1. CDW PHASE SHIFT STUDY BY UHV-LT-STM J.-C. Girard et Z.Z. Wang Laboratoire de Photonique et de Nanostructures LPN / CNRS Route de Nozay – 91460 Marcoussis, France

2. Outline 1/ Introduction : Quantum Imaging by STM imaging of ground state of quantum mechanics and of many body problem 2/ CDW in TTF-TCNQ : an ideal candidate for local phase shift  (r) studying (unique?) 3/ phase shift  a (x,y) for modulation in “a” direction (perpendicular to the chain direction): the variation is trivial in real space (commensurate pining) 4/ phase shift  b (x,y) for modulation in “b”direction (chain direction): important variation presented in our measurement, phase shift “map” 5/ Conclusion: CDW complex order parameter  e i  can be studied by STM

3. We soon find ourselves armed with wonderful new tools. The more we used them, the more applications we find; and the more applications we find, the more use of quantum theory we make. In no way do the advances of physics spread more widely to the community than in new and improved measuring devices. Is it true that « no elementary quantum phenomenon is a phenomenon until it is a recorded phenomenon »? John Archibald Wheeler Quantum Theory and Measurement measurement

4. STM might be a unique technique to study, in real space, local electronic structure with energy resolution. - the typical tip-sample resistance (tunneling gap resistance) is of 6- 10 orders higher than the sample resistance, resulting in a less significant matrix transfer element in tunneling junction. - only an insignificant electric field built inside the sample. - tunneling current is of order of 1-100pA, one electron every 1- 100ns! No question for escaping, recombination, thermalizing… Both topographic and spectroscopic measurement in nanometer scale can be performed simultaneously with a least tip-sample interaction (without destructive). Ground state in quantum mechanics is robust against the STM measurement STM measurement

5. -A lot of the phenomena that are traditionally encountered in course in quantum mechanics, such as molecular orbital, harmonic oscillators, particle in a box, eigen wave function, impurity Bohr radius, Fermi’s golden rule, chemical bonding, and the electron spin can be directly visualized with the STM. -Recently, some second quantification phenomena that are introduced in many body problems of condensed matter physics are studied by STM too. Kondo effect, Friedel Oscillation, Charge Density Wave, inhomogeneity of superconducting gap are clearly observed without ambiguity. Quantum Imaging with STM

6. Band structure

7. e0 V = 867 mV e1 V = 901 mV e2 V = 936 mV e3 V = 970 mV e4 V = 1024 mV e5 V = 1074 mV e6 V = 1124 mV e7 V = 1155 mV e8 V = 1182 mV e9 V = 1216 mV e10 V = 1255 mV e11 V = 1297 mV e6 e7 e8 e9 e10 e11 e0 e1 e2 e3 e4 e5 InAs(P)/nP(001) QD Height : 5.8 nm Lateral size : 42 nm Eigenenergy and eigenstate in QD LPN PRL 2009, APL 2010

8. Suzuki et al., Technical Review NTT (2008) Burgi et al, PRL 81, 24 (1998) Electronics states steps confinement Electronics states for InAs/GaSb (QW) C. Tournier-Colletta et al., PRL 104, 016802 (2010) Electronics states in Ag(111) islands Confined electronic states in different nanostructures by STS

9. Cleavage plane ab :quadratic cell S C H - C HN TTF-TCNQ crystallography Monocrystal: monoclinic cell a = 1.23 nm b = 0.38 nm c = 1.58 nm  = 104.6 °

10. Molecular Resolution at T = 63K on TTF-TCNQ (001) surface I t = 1nA, V bias = 50mV Acquisition time : 210 s TCNQ - TTF + TCNQ - Corrugation: TCNQ : 2 x 0.6 Å TTF : 2 x 0.2 Å h TCNQ -h TTF = 1 Å

11. CDW at 35.6K

12. Sequential images of CDW on TTF-TCNQ at T = 35.6 K Total time : 50 minutes Temp. shift: 0.4K 100 mV 1nA Time for take one Image: 210s

