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Phys. Status Solidi B 250, No. 11, 2444–2451 (2013) / DOI
10.1002/pssb.201350002 p s sbasic solid state physics
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Part of Special Issue onQuantum Transport at the Molecular
Scale
The role of the tip symmetry on theSTM topography of
�-conjugatedmoleculesBenjamin Siegert*, Andrea Donarini, and Milena
Grifoni
Institute of Theoretical Physics, University of Regensburg,
93040 Regensburg, Germany
Received 4 June 2013, revised 23 September 2013, accepted 30
September 2013Published online 30 October 2013
Keywords electronic transport, molecular electronics, reduced
density matrix, scanning tunneling microscopy
∗ Corresponding author: e-mail
[email protected], Phone: +49 941 943 2044,
Fax: +49 941 943 2038
We present an STM theory based on the reduced density
matrix(RDM) formalism which is able to describe transport
proper-ties of an STM junction for �-conjugated molecules on
thininsulating films. It combines a very popular derivation of
STMtunneling matrix elements [1], based on Bardeen’s
tunnelingformalism [2], with a generalized master equation approach
forinteracting molecular systems. We show that this method
allowsthe efficient implementation of different tip symmetries in
STMsimulations. With the example of hydrogen phthalocyanine(H2Pc),
we study the influence of s- and p-wave tip sym-metries on the
constant-height current maps of �-conjugatedmolecules.
Constant-height STM images evaluated at the cationic reso-
nance of H2Pc. Left image computed by using an s-wave tipand
right image for a linear combination of px and py states.
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction Since its invention in 1982 by Binningand Rohrer
[3], scanning tunneling microscopy has beengathering enormous
attention in- and outside the physicscommunity. There already exist
many theories describingSTM junctions [4–11], and a comprehensive
review can befound in Ref. [12]. One of the most famous theories of
STMwas initially proposed by Tersoff and Hamann [13–15] basedon the
theory of tunneling of Bardeen [2]: They modeled thetip as a
spherical potential well and showed that the tunnelingcurrent is
proportional to the local density of states of the sam-ple.
Although not capable of explaining atomic resolution inSTM
experiments, this theory laid the foundations for theunderstanding
of STM images. A more sophisticated descrip-tion of the tip-sample
tunneling was proposed by Chen [1]which took into account higher
angular momentum statesand therefore different tip symmetries. A
major benefit ofhis proposal is of rather technical nature; he
concluded in his“derivative rule” that tunneling matrix elements
involving agiven tip symmetry are proportional to corresponding
spatial
derivatives of the sample wavefunction evaluated at the apexof
the tip. He further stated that higher corrugation amplitudesand
thus atomic resolution can be ascribed to dz2 -tip symme-tries
[16]. In recent years, STM experiments with moleculeson thin
insulating films [17–19] are enjoying increasing inter-est. The
insulating layers, only few ångström thin and grownon the
conductive substrate, are effectively decoupling themolecular
electrons from the electrons in the substrate. Thus,there is
essentially no hybridization of substrate and samplewavefunction,
which allows to study the electronic propertiesof the pristine
molecule at low temperatures.
In this work we combine the reduced density matrix(RDM)
formalism [20] with the STM tunneling theory ofChen [1]. The RDM
formalism is a powerful tool to inves-tigate the transport
properties of interacting nanosystemsweakly coupled to leads. It is
applicable also to molecularquantum dots [21–23] where it can be
used to study many-body quantum interference in three-terminal [24]
and STMsetups [25, 26]. Its combination with Chen’s theory
allows
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Phys. Status Solidi B 250, No. 11 (2013) 2445
Figure 1 (Left) Artistic view of the STM setup. (Right) Sketch
ofthe potential profile of the STM setup along the direction
perpen-dicular to the molecular plane.
us to consider STM junctions containing weakly coupledmolecules
and tips with different symmetries.
