Visualization of atomic-scale phenomena in superconductors Andreas Kreisel, Brian Andersen
Niels Bohr Institute, University of Copenhagen, 2100 København, Denmark
Peayush Choubey, Peter Hirschfeld
Department of Physics, University of Florida, Gainesville, FL 32611, USA
Tom Berlijn
Center for Nanophase Materials Sciences and Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
arXiv:1401.7732 arXiv:1407.1846
Scanning tunnelling microscopy Tunnelling current: 4¼e I(V; x; y; z) = ¡ ½t (0)jM j2 ~
STM tip
eV
½(x; y; z; ²)d²
0
Local Density Of States (LDOS) of sample at given energy
sample e.g. SC J. Hoffman 2011 Rep. Prog. Phys. 74 124513 (2011)
density of states of FeSe TC=8 K
Z
J. Tersoff and D. R. Hamann, PRB 31, 805 (1985)
Topograph of Fe centered impurity in FeSe at V=6 mV
Song et al., Science 332, 1410 (2011)
Can-Li Song, et al. PRL 109, 137004 (2012)
LDOS and conductance map: Zn impurity in BiSCCO at V=-1.5 mV
Pan et al., Nature 403, 746 (2000)
Theory: State of the art methods T-matrix ●
Hamiltonian
H = H0 + HBCS + Himp
impurity scatterer (non)magnetic potential / T2 scatterer
band structure kinetic energy
H0 =
X
RR
X
cyR ¾ cR ¾
Himp =
superconductivity gap function / pairing
0 ;¾
¡ ¹0 ●
tR R 0 cyR ¾ cR 0 ¾
HBCS = ¡
R ;¾
T-matrix calculations
X
R ;R
0
X
Vimp cyR ¤ ¾ cR ¤ ¾
¾
¢R R 0 cyR " cyR 0 # + H:c:;
“resolution”: one pixel Zn impurity in BSCCO per elementary cell
–
lattice Green's function
–
Local Density of States (LDOS)
minimum on impurity, maximum at NN
T-matrix calculation + Bi-O filter function Martin et al., PRL 88, 097003 (2002)
Theory: State of the art methods Bogoliubov-de Gennes (BdG) ● ●
Hamiltonian H = H0 + HBCS + Himp self-consistent solution in real space (NxN grid, determine gaps) ¢R R 0 = ¡R R 0 hcR 0 # cR " i
●
eigenvalues En, eigenvectors (un,vn)
●
lattice Greens function
G¾ (R ; R 0 ; !) =
Xµ n
n¾¤ un¾ u R R0
! ¡ En¾ +
i0+
+
vRn¡¾ vRn¡¾¤ 0
¶
! + En¡¾ + i0+
BdG+Wannier method ●
first principles calculation – –
band structure
H0 =
Wannier functions: wavefunctions in real space
X
tR R 0 cyR ¾ cR 0 ¾
R R 0 ;¾
Fe Se ●
continuum Green's function 0
G(r; r ; !) =
X
R ;R continuum position
¡ ¹0
elementary cell of FeSe
X
cyR ¾ cR ¾
R ;¾
Wannier function with phases centered at Fe(I)
Fe(I)-dxy
G(R ; R 0 ; !)wR (r)wR¤ 0 (r0 )
0
lattice Greens function
nonlocal contributions
local density of states (LDOS)
½(r; !) ´ ¡
1 Im G(r; r; !) ¼
Application to FeSe ●
●
homogeneous superconductor
lattice LDOS (conventional: 1 pixel per Fe atom)
- 2 meV
+ 2 meV
FeSe: Results 4¼e I(V; x; y; z) = ¡ ½t (0)jM j2 ~
●
continuum density of states –
Z
eV
½(x; y; z; ²)d²
0
at Fe plane C4 symmetry! 2 meV
–
at STM tip position
-2 meV
C2 symmetry!
+2 meV
+30 meV
FeSe: Comparison to experiment STM topography on FeSe with Fe-centered impurity C2 symmetry
C4 symmetry
BdG
BdG+W
experiment Can-Li Song, et al. PRL 109, 137004 (2012)
Application to BSCCO ● ●
first principles calculation (surface) 1 band tight binding model: 1 Wannier function
Cu dxy
band structure
NN apical O tails
O Cu Bi Sr Ca
at surface: only contributions to NN
Fermi surface
BSCCO: Results ● ●
d-wave order parameter
DOS of homogeneous superconductor experiment
Zn impurity resonance at -3.6 meV
Zhu et al., PRB 67, 094508 (2003)
Pan et al., Nature 403, 746 (2000)
resonance at NN
resonance at impurity
Quasi Particle Interference (QPI) ●
Fourier transform of differential conductance maps energy integrated maps: trace back Fermi surface
¤(q) =
Z
¢0
d! Z(q; !) :
0
K Fujita et al. Science 344, 612 (2014)
octet model: 7 scattering vectors between regions of high DOS
QPI simulation no intra-unitcell information
BSCCO: weak potential scatterer
atomic scale local density of states at STM tip position
full information for all scattering vectors spots from octett model
no information beyond first BZ
Fourier transform
Comparison to experiment no large q information
Z(r; !) =
g(r; !) ½(r; !) = g(r; ¡!) ½(r; ¡!)
relative conductance map, Fourier transformation
¤(q) =
Z
¢0
d! Z(q; !) :
0
energy integrated relative conductance maps K Fujita et al. Science 344, 612 (2014)
Recapitulation: BdG+W ●
simple: just a basisX transformation of the Green's function G(r; r0 ; !) = G(R ; R 0 ; !)wR (r)wR¤ (r0 ) 0
R ;R
●
●
●
0
powerful tool for calculation of local density of states at the surface (STM tip position) of superconductors
takes into account interunitcell information and symmetries of the elementary cell and the contained atoms shown to work in –
FeSe: geometric dimer
–
BSCCO: Zn impurity resonance, QPI pattern
Summary Kreisel et al. arXiv:1407.1846
Choubey et al. arXiv:1401.7732
Acknowledgements
