Inversion Techniques for STM Data Analyses

Transcription

1 Inversion Techniques for STM Data Analyses Sumiran Pujari University of Kentucky September 30, 2014

2 Outline of the Talk Philosophy of Inversion Projects : 1 Quasiparticle Echoes 2 Quasiparticle Lifetimes 3 Bosonic Modes in High Tc Superconductors(BSCCO) Conclusion

3 Philosophy What is Inversion? STM Set-up : STM measures Local Density of States (LDOS) n( r, E) n( r, E) Im[G( r, r; E)] (1)

4 Philosophy What is Inversion? STM Set-up : STM measures Local Density of States (LDOS) n( r, E) n( r, E) Im[G( r, r; E)] (1) Extract the most general quadratic Hamiltonian that fits STM data

5 Philosophy What is Inversion? STM Set-up : STM measures Local Density of States (LDOS) n( r, E) n( r, E) Im[G( r, r; E)] (1) Extract the most general quadratic Hamiltonian that fits STM data What does quadratic imply? 1 Only one-body terms like H ij = λ ij c i c j 2 Mandy-body effects are not explicit 3 Various Order Parameters are accounted at Mean-field level, e.g. term like ij c i c j for Superconductivity

6 Example of an Inversion Technique : FT-STS Fourier-Transform Scanning Tunneling Spectroscopy : Above Data from STM on Ruthenate Sr 3 Ru 2 O 7 Jinho Lee et al, Nature Phys Extracts the Dispersion of electrons Physics : Quasiparticle Inteference

7 Quasiparticle Echoes

8 Echo Physics : Quasiparticle Interference Simplest setting : Non-interacting electrons on 2D square lattice STM Tip Sample LDOS(imp) - LDOS(clean) a) Energy in units of t Numerics for LDOS done using Recursion Method (Haydock et al, Litak et al)

9 Echo Physics : Quasiparticle Interference Simplest setting : Non-interacting electrons on 2D square lattice STM Tip Sample LDOS(imp) - LDOS(clean) 1.1 b) Energy in units of t Numerics for LDOS done using Recursion Method (Haydock et al, Litak et al)

10 Echo Physics : Quasiparticle Interference Simplest setting : Non-interacting electrons on 2D square lattice STM Tip Sample LDOS(imp) - LDOS(clean) 0.05 e) Energy in units of t Numerics for LDOS done using Recursion Method (Haydock et al, Litak et al)

11 Echo Physics : Quasiparticle Interference Simplest setting : Non-interacting electrons on 2D square lattice STM Tip Sample LDOS(imp) - LDOS(clean) 0.04 g) Energy in units of t Numerics for LDOS done using Recursion Method (Haydock et al, Litak et al)

12 Echo Physics : Quasiparticle Interference Simplest setting : Non-interacting electrons on 2D square lattice STM Tip Sample LDOS(imp) - LDOS(clean) g) Energy in units of t formally δn(r, E) = ImG 0 (r, r imp ; E)T (E)G 0 (r imp, r; E) Echoes = Energy/time counterpart of spatial Friedel oscillations Numerics for LDOS done using Recursion Method (Haydock et al, Litak et al)

13 Echolocation can use echoes to echolocate impurities LDOS(imp) - LDOS(clean) 0.05 e) Energy in units of t

14 Echolocation can use echoes to echolocate impurities LDOS(imp) - LDOS(clean) c) Energy in t units

15 Echolocation can use echoes to echolocate impurities LDOS(imp) - LDOS(clean) b) Energy in t units

16 Echolocation can use echoes to echolocate impurities LDOS(imp) - LDOS(clean) b) Energy in t units

17 Echolocation can use echoes to echolocate impurities LDOS(imp) - LDOS(clean) b) Energy in t units Anistropic Contours since Anistropic group velocities Magnitude of Echoes : measure of impurity strength

18 Echo Physics for d-wave Superconductors Two qualitatively different quasiparticle group velocities Energy Contours in Brillouin Zone N 30,30;Ω Ω