We especially focus how s-wave, p-wave, and mixed sp-wave tip
states affect the structural pattern of nodal planesand orbital
lobes (as shown in the figure in the abstract) inSTM images.
The outline of this paper is as follows: in Section 2
weintroduce our transport theory for �-conjugated moleculeson thin
insulating films in STM junctions together with theHamiltonian of
the system. We derive the tunneling matrixelements entering the
equations of motion for the reduceddensity matrix of the system and
show how to compute thetunneling current. In Secion 3, the results
of our calculationsare provided and discussed. Finally, in Section
4 we presentour conclusions.
2 Model Our system consists of a �-conjugatedmolecule on a thin
insulating film coupled to an STM tipand can be described by the
Hamiltonian
H = Hmol + HS + HT + Htun. (1)
The presence of the insulating layer allows to
neglecthybridization of the molecular wavefunction with the
wave-function of the substrate and thus to work in the weakcoupling
sequential tunneling regime. The Hamiltonian ofthe molecule Hmol =
H0 + Vee is given by
Hmol =∑
ν
�νd†νd
ν+
∑〈ν,ν′〉
bνν′d†νd
ν′ + Vee, (2)
where the multi-indices ν, ν′ include the atomic site rα =(xα,
yα, zα), the orbital index lm ∈ {1s, 2s, 2px, 2py, 2pz} of
ahydrogen-like atomic orbital and the spin index. We set
thez-coordinate of the molecule to d, where z = 0 defines
themetal–insulator interface (see Fig. 1).
The operator d†ν
creates an electron in the molecule withthe set of quantum
numbers ν with the onsite energy �ν. Thevalues for the different
onsite energies were taken from Ref.[27]. We consider the set of
all 2s (1s for hydrogen),
2px and 2py orbitals as the �-system, and consequentlythe set of
2pz orbitals as the �-system. The hopping parame-ters bνν′ are
calculated using the Slater–Koster LCAO method[28] with the overlap
parameters taken from Refs. [27, 29]and the geometrical parameters
taken from Refs. [30–32]. Forthe electron–electron interaction Vee
we are using a constant-interaction term [33]:
Vee = U2
(N − N0)2 , (3)
where U quantifies the strength of the Coulomb interaction.N =
∑
νd†
νd
νand N0 is the number of electrons occupying
the neutral molecule. Under consideration that the
grand-canonical energy of the molecule,
EG = 〈HG〉 = 〈H − μGN〉, (4)
with μG being the chemical potential of the leads in
equilib-rium, has to be minimal in its neutral state, we adjusted
theparameter U and renormalized the single-particle eigenener-gies
by a constant value. This criterion only fixes a rangeof possible U
and single particle shifts. The final valueshave been obtained by
fitting our transport calculations toexperimental results [34, 35].
Image charge effects are knownto produce such renormalizations of
onsite energies andCoulomb interactions [36]. A microscopic
evaluation of thiseffect lies outside the scope of this paper and
will be consid-ered for later work.
The substrate and the tip are assumed to act as reservoirsof
noninteracting electrons, Hη =
∑kσ
�η
kc†ηkσcηkσ , where η =
S, T denotes substrate or tip. The tunneling Hamiltonian
Htundescribes tunneling between the molecule and substrate
andbetween the tip and the molecule:
Htun =∑ηkjσ
tη
kjc†ηkσdjσ + h.c., (5)
where η again denotes substrate or tip, k = (kx, ky, kz) is
thewavevector of an electron in lead η, σ is the spin of the
elec-tron and the index j denotes a molecular orbital (MO) ofthe
�-system. The tunneling amplitudes tηkj depend on thewavevector k
and the MO index j.
