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Journal ArticleDOI

A theoretical STM study of Co n /Pt(111)

02 Feb 2011-Philosophical Magazine (Taylor & Francis)-Vol. 91, Iss: 12, pp 1747-1764

AbstractIt is shown that by ‘stacking’ together a semi-infinite sample subsystem with a semi-infinite tip subsystem, difficulties with respect to a common Fermi level for the whole system can approximately be overcome in all those cases when the substrate (serving as lead) and the second lead are different materials. Based on this procedure by means of the spin-polarized (fully) relativistic screened Korringa–Kohn–Rostocker method and an equivalent Kubo equation, theoretical spin-polarized STM spectra for Pt(111)/Co n /Cr15 W22/Cu(111) with respect to an applied external magnetic field are evaluated in terms of difference conductivities as a function of the corresponding free energy. These spectra are interpreted using layer-resolved contributions to the difference conductivities in order to indicate which parts of the sample dominate changes in the tunneling current caused by changing the orientation of the magnetization. Also shown are estimates of the time-scales to switch from perpendicular to in-plane and vi...

Topics: Magnetization (52%), Fermi level (52%)

Summary (2 min read)

INTRODUCTION

  • With the arrival of spin-polarized STM techniques, however, di¤erent theoretical approaches were needed, since it was quickly realized that for non-collinear magnetic structures (at least) spin-orbit interactions had to be included [13, 14].
  • If, however, the substrate (semi-in nite system carrying one contact) on which magnetic atoms or islands are deposited is di¤erent from the material (lead, semi-in nite system) connecting the tip to the other contact, then approximations to the condition of a common Fermi level have to be made.
  • The easy way out, namely discarding the tip subsystem from a description of the electric properties of such a system, leads to a completely wrong model of electric transport, since a semi-in nite vacuum barrier only creates re ecting boundary conditions.[17].
  • Only then the ultimate goal of the present investigations is achieved, namely displaying theoretical STM spectra in terms of di¤erence conductivities versus free energy (applied external magnetic eld), i.e., obtaining curves that directly can be compared to experimental di=dV data when applying an external magnetic eld.

STACKING SYSTEMS

  • Consider for example the system Pt(111)/Con, i.e., a Pt(111) surface covered by n complete overlayers of Co.
  • Viewed as sample system in an STM experiment, the situation becomes less straight forward, since as already was mentioned a tip with a lead has to be included.
  • In using for example a typical Cr/W tip one obviously is faced with the problem to deal theoretically with a system of the following kind, Pt(111)| {z } semi- nite /PtmConVac3Cr15W22| {z }Cut tip /Cu-lead| {z } semi- nite (1) in which the thickness parameters are given in terms of monolayers.

COMPUTATIONAL APPROACH

  • All ab initio electronic structure calculations were performed for a uniform direction of the magnetization pointing along the surface normal in terms of the spinpolarized relativistic screened Korringa-KohnRostoker method.[19].
  • The band energies in Eq. (4) are evaluated (at zero temperature) in terms of the magnetic force theorem [18] by integrating in the upper half of the complex energy plane along a contour starting at E0 and ending at the Fermi energy.
  • Of course in a real space description no restrictions caused by translational symmetry apply, however, then the size of the clusters considered matters quite a bit.
  • In Fig. 2 this continuation is displayed for the Co layer as well as for the rst three Pt layers beneath and the rst vacuum layer in Pt(111)/Co.
  • Meaning of layer-wise contributions to the total di¤erence conductivity.

Free energies and di¤erence conductivities

  • As can be seen from this gure, the tip indeed adds only a small positive contribution to the free energy.
  • Pt layer contribute most to the total di¤erence conductivity as one would intuitively expect.
  • 5 - 7 one easily can follow the various layer-wise contributions to the total di¤erence conductivity when the orientation in the sample subsystem gradually changes from perpendicular to in-plane.

Time scales

  • Peak values as indicated explicitly, also known as Bottom.
  • In Fig. 8 nally the total di¤erence conductivities viewing only the sample subsystem, zz( ; 0), for Pt(111)/Con, n 3, are displayed versus and as implicit functions of the corresponding free energy E( ; 0).
  • Pt layer grows much faster than all other contributions.

The theoretical spectra

  • The lower part of Fig. 9 is now the ultimate result of the present investigations.
  • The curves shown correspond to the positive magnetic eld part of an experimental di=dV spectrum when varying the external magnetic eld.
  • Comparison of the switching times to reach the reorientation transition with respect to the number of Co layers, also known as Bottom.
  • In order to interpret the two branches of that curve additional concepts are needed such as arguments based on the dynamics of the system during a reorientation of the magnetization.