19 Echo Physics for d-wave Superconductors Two qualitatively different quasiparticle group velocities Energy Contours in Brillouin Zone LDOS(imp) - LDOS(clean) c) Energy in t units

20 Application to Cuprates An opportunity for Gap Energy studies : complementary to Fourier Transform STS method possibly, a better analysis method for energies near gap energy where LDOS data is inhomogeneous instead use echoes to study Quasiparticle interference within one patch A check for Quasiparticle Extinction Experimental issues : O[10] lattice constants O[10] wiggles within gap if gap 40 mev energy resolution << 4 mev need current resolution (wiggle amplitude = 1-2 percent)

21 Application to Cuprates Echolocation of Impurities can be useful distinguishing Ordinary impurity from Anomalous impurity (Nunner et al, 2006) eventually correlate with cuprates chemistry

22 Quasiparticle Lifetimes

23 Motivation Alldredge et al s Experiment, Nature Phys Phenomenological modeling of Lifetime term for High Tc Cuprate, BSCCO Finite Lifetime Rounded Coherence Peaks

24 Motivation Alldredge et al s Experiment, Nature Phys Phenomenological modeling of Lifetime term for High Tc Cuprate, BSCCO Finite Lifetime Rounded Coherence Peaks Inversion Question : How to extract Lifetimes directly from STM data?

25 Lifetime Extraction : e.g. Normal Metal G 0 (k; ω) = 1 ω iη(k, ω) ɛ(k) (2) Want to extract η(k, ω) We might have ɛ(k) from other experiments, Photoemission Studies(ARPES) or FT-STS Idea : Use Quasiparticle Interference around an impurity G(r, r; ω) = G 0 (r, r; ω) + G 0 (r, r imp ; ω) T (ω) G 0 (r imp, r; ω) (3)

26 Lifetime Extraction : e.g. Normal Metal G 0 (k; ω) = 1 ω iη(k, ω) ɛ(k) (2) Want to extract η(k, ω) We might have ɛ(k) from other experiments, Photoemission Studies(ARPES) or FT-STS Idea : Use Quasiparticle Interference around an impurity G(r, r; ω) = G 0 (r, r; ω) + G 0 (r, r imp ; ω) T (ω) G 0 (r imp, r; ω) (3) Experiment measures n(r, ω) Im[G(r, r; ω)] Aim : Extract G 0 (r, r imp ; ω) from STM Data

27 Lifetime Extraction : Kramers-Kronig Step First Step : Convert n(r, ω) to G(r, r; ω) Theoretically, Kramers-Kronig Relation makes that possible Re[G(r, r; ω)] = P Λ= Λ= n(r; x) dx ω x (4)

28 Lifetime Extraction : Kramers-Kronig Step First Step : Convert n(r, ω) to G(r, r; ω) Theoretically, Kramers-Kronig Relation makes that possible Λ= n(r; x) Re[G(r, r; ω)] = P dx (4) ω x Λ= Practically, STM data only over a finite Energy bandwidth Re G LDOS Ω Ω Believe should be workable in interesting Energy Ranges, i.e. Fermi Energy in metal, Nodal Energy in Cuprates, etc.

29 Lifetime Extraction : G 0 (r, r; ω) Guess Step G(r, r; ω) = G 0 (r, r; ω) + G 0 (r, r imp ; ω) T (ω) G 0 (r imp, r; ω) A Minimization Procedure to guess c-number G 0 (r, r; ω) Based on G 0 (r, r imp ; ω) decaying to zero monotonically at large r r imp G r;ω error Window Avg. Correct guess r Good Start Guess : spatial average of G(r, r; ω) over the whole data set

30 Lifetime Extraction : Phase Reconstruction Step G(r, r; ω) = G 0 (r, r; ω) + G 0 (r, r imp ; ω) T (ω) G 0 (r imp, r; ω) With correct G 0, get G 0 (R) only upto a phase This phase crucial to get G 0 (k) upon Fourier Transforming Observation : Phase of G 0 (R) smooth, well-behaved with R Use this to reconstruct phases of G Upon Reconstruction, phase of G 0 half of G 2 0 Algorithm : if R > R then φ principal + 2πm > φ principal + 2πm.

