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Validity of the transfer Hamiltonian approach : application to the STM spectroscopic mode
Claudine Noguera
To cite this version:
Claudine Noguera. Validity of the transfer Hamiltonian approach : application to the STM spectroscopic mode. Journal de Physique, 1989, 50 (18), pp.2587-2599.
�10.1051/jphys:0198900500180258700�. �jpa-00211084�
Validity of the transfer Hamiltonian approach : application to
the STM spectroscopic mode
Claudine Noguera
Laboratoire de Physique des Solides, Université de Paris Sud, 91405 Orsay, France (Reçu le 17 avril 1989, accepté sous forme définitive le 23 mai 1989)
Résumé.
2014A partir d’une méthode de fonctions de Green à un électron, nous redérivons
l’expression perturbative du courant tunnel établie par Bardeen. Nous discutons son domaine de validité en termes d’épaisseur de la barrière et de coefficients de réflexion des ondes électroniques
aux surfaces des électrodes. A certaines valeurs de l’énergie, correspondant à des états localisés, ceux-ci peuvent devenir trop grands pour que la méthode perturbative reste applicable, et nous
soulevons le problème de l’observation de ces états par microscopie à effet tunnel (STM) en mode spectroscopique.
Abstract. 2014 Starting from a non-perturbative one electron Green’s function method, we rederive Bardeen’s perturbative expression for the tunneling current. We discuss its range of validity in
terms of barrier thickness and reflection coefficients at the surfaces of the electrodes. At special energies corresponding to localized states, these latter may become large enough to invalidate the
perturbative approach, and the question of the observability of such states in the spectroscopic
mode of the Scanning Tunneling Microscope (STM) is raised.
Classification
Physics Abstracts
68.20 - 73.20
-73.40G
1. Introduction.
In the field of Scanning Tunneling Microscopy, most topographic or spectroscopic measure-
ments have been interpreted with the help of the transfer Hamiltonian approach [1], rewritten
to account for the special geometry of the microscope [2, 3]. Based upon the simple idea that
the electronic states characterizing each independent electrode are not deeply perturbed
when they come close to each other, this approach gives a flexible expression for the tunneling
current, which has allowed one to clarify many relevant questions such as : the role of d versus s or p states [4], the resolution of the microscope [3], the nature of the corrugation at
the surfaces [5], the meaning of current versus voltage curves [6] etc.
Since it is now used as a basis to discriminate between different models of arrangement of
atoms at surfaces, it seems very important to assess its range of validity precisely. This question was already addressed many years ago in the context of planar tunneling junctions [7, 8] and it was generally recognized that the barrier thickness was an important parameter.
Yet, in establishing recently a more elaborate formalism for the tunneling current [9], we have
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180258700
met a situation, linked to the existence of surface states, in which this formalism and the transfer Hamiltonian approach predicted different results qualitatively. It thus seems that
criteria other than the barrier thickness are relevant and it is the purpose of this paper to find them.
In section 2, we recall the major steps in the derivation of Bardeen’s expression for the tunneling current and we show how it led Tersoff and Hamman to the concept of local density
of states at the center of curvature of the tip in an STM. Then, in section 3, we present a new version of the Green function formalism, within a one-electron approximation, well suited to a precise comparison with Bardeen’s approach. The rederivation of the transfer Hamiltonian
expression from the general formalism is explicited in section 4 and a thorough discussion of its implications is done in section 5.
2. The transfer Hamiltonian method.
The transfer Hamiltonian method was first introduced by Bardeen [1] to calculate the current
passing through a planar tunnel junction when a small bias V is applied between the electrodes. It makes use of a time dependent perturbation approximation for the wave
functions of the system, from which the tunneling current may be calculated. The system, consisting in two electrodes in contact with an insulating layer (or vacuum in the case of an STM), is conceptually shared into two parts, called the left and right parts, L and R, limited by a surface 1 (cf. Fig. la). It is assumed that one knows the total set of wave functions
1/1’ v and 0. characteristic of two independent systems called 1 and II, such that 1 is identical to the left part L with the insulating material extending on its right to infinity (Fig. lb) and II is
identical to the right part R with the insulating material extending on its left to infinity (Fig. lc). The probability of tunneling from say left to right is assumed to be the perturbative parameter. If at time t
=0 the electron is in the left electrode (1/1’ (t = 0) = 1/1’ v), at later times, its wave function acquires a small component characteristic of the right side :
Fig. 1.