CONCLUSION

  • It was shown that for very large systems consisting of two subsystems separated by a vacuum barrier a stacking together of the subsystems can approximately be used.
  • The full line applies when i refers to the Co layer, the dashed line to the Pt layer beneath.
  • Peak values as indicated explicitly, also known as Bottom.

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A theoretical STM study of Co_n/Pt(111)
Peter Weinberger
To cite this version:
Peter Weinberger. A theoretical STM study of Co_n/Pt(111). Philosophical Magazine, Taylor &
Francis, 2011, pp.1. �10.1080/14786435.2010.544686�. �hal-00665449�

For Peer Review Only
A theoretical STM study of Co_{n}/Pt(111)
Journal:
Philosophical Magazine & Philosophical Magazine Letters
Manuscript ID:
TPHM-10-Oct-0439.R1
Journal Selection:
Philosophical Magazine
Date Submitted by the
Author:
24-Nov-2010
Complete List of Authors:
Weinberger, Peter; Center for Computational Nanoscience
Keywords:
magnetism, magnetization dynamics, STM
Keywords (user supplied):
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Philosophical Magazine & Philosophical Magazine Letters

For Peer Review Only
A theoretical STM study of Co
n
=P t(111)
P. Weinberger
1
1
Center for Computational Nanosci ence, Seilerstätte 10/22, A 1010 Vienna , Au stria
It is shown that by "stacking" together a semi-in…nite sample subsystem with a semi-in nite tip
subsystem di¢ culties with respect to a common Fermi level for the whole system can approximately
be overcome in all those cases when the su bstrate (serving as lead) and the second l ead are derent
materials. Based on this procedure by means of the spin-polarized (fully) relativistic Screened
Korringa-Kohn-Rostocker method and an equivalent Kubo equation theoretical spin-polarized STM
spectra for Pt(111)/Co
n
/Cr
15
W
22
/Cu(111) with respect to an applied external magnetic eld ar e
evaluated in terms of di¤erence conductivities as a function of th e corresponding free energy. T hese
spectra are interpreted using layer-resolved contributions to the derence conductivities in order to
indicate which parts of the sample dominate changes in the tunneling current caused by changing
the orientation of the magnetization. Also shown are estimates of the time scales to sw itch from
perpendicular to in-plane an d vice versa. All investigated properties suggest that di¤erent situations
apply when the number of Co layers on top of Pt(111) is increased from one to three.
INTRODUCTION
In the last fe w years the number of publications
devoted to spin-polarized scanning tunnel microscopy
(STM) increased substantially, see the impressive review
article by Wiesendanger [1] and in particular Refs. [2] -
[5], claiming that by now the magnetic switching prop-
erties of single atoms or at least of very small islands
of magnetic atoms can be determined experimentally.
Now-a-days even the use of external surprisingly high
magnetic vector elds became possible, the lateral pre-
cision in moving the tip having been reduced already to
a fraction of the spacial extension of an atom on top of
a surface. The experimentally investigated samples usu-
ally consist of an ensemble of magnetic atoms or a few
monolayers of a magnetic material placed on a suitable
substrate such as for example Cu(111) or Pt(111). Very
often a so-called Cr/W-tip prepared with about 10 lay-
ers of Cr and a thick slab of W is used. This tip is con-
nected in turn to a lead. The width of the vacuum barrier
between the sample and the tip is typically b elow 10Å.
The substrate and the lead on top of the tip carry the
necessary electric contacts for the measurements. Quite
clearly, since the tunneling current is a non-local quan-
tity, in order to describe STM spectra theoretically the
whole system consisting of the substrate (serving as a
lead), the magnetic adsorbate, the tip and the second
lead ought to be taken into account.
Based on Bardeen’s suggestion [6] STM experiments
are usually interpreted theoretically in terms of the so-
called Terso¤-Hamann approach [7, 8] in which the tun-
neling current is replaced by the charge density corre-
sponding to the surface local density of states. Frequently
also approximations to Bardeen’s matrix element are in-
cluded, which, however, in its original form is very dif-
cult to evaluate since it combines nonorthogonal eigen-
states of di¤erent Hamiltonians, namely those "of the
probe" and "of the surface". These approximations are
mostly based on spherically shaped tips [7, 8] and on us-
ing an s-wave for the tip wave function. In the past the
Terso¤-Hamann approach proved to be extremely suc-
cessful in interpreting experimental data, see, e.g. Ref.
[9] and in particular Refs. [10] - [12].
With the arrival of spin-polarized STM techniques,
however, di¤erent theoretical approaches were n eed ed,
since it was quickly realized that for non-collinear mag-
netic structures (at least) spin-orbit interactions had to