The single-particle MOs of the system are obtained bynumerical
diagonalization of H0. Due to the planar geometryof H2Pc [30, 37,
38] the �-system is decoupled from the �-system and thus not
influenced by the presence of the centralhydrogen atoms. The
reduced symmetry can be recoveredalso in the �-system if we
introduce a spatial modulationof the on-site energies ��
αassociated to the electron density
of the filled �-orbitals. For this purpose, we introduce
theMulliken charge [39] relative to all occupied �-orbitals at
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2446 B. Siegert et al.: Tip symmetry and STM topography
each atomic site:
QMα
= 2e∑j, lm
|c(α,lm)j|2, (6)
where e is the absolute value of the electron charge and
c(α,lm)jis the LCAO coefficient of the orbital lm at the atomic
site αin the MO j. The Mulliken charge reflects the
non-uniformdelocalization of the MOs over all atomic valence
orbitalsin the �-system. If we exclude the electron occupying the
pzorbital, an isolated carbon (nitrogen) atom has 3(4)
�-likevalence electrons. We denote the associated charge Q0
αthe
expected Mulliken charge. From the difference between
theexpected and the actual Mulliken charge one can deduce alocal
variation of the onsite energy for the valence electronin the pz
orbital at site α:
Δ�α
= 2QMα
− Q0α
Qeffα
�r��
α, (7)
where �r is the dielectric constant of the underlying
insulator.This variation is derived as follows: We assume ��
αto be the
energy of a second shell orbital of a hydrogen-like atom
withcharge Qeff
α,
��α
=(
Qeffα
e
)2EH1 (e)
22= EH2 (Qeffα ), (8)
where EH1 (e) = −13.6 eV. The non-uniform delocalization ofthe
�-orbitals introduces a variation of the expected Mullikencharge δα
=
(QM
α− Q0
α
)and thus a screening of Qeff
α:
EH2
(Qeff
α+ δα
�r
)− EH2 (Qeffα ) ≈ 2
EH1 (e)
22e2�rQeff
αδα (9)
= 2QMα
− Q0α
Qeffα
�r��
α. (10)
Note that the variation of the charge has been screened bythe
underlying insulator. Using these energy corrections, wecan
establish an effective Hamiltonian for the �-system:
H�eff =∑
α
(��
α+ Δ�
α
)d†
αd
α. (11)
Diagonalizing this effective Hamiltonian yields the MOs ofthe
�-system:
〈r|jσ〉 =∑
α
cαj〈r|ασ〉 =∑
α
cαjpz(r − rα) uσ, (12)
where �j is the energy of the MO j and 〈r|ασ〉 denotes
apz-orbital with spinor component uσ centered at the atomicsite
rα,
pz(r − rα) = Nα(z − zα) e−Qeffα +
δα�r
2a0|r−rα|, (13)
with normalization Nα and a0 being Bohr’s radius. The tun-neling
region between the substrate and the molecule ismodelled as a
one-dimensional potential barrier. Due to thez-localization of the
molecular orbitals one can prove that forthe substrate tunneling
amplitudes it holds [25]:
tSkj = �j〈Skσ|jσ〉 = �j∑
α
cαj〈Skσ|ασ〉, (14)
where |Skσ〉 is an eigenstate of the substrate Hamiltonian.The
latter is modelled as a quantum well, only confined inz-direction,
see Fig. 1. A product ansatz for the substratewavefunction yields
plane waves in the xy-plane and the solu-tion of a one-dimensional
finite potential well in z-direction:
〈r|Skσ〉 = 1√S
eikxx+ikyy Ψkz (z) uσ (15)
with uσ being its spinor part and S the surface area of
thesubstrate. The wavefunction in z-direction reads
Ψkz (z) = Ω
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
− sin(kzz0)κ2 + k2
z
2κkzeκ(z−z0), z < z0,
cos(kzz) − κkz
sin(kzz), z0 ≤ z ≤ 0,e−κz, z > 0.
(16)
For the wavevectors, we have k2x/y/z
= 2m�2
�x/y/z and κ2 =2m�2
(�SF + φS0 − �z) for the decay constant of the evanescentparts
of the piecewise-defined wavefunction.