-Geometry of the tunneling system with two electrodes and the vacuum (or insulating) barrier :
in la : the system is shared into a left and a right part by a boundary 1 entirely located in the barrier ; in
1b : one defines a system 1 identical to the true one on the left of 1 and having vacuum (or insulator) extending to infinity on the right of Z ; in lc : one defines a system II in a way similar to 1 with a mere
interchange of right and left.
The coefficients a, (t) may be calculated by time dependent perturbation theory, and are
shown to be proportional to an effective tunneling matrix element Tvp. equal to :
The tunneling current then takes a Fermi golden rule form :
( f (E) is the Fermi Dirac function). It should be noted that the resulting current is independent of the choice of the matching surface £, provided that the Hamiltonian is identical in the part of space common to L (respectively R) and 1 (respectively II). Lang [4]
generalized this result to account for the current density in the junction. With an obvious notation, the latter is found to be equal to :
Tersoff and Hamman [2, 3] have further applied Bardeen’s result to the geometry of the STM.
They represent the tip (right electrode) as a sphere with wave functions of s character and
they assume that the second electrode has a perfect periodicity in the plane parallel to the
surface. Making a two-dimensional Fourier expansion of the wave functions, and taking advantage of their exponentional decrease in the vacuum barrier, they arrive at the conclusion
that the tunneling current is proportional to the density of states of the left electrode,
calculated at the center of curvature ro of the tip :
In such a way, they are able to account for the dominant role played by the extended states of the electrodes (s or p) compared to the d ones, which a posteriori justified the use of s wave
functions for a tungsten tip.
3. Green’s function method.
Recently, we have proposed a Green’s function method for calculating the tunneling current [9] which presents the advantage of going beyond the perturbation approximation. This
method is a generalization of treatments used by Caroli et al. [7] and Feuchtwang [8] for planar junctions, in the sense that it does not make use of specific boundary conditions to
simplify the mathematics. As a consequence the final expression for the tunneling current
includes quantities characteristic of the electrodes in contact with vacuum (or insulator), as is physically convenient, rather than those characterizing e.g. a surface bound by an infinite
barrier. In this work, we used a partitioning of the total system in three parts : left electrode,
vacuum barrier and right electrode, with two surfaces of separation. In our opinion, this partitioning is the best suited to a discussion of the contributions of the electronic states of both electrodes and of the multiple reflection events in the barrier. Yet, it is clumsy when a precise comparison with Bardeen’s approach is required, and in the following we will
reformulate the Green’s function method with the partitioning of space already described in
figure 1, with a single surface of separation Z. In doing so, it is clear that we will not be able to treat correctly the effects of the electric field in the barrier, but neither does the transfer Hamiltonian method, and it is not the purpose of this paper to look for improvements in that
direction (for a discussion of this point see e.g. Ref. [10]).
Using subscripts 1 and II for systems 1 and II respectively and small letters for their Green functions, we define two kinds of Green functions : there are first the advanced or retarded ones, which in terms of the stationary states 1/1’ JI of energies Ev of the systems are written
and the g’ Green function discussed by Keldysh [11] for out-of-equilibrium situations, which
in our case contains all the information about occupied states :
In equations (6) and (7), we have not written explicitly the dependence of the
g upon w. Yet, it should be remembered in all the following that we work in Fourier space as
concerns the time variables. The gradient of the advanced (or retarded) Green functions has the property of being discontinuous at x
=x’. This comes from the « delta function » in the
right hand side of the differential equation that governs their behaviour [11]. As a
consequence, when this is necessary, we will add a + or - superscript to indicate that the limit x’-x is taken from the right or from the left. these superscripts are unnecessary for the
gradients of the g+ functions which have no discontinuity at x
=x’.