be included [13, 14]. Unfortunately, in using a Terso¤-
Hamann approach even on an appropriate spin-polarized
ab initio level, the di¢ culties with Bardeen’s matrix ele-
ments remain, and, in particular, one of the main features
of modern experimental techniques, namely of applying
an external magne tic eld, cannot be described properly.
Because of the non-locality of the tun neling current
it was and partially still is a matter of belief to claim
that STM is an "atomic"- or "surface" speci…c experi-
mental tool, whereby even the term "su rface" is a bit
misleading, since also buried magnetic structures can be
"seen" in STM. In order to shed s ome light on the ques-
tion of what actually is "seen" in a spin-polarized STM
experiment and also to overcome the limitations of the
Terso¤-Hamann approach it was suggested [15, 16] to di-
rectly calculate the tunne ling current in terms of a Kubo
equation based on (fully) relativistic scattering theory,
i.e., based on the Dirac equation, which of course de-
scribes spin-orbit interactions correct to all orders of the
speed of light. By displaying (di¤erence) conductivities
as functions of the anisotropy energy, which in turn is
proportional to the applied external magnetic eld, it was
shown that experimentally observed di=dV curves with
respect to an applied magnetic eld can be reasonably
well reproduced. In particular layer-resolved di¤erences
in conductivities as functions of the anisotropy energy
turned out to be a useful tool to point out which parts of
the whole system contribute mos t to the total di¤erence
in conductivities.
Up to now this approach was only applied to systems
in which the substrate and the lead on top of the tip are
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the same material [15, 16], which of course determines
the Fermi level. If, however, the substrate (semi-in…nite
system carrying one contact) on which magnetic atoms or
islands are deposited is di¤erent from the material (lead,
semi-in…nite system) connecting the tip to the other con-
tact, then approximations to the condition of a common
Fermi level have to be made. Clearly, although magnetic
islands can be viewed as nanosystems, the whole system
is of course macro-sized, the substrate as well as the lead
on top of the tip serving in principle as electron reservoirs
for any kind of electric transport prope rties.
The easy way out, namely discarding the tip subsys-
tem from a description of the electric properties of such
a system, leads to a completely wrong model of elec-
tric transport, since a semi-in…nite vacuum barrier only
creates re‡ecting boun dary conditions.[17] This, simply
speaking, means that no second contact can be "welded"
on: without a second contact, however, no measurements
of electric properties can be made and therefore also no
STM experiments.
If therefore two semi-in…nite systems of di¤erent mate-
rial separated by a vacuum barrier have to be taken into
account, formally the same di¢ culties as in Bardeen’s
orginal model and in the Terso¤-Hamann approach arise
when using Density Functional Theory, namely the ne-
cessity to deal with two Green’s functions (Hamiltonians)
of di¤erent spectral properties. An approximate way to
deal with this situation is discussed in the following sec-
tion.
Furthermore, beyond the problem of two leads of dif-
ferent type the geometrical shape of the two subsystems
that are linked together via a vacuum barrier ought to
be taken into account. Since parts of both subsystems
are nano-sized in two dimensions [18] and therefore no
longer are two-dimensional translationally invariant, in
principle one ought to use a real space description not
only for the electronic and magnetic properties of the to-
tal system but also for the electric transport properties.
Although in principle this c an be achieved by using, e.g.,
the so-called Embedded Cluster Method (ECM) [18, 19]
and a real space scattering version of the Kubo equation
[20] the computational ort to be encountered is quite
substantial and therefore most likely is not suitable for
routine-like investigations.
Clearly, by approximating such a real space description
by one based on two-dimensional translational symme-
try, implying in turn that only "‡at tips" and completely
decorated atomic layers are considered, no longer partic-
ular shapes of the tip can be taken into account and also
"atom-like" features of the tunneling current cannot be
reproduced. However, in terms of such an approxima-
tion systems re‡ecting realistic thickness parameters can
be investigated on a computationally surmountable level.
For this very reason in here use is made of the computa-
tional simpli…cations provided by two-dimensional trans-
lational symmetry.
0 2 4 6 8 10 12
0
2
4
6
8
10
Cr
Pt
Co
sample
tip
c
h
a
r
g
e
s
layer
FI G. 1: (color o nline) "Stacking together" systems I and II.
Shown are the charges/layer in the vicinity of the vacuum
barrier.