Consequently, the matrix element between substratewavefunction
and a pz orbital in Eq. (14),
〈Skσ|ασ〉 =∫
d3re−ikxx−ikyy√
SΨkz (z)pz(r − rα), (17)
can be calculated as
〈Skσ|ασ〉 = Vα(kz, k‖, κ) e−ik‖·rα , (18)
where the scalar product k‖ · rα = kxxα + kyyα results fromthe
shifting of the integrand in Eq. (17) by +rα and thefunction Vα(kz,
k‖, κ), equivalent to OS(k) in Ref. [25], con-denses the results of
the integration. Ultimately, the tunnelingamplitudes for the
substrate read:
tSkj = �j∑
α
cαjVα(kz, k‖, κ) e−ik‖·rα . (19)
For an even more detailed description of the derivation of
theeigenstates of the substrate Hamiltonian and the calculationof
the tunneling amplitudes for the substrate, we refer to Refs.[25,
26], respectively.
For the derivation of the tip tunneling amplitudes, wefollow in
this paper the derivation proposed in Ref. [1] rather
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Phys. Status Solidi B 250, No. 11 (2013) 2447
Table 1 Tunneling matrix elements for the tip for l = 0 (s
state), l = 1 (p states), and l = 2 (d states).
tip state (lc) tunneling amplitude Mlckj tip state (lc)
tunneling amplitude Mlckj
s2π�2Cs
mκψ(rtip) dz2
2π�2Cz2
mκ3
(∂2
∂z2tip− 1
3κ2
)ψ(rtip)
pz2π�2Cz
mκ2
∂
∂ztipψ(rtip) dxz
2π�2Cxzmκ3
∂2
∂xtip∂ztipψ(rtip)
px2π�2Cx
mκ2
∂
∂xtipψ(rtip) dyz
2π�2Cyzmκ3
∂2
∂ytip∂ztipψ(rtip)
py2π�2Cy
mκ2
∂
∂ytipψ(rtip) dxy
2π�2Cxymκ3
∂2
∂xtip∂ytipψ(rtip)
dx2−y22π�2Cx2−y2
mκ3
(∂2
∂x2tip− ∂
2
∂y2tip
)ψ(rtip)
than the one in Ref. [26]. The advantage of this approach isthat
it is independent of the explicit form of the wavefunc-tion inside
the tip; it rather models the exponentially decayingpart of the tip
wavefunction and allows to consider differentsymmetries of tip
wavefunctions, which are reflected in theform of their vacuum
tails. It is based on Bardeen’s tunnel-ing theory [2], which states
that a tunneling matrix elementbetween two wavefunctions χ and ψ
living in two differentspatial regions can be represented by the
integral
M(χ, ψ) = − �2
2m
∫Σ
dS (χ∗∇ψ − ψ∇χ∗) , (20)
where Σ is the separation surface of the two regions. Fol-lowing
Ref. [1], we write the wavefunction of the tip χk(r)in a separation
ansatz as a linear combination of real-valuedspherical harmonics
(cubic harmonics) Ylc(θ, φ) and spheri-cal modified Bessel
functions of the third kind kl(κρ) [40]:
χk(r) =∑
l,cClcχ
lck (r), (21)
χlck (r) := Nlckl(κρ)Ylc(θ, φ), (22)
where κ =√
2m�2
(�TF + φT0 − �k), ρ = |r − rtip|. The Clc aredimensionless
mixing coefficients and the Nlc are ensur-ing that the resulting
tunneling matrix element has the rightdimension of energy. They are
calculated using a cutoff of κ−1
for the lower integration limit, since the kl(x) are not
regularat x = 0 for l > 1. In the present work we are
consideringan idealized choice of the coefficients Clc, restricting
to pures, pure p, or equally mixed s and p states. In order to
betterdescribe realistic tips the Clc can be determined from
firstprinciple calculations, following, e.g., the lines of Ref.