Our aim is to obtain the G+ Green function for the total system (capital letters refer to the
total system) with the two electrodes in contact (Fig. la), from which the tunneling current
may be derived :
A matching procedure at the surface.! allows us to relate G+ to the Green functions of the
subsystems 1 and II. This procedure, usually developed for the advanced and retarded Green function (e.g. Ref. [12]), is extended without difficulty to G+, taking into account its specific
differential equation (11), and yields the following result (unless otherwise specified,
V refers to the x coordinates and integration is performed on the surface Z ; Rydberg atomic
units are used throughout) :
It is seen that the knowledge of the advanced and retarded Green functions Ga r of the whole system is required to solve equation (9). The G a, ’ r are self consistently determined by
(u=a,r):
These systems of equations are quite complicated to solve in the general case, but they may be put under a more tractable form by eliminating the g’ functions thanks to the
relationships :
I 1 (or IL II) represents the Fermi level of system 1 (or II) in the presence of the bias V and the Heaviside function 0 (1£ 1, Il
-w ) in this equation tells that empty states of energy
úJ :> IL 1, II do not contribute to g+ , at zero temperature. Consequently one may rewrite
equation (9) in the following way :
The two systems (10) and (12) have to be simultaneously solved to determine
G’, ’ and G + . We will not try here to solve them exactly since it would only lead to the result already obtained using the three region partitioning of space [9]. Equations (10) and (12) will
constitute our basis of discussion in the next section. It should be noted that, up to that point, they are « exact » in the sense that they do not result from any perturbation approximation,
and indeed we had noticed in reference [9] that all multiple reflections in the vacuum barrier could be obtained with them. But, of course they rely upon the validity of the one electron approximation used throughout this paper.
4. New dérivation of the transfer Hamiltonian result.
To derive the expression of the tunneling current written in equations (3), (4), we make the
two following hypotheses :
a) We assume that, at all energies under consideration (ILl : úJ : ILn), the advanced or
retarded Green functions of the vacuum are real, which corresponds to a density of states
equal to zero (energies below the Fowler Nordheim regime). These Green functions (equal to
each other since they are real) will be called go : they have the usual exponentially decreasing spherical character :
with K related to the distance in energy to the vacuum level Vo : K 2 oc (Vo - m ). This first hypothesis restricts us to low bias so that the Fowler-Nordheim regime is not reached.
b) The advanced or retarded Green functions for the subsystems 1 and II may be rewritten without loss of generality under the following form (u
=a, r ) :
where the gradients of f I and fjl no longer present discontinuities at Xl
=x2. Our second
hypothesis will be that the quantites f and fIl be small compared to go whatever the positions
Xi or x2 on the surface £. We will come back to the validity of this statement in section 5, but roughly speaking, it means that the matching surface -Y has to be far enough from each surface of the electrodes.
Using these two assumptions, we are now able to solve the systems of equations (10) and (12) to successive orders of approximations, and find Ga,,(Xl@ x2 ) and G+ (xl, x2 ) for any
position of Xi and x2 on -5’:
- zeroth order :
1 st order :
At this order of approximation G+ (xl, x2) is imaginary, so that there is no tunneling current ;
it is necessary to go to the next order for G+ which makes use of the first order expression for
G a, r
.2nd order :
This immediately leads to the current density at any point y of Z (the integration has to be
performed with respect to the x variables and the scalar product dS . Vx has to be considered
for each of the four terms) :
with an obvious notation for the gradients. The last step is done by noticing, that since
go is real, the imaginary parts of either f I, II or their gradients are identical to the imaginary
parts of the gI, II or their gradients. As a consequence, they may be developed on the basis of the eigenstates of the left or right systems, as follows :
A similar expression holds for fn, with wave functions 0. and energies E, . The final result for the tunneling current reads :
This expression is exactly similar to the transfer Hamiltonian one. We have thus proved in this
section that, provided that the partitioning of the system in two parts is legitimate, a) and b)
are sufficient conditions to get Bardeen’s expression.
