Wednesday, 8th August
Dimitri Basov (University of California, San Diego)
An infrared probe of electronic correlations and many body effects in solids: a case study of high-Tc pnictides and graphene
Blogged by Piers Coleman
Good morning folks. Andre Marie is introducing Dmitri Basov. He remembers Dimitri from his Toronto days, when, with Tom Timusk, he discovered the optical signatures of the pseudogap in the cuprates. Andre Marie mentioned how much new insight into high temperature superconductivity have come from Dimitri's work.
Dimitri mentions how many of his collaborations began here at the ICTP. He has recently written a review on optics in RMP, cited below
D. N. Basov, R. D. Averitt, D. van der Marel, M. Dressel, K. Haule “Electrodynamics of correlated electron materials” Rev. Mod. Phys. 85(2), 471 (2011).
Dimitri begins with a review of frequencies, emphasizing the breadth of energies that govern cuprates and graphene, from 10 to 30,000cm^-1. (1cm^-1 = 0.124 meV = 1.4K). Dimitri reminded us that j = sigma E, that the dielectric constant is related to the optical conductivity
epsilon = 1 + (4 pi i) sigma/omega
Sigma obeys the remarkable sum rule
This sum rule, he said, derives from the fact sigma is a response function (blogger: its really a statement of impulse - this integral is the instantaneous current that responds to the impulse from a pulse of electric field - at short times, the acceleration is given by Newtons law).
Dimitri discusses the Drude model - which he says plays the same roll to electrodynamics as Shakespeare does to literature! Drude's model came out just three years after the discovery of the electron. Here is a summary of the main points:
DB shows the optical conductivity of a weakly interacting metal. The area under the Drude peak is just the electron density (see above). The relaxation rate tau(\omega) ~ omega^2 is frequency dependent. In an electron phonon system, the scattering rate is determined by the phonon spectrum, alpha^2 F(omega), and one can actually invert the expression. C-60 is an example of an electron-phonon superconductivity with a relatively high Tc. But what about strongly interacting systems - here the Drude weight is dramatically reduced below the band-structure value. The frequency dependence is substantially different to a simple Lorentzian - the coherent part drops away more rapidly, there is also a large incoherent background.
Question - what is the connection of sigma(omega) to the reflection of IR light?
DB - what we actually measure is the reflectance R(omega). From this we can extract both the real and imaginary part of the conductivity using Kramers-Kronig. If you have a transparent system, you can directly pull out the real and imaginary parts of sigma.
Now he turns to the optical conductivity as a probe of correlations in cuprate and iron-based superconductors. These pose many interesting questions:
(1) Exotic superconductors - are they all alike - at least within one family? (cf Tolstoy "Happy Families are all alike")
(2) Unusual normal state properties - are they a pre-requisite for Hi Tc?
(See D. N. Basov and A. V. Chubukov, Nature Phys 7, 272, (2011) ).
Dmitri contrasts the phase diagram of the two systems. Dimitri points out that both have a pseudo-gap. Doping has a different meaning in the two types of system. The Co and P doped Ba(Fe2As2) show a nice Drude peak, with a spectral weight that does not change a lot with doping. (See below, rhs - note that the log scale exaggerates the difference between doped and undoped).
By contrast, the doped cuprates show a strong growth of the Drude peak with doping - because you are doping an insulator. He notes that the scattering rates must be driven by interactions.
Q: How does one know where the interband processes begin?
DB: I know that most of the low energy weight must come from intraband weight (in the iron-based). This is the only way to get consistency with other probes.
Q: Can the changes in the optical conductivity be connected to electron electron interactions
DB: Yes - the spectral weight is much smaller than non interacting theories predict. Secondly the form of the conductivity is quite different that predicted by non-interacting theories.
DB replots the data on a linear scale to contrast the cuprates and iron sc. The low frequency part is coherent, the high frequency part, with a large excess over and above the Drude, is the "incoherent"
part
Intraband = coherent + incoherent
Now we turn to SC dynamics. DB notes that Michael Tinkham measured the energy gap before BCS theory (1956), with an onset of conductivity (absorption) at twice the energy gap. In the early days, it was not known how to connect the absence of absorption at low energies, with the unchanged spectrum at high energies. Tinkham showed there is a threshold. Now what about the cuprates and iron sc? In
Q: The shape of the curve - does it depend on the symmetry of the gap?
DB: Yes, the symmetry of the gap can impact the form of the optical conductivity. d-wave sc have certain peculiarities - where there is a node - but the gross features are not that different from an s-wave sc. The dominant effect is the suppression of absorption below the gap.