31 Lifetime Extraction : Results From extracted G 0 (R) Fourier Transform G 0 (k) η(k) a) b) c) d) Above a) Input G 0 (k), b) Ideal Extraction (Note blockiness due to finite window of data) Extraction with error added, c) 0.1 % and d) 1 %

32 Remarks There might be data sets out there... Above Expt data set from STM on InAs (semiconductor) surface, K. Kanisawa et al, PRL 86, 3384 (2001).

33 Remarks There might be data sets out there... Above Expt data set from STM on InAs (semiconductor) surface, K. Kanisawa et al, PRL 86, 3384 (2001). Scheme Extendable to Superconducting case...

34 Bosonic Modes in High Tc Superconductors

35 History of Bosonic Modes in Superconductivity 1911 : Superconductivity discovered in Kamerlingh Onnes lab 1957 : BCS Theory BCS Theory didn t get Tc right always! For strong electron-phonon coupling, Renormalization effects were important to estimate correct Tc

36 History of Bosonic Modes in Superconductivity 1911 : Superconductivity discovered in Kamerlingh Onnes lab 1957 : BCS Theory BCS Theory didn t get Tc right always! For strong electron-phonon coupling, Renormalization effects were important to estimate correct Tc Eliashberg Theory : a Self-consistent Strong-coupling theory Got Tc right! Also, consistent α 2 F (ω) of phonons extracted!

37 Bosonic Modes in Bi 2 Sr 2 CaCu 2 O 8+δ Inversion questions : Above typical spectrum from Lee et al, Nature 2006 : STM on BSCCO. Bosonic Mode shown to be O phonon through Isotope effect

38 Bosonic Modes in Bi 2 Sr 2 CaCu 2 O 8+δ Above typical spectrum from Lee et al, Nature 2006 : STM on BSCCO. Bosonic Mode shown to be O phonon through Isotope effect Inversion questions :? Extraction of Boson Frequency? Lee et al : Inflection point before hump - Coherence Peak Energy Above motivated via Inelastic Tunneling Spectroscopy (IETS) Also, feature suggested to be possibly Strong-Coupling SC-like? Extraction of electron-boson coupling strength?

39 Our Model : Weak-Coupling Phenomenology Assumptions : d-wave SC/Bogoliubov Quasiparticles already established H d wave = ɛ(k)c k,σ c k,σ + (k)c k, c k, + h.c. ( G 0 (k; iω n ) 1 iωn ɛ(k) (k) = (k) iω n + ɛ(k) (k) = 0 2 (cos(k x) cos(k y )) Mechanism Unknown : No Speculations Similar to Zhu et al s numerical work (also O. Fisher s group) ).

40 Our Model : Weak-Coupling Phenomenology Assumptions : d-wave SC/Bogoliubov Quasiparticles already established H d wave = ɛ(k)c k,σ c k,σ + (k)c k, c k, + h.c. ( G 0 (k; iω n ) 1 iωn ɛ(k) (k) = (k) iω n + ɛ(k) (k) = 0 2 (cos(k x) cos(k y )) Mechanism Unknown : No Speculations Similar to Zhu et al s numerical work (also O. Fisher s group) Introduce Boson in simplest possible way Boson : Einstein Oscillator Ω(k) = Ω 0 Momentum-Independent Electron-Boson Coupling g(k) g ).

41 Bosonic Self-Energy Effect of Boson on d-wave quasiparticle? Calculate to lowest order in g Σ(iω n ) = T N L p,ω m g 2 D(iΩ m )G 0 ( k q; iω n iω m ) Simplification : Momentum Independence

42 Bosonic Self-Energy Effect of Boson on d-wave quasiparticle? Calculate to lowest order in g Σ(iω n ) = T N L p,ω m g 2 D(iΩ m )G 0 ( k q; iω n iω m ) Simplification : Momentum Independence Unlike Eliashberg... no self-consistency above!