5. Discussion.
The whole validity of the transfer Hamiltonian method thus relies on the comparison between 90(Xl, X2) and both fI(xl, x2) and fII(xl, x2) (Eq. (14)) when Xi and x2 lie on the matching
surface 1. To understand under which circumstances f I and fn are small compared to
go, we first observe that they represent the contribution to gl, Il of the vacuum electronic waves
reflected either on the surface Si or S2, if SI and S2 are defined as the surfaces of contact between the electrodes and the insulator (or vacuum) (Fig. 2). Let us write for example the matching equations on SI which allow gi to be calculated once the vacuum Green functions go and the Green function gb of the left bulk electrode 1 are known (the integration is performed on the Si surface, and all the Green functions are either advanced or retarded) :
if Xi and x2 on the right of S 1
if Xi on the left of S 1 and x2 on its right . (21)
From this equation, it may be inferred that, very generally, for Xi and x2 belonging to the surface Z completely located on the right of Si, fI (Xl, x2 ) takes the following form :
where the non-local functions A and B contain information on the reflectivity of the surface for waves travelling in the vacuum. A similar expression holds for fn. Since at a given energy
w below the vacuum level (assumption a)) go has an exponentially decreasing behaviour, it
Fig. 2. - Same system as in figure 1 : Si and S2 represent the surfaces of contact between the two electrodes and the vacuum (or insulator).
turns out that, whatever the positions of x, and x2 on Z, fj is weighted by a prefactor of the
order of e- 2 Kd1 where dl is the distance of closest approach between Z and Si . Qualitatively, it
thus appears that the perturbative development with respect to f and fIl is valid provided
that two requirements are fulfilled :
i) the distance between the electrodes has to be large compared to 1 / K ; ii) the matching surface Z should not be chosen too close to either electrode.
The first requirement is not very restrictive as long as large biases are not applied to the
electrodes : for electrons close to the Fermi level in usual materials, V - w is of the order of a
few eV ; consequently, 1 / K is about one Angstroem, while most often the microscope is operated at tip-to-surface distances of the order of ten Angstroems. Yet, this would no longer
be true at smaller distances or close to the Fowler-Nordheim regime, where 1 / K becomes large. This is indeed what we found in simple one-dimensional models [13]. The second requirement should just be kept in mind to temper the general belief that the choice of the
matching surface is completely free in Bardeen’s theory : the preceding discussion shows that it would be wrong to calculate the transfer Hamiltonian matrix element for example on one of
the surfaces of the electrodes, since, there, the two Green functions go and f I, for example,
can be of similar importance.
Actually, the two preceding conditions are not sufficient to assess the validity of the transfer Hamiltonian approach, since f I also depends on the values of the reflection coefficients A and B at the surface of the electrodes. The imaginary parts of these coefficients, relevant in order to obtain a non zero tunnel current (see Eq. (17)), are non vanishing for energies
w corresponding to propagating states in the bulk band structure of the electrodes, but also
for discrete energies in the gaps of the band structure ; at these energies, a special matching
between evanescent waves inside and outside the crystal may be performed depending upon the specific orientation of the surface ; this is the general criterion which reveals the existence of surface states, well recognized and observed e.g. in photoemission. From equations (18), (20), we expect them to contribute also to the tunneling current.
In the following, we would like to show that, at these energies, not only does the
perturbative treatment give quantitatively wrong results, but also that it predicts a contribution
to the current from these states which does not exist. For this, we will develop the complete
resolution of the systems of equations (10) and (12) in one dimension, and then, we will generalize the results in three dimensions.
