3.3 Investigating the band structure of surfaces using quasiparticle interfer-
3.3.3 Joint density of states and stationary phase approximation
The formation of standing waves can be described in the reciprocal space picture as quasiparticle interferences (QPIs). A standing wave with scattering vector~q forms after elastic scattering between quasiparticle state~kand quasiparticle state~k0:
~q=~k0−~k (3.17)
Figure 3.8 |Quasiparticle interferences. (a) CEC of a parabolic surface state including a possible backscattering vector~qbetween Bloch states with~kand~k0. (b)All backscattering vec-tors end on a circular contour in FT-LDOS.
(a) CEC
In the case of noble metal surface states, which are isotropic and centered at theΓpoint of the Brillouin zone, backscattering processes entail~k = −~k0and result in an interfer-ence pattern with periodicityλ=2π/q=π/k. Therefore, considering backscattering in all directions within the isotropic surface state, the Fourier-transformed local density of states (FT-LDOS) will give rise to a circle in the middle with a radiusq=2k[Figure 3.8].
Understanding the standing wave patterns of more complex band structures requires a more general description beyond simple backscattering processes. For the prediction of FT-LDOS features the joint density of states (JDOS) concept [171, 172] will be briefly explained, which can be used to understand QPI patterns based on the constant energy contour (CEC) of arbitrary band dispersions.
Following the derivation of Simonet al.[172], LDOS oscillations due to quasi parti-cle interferences can be described by linear combinations of the quasipartiparti-cles in Bloch states before and after the scattering event leading to the expression
ρ(E,~r)∝
Here the interference term¯
¯ψ~q
¯
¯
2 is formed by linear combinations of initially unper-turbed Bloch states with wavevectors~kand~k0. The interference term can be written as a Fourier coefficient of the LDOS with
g(E,~q)= Ï
E(~k)=E(~k0)=E
f(~k,~k0,G)~δ(~q−~k+~k0±G) d~ 2kd2k0. (3.20)
G~is a reciprocal lattice vector andf is a weighting factor which controls the scattering probability between two states based on the precise configuration and nature of defects.
If various scatterers of different kind are present, one can assumef to be smoothly vary-ing as a function of~kand~k0andg(E,~q) coincides with the JDOS, which in principle is constructed by counting the number of scattering vectors~q=~k−~k0±G~with the initial
3.3 Investigating the band structure of surfaces using quasiparticle interference mapping 47
kG
kG
Γ
(a) CEC (b) JDOS
qinter qintra
qinter qintra
(c) JDOS
SP-Figure 3.9 | Scattering in graphene.(a)CEC at a binding energy ofE = 1 eV calculated with the tight binding solution in equation2.5. The dashed hexagon resembles the Brillouin zone of graphene.
(b)JDOS calculation according to equation3.21.(c)JDOS with additional stationary phase condition according to equation3.25.
and final states lying on the CEC [172]:
JDOS(E,~q)= Ï
E(~k)=E(~k0)=E
δ(~q−~k+~k0±G) d~ 2kd2k0 (3.21)
The JDOS is connected to the FT-LDOS as it can be measured by STM. However JDOS neglects the scattering probabilities and therefore JDOS gives only a first approximation of the FT-LDOS features. Differences between JDOS and FT-LDOS in experiment will be discussed in the following.
A typical JDOS calculation for graphene scattering at energies away from the Dirac point, where trigonal warping already plays a dominant role, is presented in Figure 3.9 (a-b). The features in JDOS are formed by scattering processes between two points within the same valley (intravalley) or between two adjacent valleys at K and K’ (intervalley).
Features in the JDOS show a sharp contour with filled intensity within the contour. The filled contour in the JDOS calculations in Figure 3.9 (b) depicts differences compared to real experiments where only the contour is observed [72] showing the limitation of a JDOS calculation according to equation 3.21.
In a more elaborate formulation of the scattering problem, one can write the LDOS ρ(E,~r) using the retarded Green’s functionGr(E,~r,~r0) [173]:
ρ(E,~r)= −1 πIm£
Tr£
Gr(E,~r,~r)¤¤
(3.22) Here, the retarded Green’s functionGr(E,~r,~r0) can be viewed as the relative probability amplitude of a particle appearing at~rvia different paths which was previously created at
~r0(by the tip) [173]. Using theT-matrix approximation one can solve the problem includ-ing the precise scatterinclud-ing potential of the impurity and the effects of multiple scatterinclud-ing, given the Hamiltonian of the system is known and the impurity potential is localized
[160]. Finally, the change in LDOSδρcompared to the background valueρ0is given by [174]
δρ(E,~r)=ρ(E,~r)−ρ0(E) (3.23)
= −1 πIm
Ï d2kd2k0
(2π)4 ei(~k−~k0)·~rTrh
Gr0(E,~k)T(E,~k,~k0)G0r(E,~k0)i
. (3.24) HereGr0(E,~k) is the free retarded Green’s function in momentum space andT theT -matrix containing the precise scattering potential. The above expression can be used to calculate the decay of the LDOS oscillations [174]. Instead of giving a comprehen-sive review of the calculations here, the focus is on the integral with the oscillatory term ei(~k−~k0)·~rwhich will only give a contribution to the LDOS oscillations if first the scattering condition~q =~k−~k0±G~is fulfilled and second the phase in the integral is not strongly varying, since then the LDOS oscillation will vanish as a result of destructive interference.