Finally, one has to realize that with all reorientations
of the magnetization enforced by an e xternal magnetic
eld a particular dynamics is connected, implying, e.g.,
that certain parts of an experimental STM spectrum
(di=dV versus applied magnetic eld) correspond to fast
reorientations of the magnetization and others to slower
processes. For this reason also estimates of the switch-
ing times are presented as based on the Landau-Lifshitz-
Gilbert equation by using internal elds calculated on an
ab-initio level.
Clearly, a theoretical description of experimental
di=dV curves with respect to the applied sample bias
(voltage) is even more complicated, since then also
current-induced changes in the orientation of the magne-
tization have to be taken into account. Up-to-now only
a formal discu ss ion of how to deal with this problem in
the presence of spin-orbit cou pling was prese nted. [21]
In the following sections rst the construction of the
applied scattering potentials is described, then the evalu-
ation of the hypersurfaces of the free energy and of di¤er-
ence conductivities (contrast, di=dV ) is discussed, includ-
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ing a formal scheme of interpretation of these di¤erence
conductivities and an estimate for the dynamics involved.
Only then the ultimate goal of the present investigations
is achieved, namely displaying theoretical STM spectra
in terms of di¤erence conductivities (contrast) versus free
energy (applied external magnetic eld), i.e., obtaining
curves that directly can be compared to experimental
di=dV data when applying an external magnetic eld.
STACKING SYS TEMS
Consider for example the system Pt(111)/Co
n
, i.e.,
a Pt(111) surface covered by n complete overlayers of
Co. The magnetic properties (spin and orbital moments,
magnetic anisotropy energy) of a free surface of Co ad-
layers or of Co clusters on Pt(111) were already studied
extensively in the past [22–24]. However, viewed as sam-
ple system in an STM experiment, the situation becomes
less straight forward, since as already was mentioned a
tip with a lead has to be included. In using for example a
typical Cr/W tip one obviously is faced with the problem
to deal theoretically with a system of the following kind,
Pt(111)
| {z }
semi-nite
/Pt
m
Co
n
Vac
3
Cr
15
W
22
| {z }
Cu
t
tip
/Cu-lead
| {z }
semi-nite
(1)
in which the thickness parameters are given in terms of
monolayers. Fortunately, because of the vacuum barrier
the sample subsystem is only very weakly coupled to the
tip subsystem. It is therefore tempting to rst calculate
selfconsistently both subsystem as free surfaces,
Pt(111)/Pt
m
Co
n
Vac
r
| {z }
free surface I
, Vac
s
Cr
15
W
22
Cu
t
/Cu(111)
| {z }
free surface II
(2)
m 12; r; s 3; t 15
for an illustration see Fig. 1, and then "stack" them to-
gether in the following manner
Pt(111)/Pt
m
Co
n
Vac
2
| {z }
I: sample subsystem,
1
/ Vac
1
Cr
15
W
22
Cu
t
/Cu(111)
| {z }
II: tip subsystem,
2
(3)
Independent calculations for free surfaces imply that, dis-
regarding possible relaxation ects in the surface-near
region, for the sample subsystem the lattice spacing and
Fermi energy of bulk fcc Pt has to be used, while for the
tip subsystem the corresponding quantities for fcc Cu ap-
ply. In terms of charges in the vacuum barrier the error
to be encountered by s uch a stacking procedure is of the
order of a few thousandth of an electron. For example,
treating the subsystems of the system listed in (2) as
free surfaces one nds that the charge in the second vac-
uum layer of Pt(111)/Pt
m
Co
n
Vac
r
amounts to 0.00115
electrons, while for Vac
s
Cr
15
W
22
Cu
t
/Cu(111) the corre-
sponding charge is 0.00556 electrons. Clearly, for r; s > 2
the corresponding charges are substantially less.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
-150
-100
-50
0
50
100
150
200
250
300
σ
i
z
z
(
9
0
,
0
)
(
S
I
)
x
1
0
4
δ (mry)
FI G. 2: (color online) Numerical continuation of local der-
ence conductivities
i
zz
(
1
; 0),
1
= 90,
(r)
1
;
(r)
2
= 0, see
Eq. (9), to the real axis for Pt(111) covered by a single layer
of Co. Diamonds de note the Co layer, squares, up- and down
tri angles in turn the rst, second and th ird Pt layer beneath,
circles refer to the rst vacuum layer ("surf ace state").
To consider as indicated in (3) the selfconsistent scat-
tering potentials of the two su bsystems as a single set of
such scattering potentials to be used for an evaluation of
electric properties in terms of the Kubo equation and of
the free energy has to be regarded as a simple approx-
imation for the joining up of two Green’s functions of
di¤erent spectral properties to a single one for the whole
system; as an attempt to cope with spin-p olarized STM
experiments in which the two leads are of di¤erent mat-
ter.
FREE ENERGIES
Based on the assumption that the tip system is su p-
posed to add little to the free energy (at zero temperature
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