43 Self-Energy Calculation continued... Using Matsubara trick, the Matsubara sum done and we get Σ(z) = g 2 { (ω + 2 Tr Ω0 )I + ɛ(k)τ 3 (k)τ 1 k (ω + Ω 0 ) 2 E(k) 2 + Ω 0(I + ɛ(k) E(k) τ 3 (k) E(k) τ } 1) [ω E(k)] 2 Ω 2. 0 E(k) = + ɛ(k) 2 + (k) 2 Anticipating feature around z = (E coherence + Ω 0 ) + iη η : Small imaginary part

44 Self-Energy Calculation continued... Using Matsubara trick, the Matsubara sum done and we get Σ(z) = g 2 { (ω + 2 Tr Ω0 )I + ɛ(k)τ 3 (k)τ 1 k (ω + Ω 0 ) 2 E(k) 2 + Ω 0(I + ɛ(k) E(k) τ 3 (k) E(k) τ } 1) [ω E(k)] 2 Ω 2. 0 E(k) = + ɛ(k) 2 + (k) 2 Anticipating feature around z = (E coherence + Ω 0 ) + iη η : Small imaginary part Feature due to Saddle Point Singularity in Second Sum above.

45 Self-Energy Calculation continued... Using Matsubara trick, the Matsubara sum done and we get Σ(z) = g 2 { (ω + 2 Tr Ω0 )I + ɛ(k)τ 3 (k)τ 1 k (ω + Ω 0 ) 2 E(k) 2 + Ω 0(I + ɛ(k) E(k) τ 3 (k) E(k) τ } 1) [ω E(k)] 2 Ω 2. 0 E(k) = + ɛ(k) 2 + (k) 2 Anticipating feature around z = (E coherence + Ω 0 ) + iη η : Small imaginary part Feature due to Saddle Point Singularity in Second Sum above. Form of Singularity Σ(z) log(z) (cf. Van Hove in 2d) Note Σ 12 = 0 here.

46 Self-Energy Calculation continued... a) Numerical Confirmation of Log Singularity c) parameter fit (t 1 150meV) to dispersion from ARPES (Norman et al) = 0.2t 1 and Ω 0 = 0.25t 1 η = 0.005t 1, 0.01t 1 and 0.02t 1 Peak Height Change Linear vs. Geometric Change in η log Singularity

47 Boson Feature Boson feature : δn(e) Im[I 1 (E)Σ 11 (E) + I 2 (E)Σ 22 (E)] n Ω mev mev Re Im 11 Ebos E 0.0 bos E bos Ecoh Ebos Ω mev Boson Freq. Inflection point before hump - Coherence Peak I 1, I 2 : Band-structure dependent functions

48 Fitting Scheme for Experiment n Ω arb. units Ecoh Ebos Ω mev Coherence Peak also a similar Log singularity Calibration using Coherence peak, i.e. ñ(e) n(e) Then fit the Boson Feature

49 Fitting Scheme continued... Fitting Form for Coherence Peak ñ(e) = β cal n(e) 0.9 n Ω E coh Ω mev

50 Fitting Scheme continued... Fitting Form for Coherence Peak ñ(e) = β cal n(e) n(e) = n reg (E) + n sing 0 (E) 0.9 n Ω E coh Ω mev

51 Fitting Scheme continued... Fitting Form for Coherence Peak ñ(e) = β cal n(e) n(e) = n reg (E) + n sing 0 (E) n reg (E) = a coh E + b coh 0.9 n Ω E coh Ω mev

52 Fitting Scheme continued... Fitting Form for Coherence Peak ñ(e) = β cal n(e) n(e) = n reg (E) + n sing 0 (E) n reg (E) = a coh E + b coh n sing 0 (E) = a2 m Re[f (E E π 2 coh + iη coh )] f (z) = Log[m z/4k x K y ] m, K x, K y - Band structure Saddle Point properties n Ω E coh Ω mev

53 Fitting Scheme continued... Fitting Form for Coherence Peak ñ(e) = β cal n(e) n(e) = n reg (E) + n sing 0 (E) n reg (E) = a coh E + b coh n sing 0 (E) = a2 m Re[f (E E π 2 coh + iη coh )] f (z) = Log[m z/4k x K y ] m, K x, K y - Band structure Saddle Point properties 0 = 44.23meV, η coh = 10.7(9)meV 30 E coh Ω mev n Ω