[41].By using the recurrence relations of the spherical
modifiedBessel functions and relating k0(κρ) to the Green’s
functionof the spherical Helmholtz equation, Ref. [1] arrives at
his“derivative rule,” stating that a tunneling amplitude
involvingtip states with symmetries different from s-wave
symmetriescan be related to the respective spatial derivatives of
the sam-
ple wavefunction, evaluated at the position of the tip.
Finally,by inserting Eq. (22) into Eq. (20) and following the
pro-cedure in Ref. [1], the tunneling amplitudes for the tip
areobtained as:
tTkj =∑
l,c
ClcMlc
kj =∑
l,c
ClcM(χlc
k , ψj). (23)
Tunneling amplitudesMlckj for s states (l = 0), p states (l =
1),and d states (l = 2) are listed in Table 1.
The dynamics of our system is calculated using thereduced
density matrix approach [20]. We give in this workonly a short
overview over the derivation of the equa-tion of motion for the
reduced density matrix (RDM). Amore detailed discussion can be
found for example in Refs.[23–25]. The starting point is the
density matrix of the com-plete system in the interaction picture.
Its time evolution isdescribed by the Liouville–von Neumann
equation,
i�dρI(t)
dt= [HItun(t), ρI(t)] (24)
with the tunneling Hamiltonian acting as a perturbation. Eq.(24)
has the formal solution
ρI(t) = ρI(t0) − i�
∫ tt0
dt1 [HI
tun(t1), ρI(t1)]. (25)
By reinserting Eq. (25) into Eq. (24) and tracing out thedegrees
of freedom of the substrate and the tip, we obtainthe generalized
master equation:
ρ̇Ired(t)
= − i�
trS,T{
[HItun(t), ρI(t0)]
}
+(
i
�
)2 ∫ tt0
dt1 trS,T{ [
HItun(t), [HI
tun(t1), ρI(t1)]
] },
(26)
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2448 B. Siegert et al.: Tip symmetry and STM topography
which describes the time evolution of the RDM up to secondorder
in the tunneling Hamiltonian. In the secular approxima-tion, after
retransforming the reduced density matrix to theSchrödinger
picture we project it onto subblocks of particlenumber N and energy
E as
ρNE = PNE ρSred PNE, (27)
where we used the projection operator
PNE =∑
j
|NEj〉〈NEj|. (28)
Equation (26) finally becomes:
ρ̇NE = −∑
η
∑i, j
∑E′
{
Γ η+ij
(E′ − E)PNE di PN+1E′ d†j ρNE
+ Γ η−ij
(E − E′)PNE d†i PN−1E′ dj ρNE
− Γ η−ij
(E′ − E)PNE di ρN+1E′ d†j NE
− Γ η+ij
(E − E′)PNE d†i ρN−1E′ dj NE
}. (29)
Furthermore, we restrict our transport calculations to
themany-body ground states of the molecule. The rate Γ η±ij (�E)in
Eq. (29) is given by
Γ η±ij
(�E) = 2π�
∑k
(tη
ki
)∗tη
kjf±η
(�E)δ(�ηk − �E), (30)
where f +η
(�k) is the Fermi distribution of an electron inlead η = (S, T )
with chemical potentials μT = −φT0 + ceVb,μS = −φS0 − (1 − c)eVb
and f −η (�k) = 1 − f +η (�k). Notethat in STM setups there is an
asymmetric bias drop c acrossthe junction.
Its value can be estimated as follows: Let us con-sider an empty
capacitor (substrate–tip) with capacitanceC0. After replacing half
of the vacuum inside the capaci-tor by a dielectric with relative
permittivity εr (substrate +insulatinglayer − tip), the effective
capacitance of this sys-tem then reads Ceff = εr1+εr C0, yielding
the parameter for thebias drop as c = Ceff
C0= εr
1+εr . This estimation relies on theassumption of equal
tip–molecule and molecule–substratedistances. They are comparable
in typical STM setups withinsulating films. Taking the relative
permittivity of NaClεNaClr = 5.92 [42] yields c = 0.86. At present
we neglect apossible dependence of the bias drop on the tip and
substrateworkfunction. We believe that this effect would be
capturedby a self-consistent solution of the Poisson equations for
thejunction that goes beyond the scope of this publication.
