Imaging the impact on cuprate superconductivity of varying the interatomic distances within individual crystal unit cells

Slezak et al. 10.1073/pnas.0706795105.

Supporting Information

Files in this Data Supplement:

Fig. 5. Topograph and corresponding phase map exhibiting modulations with the same wavevector q_SM. (A) A 48-nm-square topograph of Bi₂Sr₂CaCu₂O_8+d, showing the BiO layer exposed by cleaving. Red lines identify the top of the c-axis supermodulation in a central region of the topograph. An area in which the periodic supermodulation undergoes a noticeable phase slip is indicated by an arrow. The red lines are extracted from the phase map in B, highlighting points for which f(r) = 0, corresponding to the "top" of the supermodulation, looking down at the image. (B) Phase map f(r) of the supermodulation in A, corresponding to the identical field of view, with the phase slip in the same location, indicated by the arrows.

SI Text

In generating the phase map, the topographic images z(r) of the BiO surface are first Fourier-filtered to remove extraneous modulations other than those around the first harmonic of the supermodulation, giving a filtered image z'(r) in which the supermodulation appears as a sinusoidal modulation of nonconstant amplitude with the same phase shifts as the original topograph. Product images X(r) and Y(r), analogous to the X and Y channels of a lock-in amplifier, are then obtained by multiplying z'(r) by reference images A(r) = sin(q_SM · r) and B(r) = cos(q_SM · r), respectively. X(r) and Y(r) are then "low-pass filtered" to give X'(r) and Y'(r), removing all Fourier components with q-vector magnitudes greater than or equal to ≈2q_SM (a process analogous to the averaging function of a lock-in amplifier). A phase shift map d(r) can then be produced using the relationship d(r) = arctan[Y'(r)/X'(r)], which represents the phase difference between the perfectly regular reference images and the actual supermodulation at each point r. The phase map f(r) is then given by simply adding this phase shift to the phase of the reference, i.e., f(r) = q_SM · r + d(r), mod360^o.

This new phase map f(r) accurately and automatically tracks the shifts of the supermodulation phase in each region, as can be confirmed by direct comparison with the original topograph (the results of this process for one topograph are presented in SI Fig. 5. Using these techniques, we successfully determine the unit cell dimensions throughout and label them by f(r).

• Early Edition
• Archives
• Online Submission

• Feature Articles

  (Expanded research articles of exceptional breadth)

• Commentaries

  (Expert commentary on leading research)

• Letters

  (Online-only letters to the editor)

• Inaugural Articles

  (Articles by recently elected NAS members)

• PNAS Profiles

  (Biographical profiles of Academy members)

• Sustainability Science

  (Interactions of human and environmental systems)

• Collected Papers

  (Editorials, Perspectives, Reviews, Colloquia, etc.)

• Special Features

  (Solicited articles on exceptional research topics)

• Sackler Colloquia

  (Scientific reports held under NAS auspices)

• Podcasts

  Brief podcasts with leading scientists