This circumstance is connected to thestationary phase approximationand is discussed in [175, 174]. For large distancer from the impurity the phase oscillates rapidly leading to remaining contributions only for pairs ofstationary points~kand~k0. The condition for stationary point pairs in a measurement with many scatterers can be given in the form [174]
∇~kE(~k)¯
¯
¯~k= − ∇~kE(~k)¯
¯
¯~k0 . (3.25)
The condition requires an antiparallel orientation of gradients, which is reasonable when considering that the gradient describes the group velocityv⊥= ħ−1|∇~kE(~k)|of the bands in solid state physics. In the JDOS calculations the stationary phase condition can be implemented in equation 3.21. An example for the calculation of graphene including the stationary phase approximation (JDOS SP-) is shown in Figure 3.9 (c). One can see that compared to the JDOS calculation without stationary phase condition, the intensity inside the scattering features is suppressed and the result reproduces the experimental situation [72]. The stationary phase approximation gives a more precise prediction of FT-LDOS features.
In the case of parabolic surface state bands the standing waves are formed by backscat-tering with~q=2~kwhere the involved Bloch states~kand~k0are antiparallel as depicted in Figure 3.10 (a). Other scattering vectors with~k06= −~kare not possible due to the violation of the gradient condition in equation 3.25.
Scattering between states with lifted spin degeneracy
JDOS has to be understood as a first approximation of the FT-LDOS, since it does not con-sider the precise configuration of the scatterers and the scattering probabilities between single states. In particular it was shown that scattering can be suppressed due to lifted spin degeneracy. A scattering between points on the CEC with different spin
orienta-3.3 Investigating the band structure of surfaces using quasiparticle interference mapping 49
(a) ∇kE(k)
q1 q2
(c) (b)
q1
q2
qintra
Figure 3.10 | Scattering vectors in systems with and without spin degeneracy. Examples of possible scattering vectors~q1 and suppressed scattering vectors~q2 for the CECs of(a) spin de-generate parabolic surface states including stationary phase approximation and(b)parabolic surface states with Rashba spin splitting. (c) In monolayer graphene with pseudospin (orange arrows) the intravalley scattering~qintrais suppressed.
tion, as they occur in topological insulators or Rashba-split surface states, does not occur since the wave functions before and after the scattering are orthogonal [171, 176, 177].
In topological insulators and Rashba split surface states a fixed relation between the spin and the CEC is found. An example of the CEC for a Rashba spin-split parabolic surface state is depicted in Figure 3.10 (b). For the scattering between states on the CECs, all restrictions from JDOS as discussed above have to be obeyed. In addition, considerable intensity is only expected for scattering between points on contours, where the spin of the involved states~k and~k0 is aligned parallel. In the Rashba-type CEC this is the case for scattering vectors~q1between circles with opposite direction of spin rotation.
A similar selection rule applies for graphene due to the pseudo-spin [72, 77]. Here, backscattering within one valley, referred to as intravalley backscattering is not expected due to the antiparallel alignment of the pseudo-spin texture [Figure 3.10 (c)]. Conse-quently, the intravalley scattering reflex in FT-LDOS predicted by JDOS in Figure 3.9 (c) is expected to be suppressed for perfect monolayer graphene [72, 77].
4 | Experimental setup, additional experimental techniques and sample preparation
4.1 Experimental setups used in this work
The experiments were performed under ultra high vacuum (UHV) conditions in two STM setups from Omicron Nanotechnology: A VT STM/AFM setup [Figure 4.1] and a Cryo-genic STM setup [Figure 4.2]. Both setups consist of independent analysis and prepara-tion chambers. In situsample preparation was performed in the attached preparation chamber. The preparation chambers are equipped with identical instrumentation for
Figure 4.1 | VT STM/AFM by Omicron Nanotechnology. (a) UHV setup includ-ing independent analysis and preparation chambers. Sample characterization meth-ods are STM (1), LEED (2) and PES (3).
(b) Zoom of analysis chamber with STM (1) and LEED (2) equipment.
(a)
Analysis Preparation 1
3
(b)
2
1
51
Figure 4.2 |Cryogenic STM by Omicron Nanotechnology. (a)UHV setup includ-ing 4He cryostat (1), passive pneumatic damping system (2), STM chamber (3) and preparation chamber (4). (b) Zoom of the independent analysis and prepara-tion chambers. The preparaprepara-tion chamber is equipped with a LEED system (5).
(a)
(b)
Analysis Preparation
5
3
1
2 3 2 4
sample preparation including a sample flashing station (electron beam heating) for an-nealing at high temperatures and resistive heating for intermediate anan-nealing tempera-tures. An Ar+ sputter gun for surface bombardment, effusion cells and electron beam evaporators for metal deposition and leak valves for a precise gas inlet are available.
Both setups allow for a quick LEED characterization. The Omicron VT STM/AFM setup is furthermore equipped with a hemispherical electron energy analyzer and excitation sources for PES experiments.