Equation (29) can be put in a shorter form,
ρ̇red = Lρred. (31)
The operator L is called the Liouvillian of the system.
Wecalculate the stationary solution ρstatred which is given by
thenullspace of L. Ultimately, the stationary current
throughcontact η is calculated as
〈Iη〉 = trsys{
Iηρstatred
}, (32)
where the current operator Iη can be obtained from the
time-derivative of the average charge on the system and is
definedas [25, 26]:
Iη = e∑i, j
∑N
∑E, E′
PNE{
diPN+1E′ d†j Γ η+ij
(E′ − E)
− d†iPN−1E′ dj Γ η−ij
(E − E′) }PNE. (33)
3 Results In all calculations we used a substrate–molecule
distance of d = 7 Å and the workfunctions of thesubstrate and the
tip were chosen to be equal: φT0 = φS0 =−μG, see also Fig. 1. In
Fig. 2, we show the I–V characteris-tics and a stability diagram of
H2Pc. From Fig. 2a, we are ableto inhere the state of the system
according to the values of theworkfunction and the bias voltages.
In analogy with stabilitydiagrams associated to single electron
transistors, what wesee in the central part of this figure is a
Coulomb diamond.Varying the workfunction acts as ramping an
effective gatevoltage. Transitions occur if the chemical potential
of thesubstrate μS = μG − (1 − c)eVb or the chemical potential
ofthe tip μT = μG + ceVb matches the chemical potential of
themolecule μN = EN − EN−1. At zero bias, going left throughthe
diamonds increases the particle number in the system,while going
right decreases it. Due to the asymmetric biasdrop c, the Coulomb
diamond is tilted. Its steeper bordersmark transitions where the
chemical potential of the substratealigns with the chemical
potential of the system, while its flat-ter borders indicate a
resonance with the chemical potentialof the tip.
The bias trace in Fig. 2b, taken at μG = −2.9 eV,shows two
resonances at Vb = 0.13 V and Vb = −0.8 V. Thecorresponding
constant-height STM images in both casesresemble the LUMO orbital
of H2Pc. The reason for thisis the transition from an N to an N +
1-particle state (seeFig. 2a), which is mediated by the LUMO single
particleMO. The I–V characteristic in Fig. 2c corresponds instead
toμG = −4 eV. The constant-height STM images correspond-ing to the
two resonances now resemble different orbitals.The current map at
positive bias still reflects the LUMO asthe system undergoes an N
to N + 1 transition, while at neg-ative bias it reflects the shape
of the HOMO consistently withthe corresponding N − 1 to N
transition. This regime, wherethere is an anionic resonance at
positive bias and a cationicresonance at negative bias is usually
the most common inSTM experiments with molecules on thin insulating
films[17, 18]. In the following we will discuss the influence
ofdifferent tip symmetries on the STM topography of H2Pc,
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Phys. Status Solidi B 250, No. 11 (2013) 2449
Figure 2 Stability diagram (a) and I–V characteristics (b, c)
for H2Pc on a thin insulating film of thickness d = 7 Å and a tip
height of4.3 Å. In the central region in panel a transport is
blocked due to Coulomb blockade and the molecule is neutral, with N
= 42 electronspopulating the �-system. The current traces shown in
panels b and c correspond to the two dashed lines in panel a. The
small insets showconstant-height STM images computed at the bias
voltage of the respective resonance, indicated by the arrows.
with the workfunctions of the tip and the substrate chosen tobe
φT0 = φS0 = −μG = 4 eV.
For the s-wave tip symmetry the recorded constant-height STM
images, Fig. 3a and d, are showing the
characteristic patterns of nodal planes and orbital lobes ofthe
corresponding MOs.