Let us call xL and XR the positions of the surfaces of the electrodes and xo the position of 4, in one dimension (XL : Xo : XR) and let us first calculate gI(xo, xo). Denoting
X the logarithmic derivatives of the different Green functions :
we find successively the following equalities :
It turns out that most quantities of interest are simply expressed in terms of the logarithmic
derivatives of the Green functions : these are the right quantities to work with when a
matching procedure is performed (cf. Eq. (23b)), because they express the conservation of the electron flux at the boundary. The local density of states at the surface of the left electrode is given by the imaginary part of gl(XL, xL ). It contains two kinds of contributions at energies
below the vacuum level : the first one is found when Im Xb is non-vanishing ; it represents the propagating states of the electrode, which density has been modified due to the reflection at the surface. But there is also a second contribution at discrete energies where Xo + Xb
=0,
due to surface states. Let us call Es, such an energy : for llJ close to Es,, one can develop
The local density of states at the surface of the electrode presents a divergence at
ú)
=Ess of weight W,,. A similar divergence characterizes the density of states at the position
of the matching surface X (at xo) : if we introduce the function f as in equation (14), we
obtain :
An exponential factor e - 2 K (xo - xL) >
originating from the propagator gO(XL, xO)2 weights the
contribution from extended as well as surface states (the distance of closest approach between
Z and Si is here xo - xL) . For energies close to Ess, the imaginary part of fI, (xo, xo) is large and equal to :
clearly invalidating the use of perturbation theory to treat it.
Furthermore, when the complete resolution of the systems of equations (10) and (12) is performed, it is found that the current in the junction takes a very simple form :
where XI and XI, represent respectively the logarithmic derivatives of the Green functions of the systems 1 (resp. II) evaluated at the matching surface £. The tunneling current thus
contains contributions only from the electronic states of 1 and II which are associated with a
non-zero value of Im XI or Im XII respectively. If we express Im XI for example, we find
At energies close to E,,, this quantity is real and does not present any anomalous
behaviour ; more precisely, it is equal to XI
= -K. This means that the surface states o f the left
electrode do not contribute to the tunneling current, nor of course do those of the right
électrode. This comes from a cancellation effect between the numerator and the denominator in equation (28), while, when one does perturbation theory on this quantity, one approxi-
mates XI by :
rthus destroying the cancellation effect and restoring the divergence in Im XI. This is
reminiscent of the cancellation effect stressed by Harrison [14] which forbids to express the
tunneling current uniquely in terms of density of states.
It is not possible to make as a simple derivation in the case of a three-dimensional system.
Yet one can note the following points : first, the matching equations may be put in a form reminiscent of the one-dimensional case by performing a Fourier analysis of the Green functions with respect to the space variables parallel to the surface. Secondly, at a given kll value, one can define the logarithmic derivative of the Green function, the gradient being
taken with respect to the coordinates normal to the surface. This quantity X involves the ratio between the Green function and its gradient taken at the same kjj value. Finally, the equation
of definition of the surface states that we have found has its exact counterpart as Xo + Xb
=0 and the surface states again appear as discrete poles, yielding divergences both in
the local density of states, and its gradient at a given kll value. This induces the same
cancellation effect in lm X as in one-dimension, thus showing that these states do not
contribute to the tunneling current.
Actually, such a conclusion can be reached, not only for surface states, but also for any state (due to defects...) which is associated with a discrete pole in the Green functions 91, II, at a given kll value. We can physically assign this result to the fact that the tunneling
current does not depend only on the density of the states available for tunneling, but also on
the group velocity, perpendicular to the surface, of the electrons in these states. Although this
group velocity does not appear explicitly in our derivation, one should remember that in one
dimension, or in three dimensions at a given kp value, the group velocity in the perpendicular
direction is proportional to the inverse of the density of states. A discrete pole in the Green function is thus necessarily associated with a non-dispersive state with respect to
ki , i.e. to a state localized in real space, having a zero group velocity perpendicular to the
surface. The non-perturbative treatment thus tells that these states do not contribute to the
tunneling current, and, due to the divergence of the Green function, the perturbative development is never valid in such a situation.