54 Fitting Scheme continued... Fitting Form for Boson Feature n(e) = n reg bos (E) + δnsing (E) 7. n Ω 6. E bos Ω mev 135

55 Fitting Scheme continued... Fitting Form for Boson Feature n(e) = n reg bos (E) + δnsing (E) n reg bos (E) = a bose + b bos 7. n Ω 6. E bos Ω mev 135

56 Fitting Scheme continued... Fitting Form for Boson Feature n(e) = n reg bos (E) + δnsing (E) n reg bos (E) = a bose + b bos δn sing (E) = 2a2 m (2π) 4 ig 2 Im[I(E + iη bos )f (E E bos + iη bos ] f (z) = Log[m z/4k x K y ] n Ω E bos Ω mev 135

57 Fitting Scheme continued... Fitting Form for Boson Feature n(e) = n reg bos (E) + δnsing (E) n reg bos (E) = a bose + b bos δn sing (E) = 2a2 m (2π) 4 ig 2 Im[I(E + iη bos )f (E E bos + iη bos ] f (z) = Log[m z/4k x K y ] Ω 0 = 56(1)meV, g = 36(16)meV, η bos = 11(2)meV n Ω E bos Ω mev 135

58 Weak Coupling Confirmation Idea : compare strength of the two Log singularities Coherence Peak : due to bare propagator Boson Feature : due to Self-Energy corrections O(g 2 ) Dimensionless Ratio : λ log = 0.06

59 Weak Coupling and Momentum Dependence What when g(k)??

60 Weak Coupling and Momentum Dependence What when g(k)?? Boson Feature arguments goes through as before... Only change : I 1 (E), etc. gets an additional weight during Tr k

61 Weak Coupling and Momentum Dependence What when g(k)?? Boson Feature arguments goes through as before... Only change : I 1 (E), etc. gets an additional weight during Tr k Gap gets Renormalized

62 Weak Coupling and Momentum Dependence What when g(k)?? Boson Feature arguments goes through as before... Only change : I 1 (E), etc. gets an additional weight during Tr k Gap gets Renormalized Calculate Maximal Gap renormalization given extracted g 2 12 k Π,0 ;Ω Ω mev

63 Weak Coupling and Momentum Dependence What when g(k)?? Boson Feature arguments goes through as before... Only change : I 1 (E), etc. gets an additional weight during Tr k Gap gets Renormalized Calculate Maximal Gap renormalization given extracted g 2 12 k Π,0 ;Ω Ω mev 5 mev 0 = 40 mev electron-boson coupling weak

64 Weak Coupling and Momentum Dependence What when g(k)?? Boson Feature arguments goes through as before... Only change : I 1 (E), etc. gets an additional weight during Tr k Gap gets Renormalized Calculate Maximal Gap renormalization given extracted g 2 12 k Π,0 ;Ω Ω mev 5 mev 0 = 40 mev electron-boson coupling weak Observed phonon can not be mechanism

65 Thank you! Acknowledgements : 1 Chris L. Henley 2 J. C. Davis and group (especially Jacob Alldredge, Milan Allan)

66 Technical 1 : Matsubara trick

67 Technical 2 : Log Singularity

68 Echoes : Analytical Calculations for ( r r imp 1) G 0 (r r ; E) = const r rimp i 1 E(k i ) ( cos k i.(r r imp ) 1 + ɛ(k i ) E Curvature of E-contour (k i ) E (k i ) E 1 ɛ(k i ) E ) Salient features : Cosine terms give echoes/qp interference k i are wave-vectors at which group velocity is along r r imp 1/ R dependence on distance from impurity (1/R for LDOS) rest is dispersion/material details

69 Steps of the calculation : G 0 (r r ; E) = BZ d 2 k eik.(r r imp) ( E + ɛk ) k E 2 Ek 2 k E ɛ k Vanishing denominator E 2 Ek 2 reduces it to an integral over E = E k contour e ik.(r r imp) term : use stationary phase approx for large r r : stationary for k i s