ) self - energies , by using langreth s theorem for the time ordering over the defined contour @xcite . after including the elastic contribution we obtain @xmath183^{(2)}_{nm}=-2j^2_\mathrm{sd}\sum_{\alpha,\beta}[g^{\lessgtr}_0(t_1,t_2)]_{\beta\beta}[g^{\gtrless}_0(t_2,t_1)]_{\alpha\alpha } \nonumber \\ & \times \sum_{l}[d^{\lessgtr}_0(t_1,t_2)]_{ll}\nonumber \\ & \times\sum_{i , j}\big(s^i_{nl}s^j_{lm}[{\sigma}^i]_{{\alpha}{\beta}}[{\sigma}^j]_{{\beta}{\alpha}}+\delta_{ij}\delta_{\alpha\beta}\chi s^i_{nm}[{\sigma}^i]_{{\alpha}{\beta}}\big)\:.\end{aligned}\ ] ] on computing the fourier transform we note the two different expressions for the lesser and greater green s functions are @xmath184^{(2)}_{nm}=-\frac{j^2_\mathrm{sd}}{\pi}\sum_lp^{\lessgtr}_l\times \nonumber \\ & \sum_{\alpha,\beta}\int\limits_{-\infty}^{+\infty}{d}{\omega}[g_0^{<}(\omega)]_{\beta\beta}[g_0^{>}(\omega\pm(e-\varepsilon_l))]_{\alpha\alpha}\times\nonumber \\ & \sum_{i , j}\big(s^i_{nl}s^j_{lm}[{\sigma}^i]_{{\alpha}{\beta}}[{\sigma}^j]_{{\beta}{\alpha}}+ \delta_{ij}\delta_{\alpha\beta}\chi s^i_{nm}[{\sigma}^i]_{{\alpha}{\beta}}\big),\end{aligned}\ ] ] where we have defined @xmath185 and @xmath186 and @xmath187 $ ] . by assuming that the spin system is in thermal contact with a heat bath kept at the temperature @xmath46 , the energy levels @xmath188 should be broadened by the factor @xmath189 . this can be neglected for the ease of the calculation since in general @xmath190 . however we do not disregard the broadening in the electronic green s function due to contact to tip and substrate as this is pivotal to the calculation of the non - equilibrium spin populations . j. j. parks , a. r. champagne , t. a. costi , w. w. shum , a. n. pasupathy , e. neuscamman , s. flores - torres , p. s. cornaglia , a. a. aligia , c. a. balseiro , g. k .- l . chan , h. d. abrua and d. c. ralph1 _ et al . _ , science * 328 * , 1370 ( 2010 ) .
the conductance profiles of magnetic transition metal atoms , such as fe , co and mn , deposited on surfaces and probed by a scanning tunneling microscope ( stm ) , provide detailed information on the magnetic excitations of such nano - magnets . 
 in general the profiles are symmetric with respect to the applied bias . 
 however a set of recent experiments has shown evidence for inherent asymmetries when either a normal or a spin - polarized stm tip is used . in order to explain such asymmetries here we expand our previously developed perturbative approach to electron - spin scattering to the spin - polarized case and to the inclusion of out of equilibrium spin populations . in the case of a magnetic stm tip 
 we demonstrate that the asymmetries are driven by the non - equilibrium occupation of the various atomic spin - levels , an effect that reminds closely that electron spin - transfer . 
 in contrast when the tip is not spin - polarized such non - equilibrium population can not be build up . in this circumstance 
 we propose that the asymmetry simply originates from the transition metal ion density of state , which is included here as a non - vanishing real component to the spin - scattering self - energy .