6. Conclusion.
We have presented a non-perturbative Green function approach, relating the value of the tunneling current in a junction to quantities characteristic of the electronic structure of each electrode in the absence of the other, thanks to a matching procedure at a boundary
£ located in the barrier (vacuum for an STM or insulator in a traditional junction). This
method is a good starting point to rederive Bardeen’s expression for the tunneling current.
We found that three assumptions are needed :
- the Green function in the barrier has to be real. This is achieved when the bias applied
to the electrodes is not too large, so that the tunneling electrons remain under the vacuum
level. If this were not the case, it is clear that the approach with two surfaces of matching
would be better suited to describe the discrete states in the barrier which give rise to the
Fowler-Nordheim regime. Another situation in which the Green function in the barrier is not
real, is the case of the resonant tunneling. In such circumstances, two possibilities are open : either include the impurity in one of the electrodes, if it is more or less adsorbed on it, or leave
its contribution in go : in the latter case, it is necessary to reconsider the successive orders of
approximations in equations (15-18) and the final result for the current, given in the appendix, has no longer the usual form (Eq. (4)) ;
- the second assumption is linked to the barrier thickness. Due to the exponential
decrease of the vacuum Green function, in the regime where assumption n° 1 is fulfilled, each
of the two « small parameters » in the perturbative development is weighted by a factor of the
order of e- 2 Kd , where d is the distance of closest approach between the electrode upon consideration and the matching surface 1:. As a consequence, the barrier thickness D has to be large compared to 1 / K , which is not a restrictive condition except at large bias,
and the choice of the matching surface X is free provided that it does not come too close to
any of the electrodes ;
- the last assumption concerns the existence of localized states in the electronic spectrum of the electrodes. We have shown that the small parameters in the perturbative development
are equal to the difference between the Green functions gI or gII and the vacuum Green function go. If it occurs that, at a given kll , one of the Green function gI, II presents a discrete pole, then the perturbative coefficient will diverge and the whole procedure will break. Such discrete poles exist when surface states are present or, more generally, when states of
whatever origin (defect... ) in one of the electrodes have their wave functions localized in the direction perpendicular to the surface. For such states, the absence of dispersion in the k1 direction in the band structure, and the absence of connection to any propagating state (at
zero temperature and in a one-electron approach which have been assumed throughout this paper), are associated with a zero group velocity in the perpendicular direction. We
emphasize that, in this case, there is no contribution to the tunneling current, contrary to the predictions of the perturbative treatment. For surface resonant states, we expect the problem
to be less drastic : these states are associated with a high density of states, but the life time of the electron at the surface is not infinite, due to a small admixture of the wave functio‘n with
propagating states. As a consequence the transfer Hamiltonian approach might be quantitat- ively, but not qualitatively, wrong for these states.
It was the goal of this paper to assess under which circumstances the transfer Hamiltonian
approach can be safely used to interpret experimental results. More specifically, it is now well recognized that surface states do contribute to the tunnelling current measured with a scanning tunneling microscope, in the spectroscopic mode. This is generally accounted for by relying upon Tersoff and Hamann’s theory. We have proved here that the observation of surface states is definitely due to interaction processes, beyond any one-electron approach.
Auger type processes have already been proposed by Louis et al. [16], and we mentioned phonon mediated processes [9]. Work in this direction is currently under progress.
Acknowledgements.
1 am particularly grateful to the Commissariat à l’Energie Atomique for the year spent in Saclay in the group of J. Lecante (Service des Atomes et des Surfaces) during which a large
part of this work has been done.
Appendix.
We present here the specific form that the perturbative development takes when the Green function in vacuum is not real. This may happen either when an atom is adsorbed on one of the electrodes, or when there is an impurity in the barrier characterized by an energy level in the range of the relevant energies for tunneling : ul -- (o g II. Starting from equations (10)
and (12), one finds successively :
- zeroth order :
-