A striking difference to the STM images with the s-wavetip can
be seen in Fig. 3b and e, where constant-height STM
Figure 3 Constant-height STM images of H2Pc for different tip
configurations at a tip-sample distance of 5 Å and a thickness of
theinsulating layer d = 7 Å. Panels a–c show current maps recorded
at the cationic resonance at Vb = −1.1 V, while panels d–f show
currentmaps recorded at the anionic resonance at Vb = 1.4 V. The
white bar indicates a length of 5 Å.
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2450 B. Siegert et al.: Tip symmetry and STM topography
images are shown for a tip with px and py symmetry. Thenodal
plane patterns of the MOs disappeared; instead thecurrent now has
maxima and minima at the positions ofthe nodal planes and of the
orbital lobes, respectively. Byconsidering the tunneling matrix
elements as the derivativesof the molecular wavefunction in the x-
and y-directions,this result is not surprising. However, as put
forward inRef. [19], there exists a nice empirical explanation of
thesefindings, namely that if the tip is directly placed above
apz-orbital of the molecule, the contributions of the p-orbitalin
the tip are cancelling each other due to their phases,differing in
their signs. By placing the tip above a nodalplane, that is between
two pz orbitals which differ in theirsigns, there is no such
cancellation of the contributions ofthe p-orbital in the tip and
tunneling is occuring.
Finally, in Fig. 3c and f we show the results of a simu-lated
mixed s–p wave tip, which we modeled as an equallyweighted linear
combination of s, px, and py orbitals. Theyshow the non-uniform
competition of the contributions ofthe s and the p orbitals in the
tip. In the inner parts of themolecule there are still maxima in
the current occuring atthe positions of the nodal planes. In the
outer parts, how-ever, the contribution of the s-wave tip state
outweighs thep-wave contribution, and there are maxima at orbital
lobes.Additionally, in contrast to Fig. 3b and e, the overall
imageappears more smeared out, what also can be attributed to
thes-wave part of the tip state.
4 Conclusions In this work we have presented a semi-quantitative
transport theory for STM-setups involving �-conjugated molecules on
thin insulating films. To this endwe derived tunneling matrix
elements that account for theconditions under which tunneling
between the molecule andthe contacts takes place. For the substrate
we started froma microscopic point of view [26], using the
solutions of aSchrödinger equation for a potential well in the
z-direction.For the tunneling matrix elements of the tip we used
theapproach in Ref. [1], which only considers the shape of thetip
wavefunction in the vacuum between tip and sample, butallows for
the inclusion of spatially different tip states.
Finally, we derived the generalized master equation forthe
reduced density matrix in secular approximation withthe tunneling
Hamiltonian treated in second order. The cur-rent through the
system was calculated using the stationarysolution for the reduced
density matrix.
In order to model the investigated molecule we used
theSlater–Koster LCAO approach [28] to get its
single-particleeigenstates and a constant-interaction model to
calculate theenergies of the many-particle ground states of the
molecule.
We presented our numerical calculations for the case ofhydrogen
phthalocyanine. As in a recent experimental work[19], which
addressed the same question, we confirm thatdifferent tip
symmetries have strong influence on the topog-raphy of STM
images.
All in all we showed that our formalism is able tomirror
experimental findings quite well, despite the limi-tations induced
by our applied approximations. A natural
improvement of our model is a better treatment of screeningand
image charge effects induced by the contacts, as theycould be
responsible for negative differential conductanceand the breaking
of molecular symmetries [36]. Also therole of vibronic excitations
would be an interesting topic toaddress. For example, it has been
experimentally shown inRef. [18] for the case of a naphthalocyanine
that vibrons canbe crucial in the current-induced switching of the
positionsof the central hydrogen atoms of the molecule.
Acknowledgements Sincere thanks are due to Jascha Reppfor
fruitful discussions. We also acknowledge financial support bythe
DFG within the research programs SPP 1243 and SFB 689.
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