13. dimensions : 14.89 nm x 16.09 nm 41.b 12.a y: chain dir. x a=1.22nm b=0.38nm a b analysis on 12 rows X 41columns LPN PRB 2003 T = 35.6K

14. Fourier Analysis : bi-q modulation two generating vectors for CDW at 35.6K : q 1 = 0.25 a* + 0.295 b*  q a. a* + q b b* q 2  -0.25 a* + 0.295 b*  - q a. a* + q b b* In real space, the wave vectors are:  = a a + b b with a =  q a ) = 4 (commensurate)  = - a a + b b b =  q b ) = 3.39 (incommensurate) (LPN PRL 2009, PRB 2008, PRB 2006, PRB2003)

15. CDW is a quantum condensate state of electronic state in low- dimensional materials. It can be presented as a complex order parameter  e i   determines - the size of the electronic energy gap - the amplitude u 1 of the atomic displacements - the amplitude of the electron density modulation  (r) determines the position of the CDW relative to the underlying lattice.  (r) =  0  cos [  (r) ]  with  (r) = q CDW. r +  r)  r) represents local deformation of CDW which is related to the elastic energy Phase shift  r)

16. Weak / Strong CDW pining and local phase shift CDW pinning : Coulomb interaction between the CDW and impurities Perfect one-dimensional lattice:  r) is constant (superconducting state) r)r) No impurities r weak pinning (Fukuyama, Lee, Rice): smooth variation in  r)over a distance containing several impurities r  r) iiiiiiiii r strong pinning: abrupt variation in  r) at each impurity site  r) iiiii -Phase determination is crucial to understand the physics of a complex order parameter - Importance in the understanding of the static and dynamic properties of the CDW state.

17. Phase shift in modulation bi-q : a unit : b unit  (x,y) = 2A. cos[(  a ). x +  a ]. cos[(  b ).y +  b ] (2) the phase shift  a and  b can be calculated separately: Perpendicular to chain, in “a” direction (commensurate) y = const ;  (x) = 2A(y). cos[(  a ). x +  a ] (3) For a chain, in “b” direction (incommensurate) x = const ;  (y) = 2A(x). cos[(  b ).y +  b ](4)  (x,y) =  0 [1+  (x,y) ]  (x,y)= A 1 cos(q 1.r +   ) cos(q 2.r +   ) q 1 = 0.25 a* + 0.295 b*  q a. a* + q b b* q 2  -0.25 a* + 0.295 b*  - q a. a* + q b b*

18. CDW phase transition (T P = 53K) Semiconductor at low temperature (condensate state) an ideal candidate to study the local phase shift  (r) - Quadratic unit cell in the ab plane. (a.b = 0) - Low temperature CDW phase (T<38K): Commensurate in the “a” direction Incommensurate in the “b” direction bi-q modulation -the phase shift  a and  b can be treated as independents parameters both  a and  b are fonction of x and y TTF-TCNQ: an ideal candidate to study the local phase shift  (r)

19. dimensions : 14.89 nm x 16.09 nm 41.b 12.a y: chain dir. x a=1.22nm b=0.38nm a b analysis on 12 rows X 41columns LPN PRB 2003 T = 35.6K

20. phase shift  a for modulation in “a” direction In “a” direction (commensurate), for a TCNQ molecular row (y = const), CDW modulation  (x,y) = A(y)  y cst..cos((  a ).x +  a ) (4) with (  a ) = 2  =  analysis method: 1 st step: cross section profile in “a” direction (perpendicular to chain ) 2 nd step: determination of the CDW maxima at the lattice position 3 rd step: best fit from sinusoidal function A.cos((  a ).x +  a ) Origin’s non linear regression method : Levenberg- Marquardt algorithm (  2 minimization)  a is the only fitting parameter (  a  average on  3 CDW wavelengths) A = Max(Z i ) – Min(Z i ) Distance (nm) Amplitude (nm)

21. “a” direction phase shift  a Agree with X-ray analysis result reported:  /4 J.P. Pouget The change of  a is trival in real space, we concentrate on the local variation of  b Commensurate pining by the lattice  a = - 0.76  - (  /2 +  /4) Distance (nm) Amplitude (nm)

22. phase shift for modulation in “b” direction a single chain analysis In “b” direction (in commensurate), for a TCNQ chain (x = constant) CDW modulation is  (x,y) = A(x)  x cst..cos((  b ).y +  b ) With (  b ) = 2  The fitting function is  (x,y) = A..cos((  b ).y +  b )  b is the only fitting parameter (average on  12 CDW wavelengths) amplitude (nm) distance (nm) We find  b = -258 

23. The fitting function is  (x,y) = A..cos((  b ).y +  b )  b is the only fitting parameter (average on  12 CDW wavelengths) amplitude (nm) distance (nm)

24. The phase shift on four adjacent chains

25. As we measured  a =  /4 for the CDW commensurate a- component First chain:  (y, x = 0) = 2Acos(  a ).cos(q b.y +  b ) = 2 A 1 cos(q b.y +  b ) Second chain  (y, x = a) = 2Acos(  /2 +  a ).cos(q b.y +  b ) = 2 A 2 cos(q b.y +  b ) Third chain  (y, x = 2a) = 2Acos(  +  a ).cos(q b.y +  b ) = 2 A 1 cos(q b.y +  b  ) Fourth chain  (y, x = 3a) = 2Acos(3  /2 +  a ).cos(q b.y +  b ) = 2 A 2 cos(q b.y +  b  )

26. Variation on each chain of  b (n) Average  b (n)>= 0.45  b unit  b (n) = 0.15  b (n) = 0.22  b (n) = 0.12  b (n) = -0.11  b (n) = -0.04  b (n) = 0.73  b (n) = -0.77  b (n) = -0.94  b (n) = -0.36  b (n) = -0.72  b (n) = 0.51  b (n) = -0.38

27. Variation of the phase shift  b in the “a” direction the phase shift has a variation of 4 degrees per a unit length in the a direction. With  correction  b for 12 chains

28. Local phase shift: the variation of  b (n) in the “b” direction We have previously measured the average phase shift over 12 CDW for each TCNQ chain.On a single chain, the local variation of the phase shift  b (n) can be determinedby selecting only 6 CDW wavelengths centred at y = n.b  b (10) = -262°  b (30) = -252°  b (10) = -262°  b (30) = -252° We measured a  b phase shift of 0.15 degrees per b unit length along a single chain in the b direction

29. Local phase shift  b (n) in next image in “b” direction

31. CDW Domains Size Estimation in TTF-TCNQ In 1980’s, Fukuyama, Lee, Rice postulated the existence of domain in the CDW condensate in the weak pinning case. Inside the domain the variation of phase shift is less than  So far, the experimental observations of domains is difficult (Steed and Fung;? Fleming; ?) We have measured variations of the phase shift  b for modulation in b direction Transverse to the chains (in a direction)  = 4 degrees / a unit length Parallel to the chains (in b direction) 0 <  b (n) < 0.45 degrees / b unit length If we defines size domains with L  and L  as the lengths on which the phase shift varies by  degrees: L     a = 2470 nm  2.5  m L     b = 61120 nm  60  m

32. Fourier Analysis Selection d’un spot dans le spectre de Fourier Transformation de Fourier inverse écart a la périodicité Phase shift map  <   F. Pailloux, LMP, Univ Poitiers) 0 22 Methode d’analyse d’images HR-TEM M.J. Hyttch ( Ultramicriocopy 74 (1998) 131)

33. Summary in “a” direction ( commensurate ), CDW is pinned by the lattice with a phase shift  a close to   relative to the underlying lattice - in “b” direction ( incommensurate), CDW is weakly pinned by impurities as the phase shift  b varies smoothly along the chain -the existence of the  b correlation between chains and the  b vary slowly from chain to chain., however the change of  b is more important in “a” direction than in “b” direction the size of the CDW domains in TTF-TCNQ  can be estimated as: L  x L  = 2.5  m x 60  m STM ability to determine both atomic structure and CDW structure  resolution of  complicated structural details of CDW CDW complex order parameter  e i  can be studied by STM