Q: Optical conductivity averages all k, so what gap is measured?
DB: The strongest absorption measures the maximum gap.
However the situation is more complicated than in lead. There is a lot of extra low energy absorption
associated with multi-band behavior.
DB shows the optical spectrum of pseudo-gap materials, where there are signs of the pseudo-gap even in the normal state, above Tc. This gap-like behavior is called the "pseudogap". One is gapping part of the Fermi surface, suppressing some scattering, which leads to a narrowing of the remaining Drude peak.
Its particularly interesting to look at the frequency-dependent scattering rate. The scattering just above Tc has all the features of the fully developed sc. Amazingly, there is a similar, though slightly less marked pseudo-gap feature in the iron optical conductivity. The pseudogap scale determines a crossover from weak low frequency scattering to strong high-frequency scattering rate. In the iron-sc, the pseudogap is related to the SDW.
Q: How is this related to transport?
DB: Transport is the dc limit. But we get more information from the frequency dependence - it shows us how we are deviating from Drude. These spectra highlight what is significant in the response.
Q: Can one probe stripe order or phase coexistence using optical c?
DB: Yes, we tried to probe stripe order in La 214, and you can look at the anisotropy of the conductivity parallel and perpendicular to the stripes using polarized light. This has also been done in the pnictides, where recently Leo De Georgi is investigating anisotropies associated with nematicity.
R: You associated the pgap in the pnictides with the SDW, whereas AM Tremblay associated it with Mott physics. Could you contrast the two in the second lecture?
DB: Yes I will.
Lecture 2.
DB returned to the issue of the origin of the pseudogap. There is clear similarity in 1/tau (w) data.
In pnictides, pseudogap physics disappears beyond optimal doping.
DB then turned to the discussion of the coupling to the bosonic spectrum.
Tutorial 3: the pairing glue.
Discussion of the Allen formula and how to extract \alpha^2 F(w) from the second derivative of 1/tau.
The discussion first focused on the cuprates, but then was extended to pnictides.
Back to the pseudogap behavior of 1/tau in cuprates and pnictides. In pnictides, the structure associated with the pseudogap is quite different from the SC gap. New aspect of the pseudogap story -- presence of SDW. The pseudogap scale in the pnictides == the same SDW scale as extracted from ARPES.
Q. How \alpha^2 F was extracted from the data?
A. By using the theoretical formula with phenomenological pseudogap parameter and a coupling to "some" mode.
Tutorial 5 (Tutorial 4 was probably eliminated) -- condensate formation below T_c and from what frequencies the condensate is formed.
DB discussed Homes Law - the scaling of rho_s with sigma_DC * T_c. This can be understood from the sum rule - an area sigma_DC * 2 Delta ~ Sigma_DC * Tc condenses. This indicates the presence of strong inelastic dissipation at Tc - dissipation that does not depend on disorder.
Q: Does the ratio of Delta/T_c remain constant.
DB: No - there is quite a bit of spread of this ratio, but within the log-log scaling, this doesn't matter.
Q: Wouldn't this also work for say lead.
DB: Yes it would, but the resistivity would be driven by dirt, not inelastic scattering.
Tutorial 6 Electronic Kinetic Energy and Correlations.
At the two extremes, we have Mott insulators - localized, Coulomb arrests metallic transport, at the other fully itinerant simple metals. What we are dealing with is "correlated metals" that lie between these two extremes. One way to characterize this is using Kinetic energy, in ref to the KE from band calculations.
Exptal KE = integral of sigma(omega)domega
Kband <---- band theory
The ratio of the two is a measure of correlation strength in the material.
Q: What cut-off do you use in obtaining these plots?
DB: We integrate up to the bottom of the inter-band transitions, trying to catch the coherent component of the sigma.
Remarkably, the iron-based superconductors lie around 0.4 on this scale - comparable with the cuprate cousins. Ba 122 is around 0.3, LeFePO around 0.5. LasCO is around 0.2. MgB2 and doped C60 are at 0.8 and 1.0 respectively. You can conclude that electron phonon does not renormalize the electron Kinetic energy.
KE is suppressed in all exotic superconductors. Iron, cuprate, ruthenate, heavy fermion and organics all experience this. This implies that there are changes in the spectrum at high energies - appears to be a condition for correlated superconductors. High Tc systems are not in general, super-correlated, because of course, strong e correlations kills the Drude peak, which also implies a reduced superfluid density and thus suppresses the superconductor. This appears to be going on in the under-doped side of the cuprate phase diagram.
Q: Is there a sharp transition from correlated to conventional metals?
DB: Not as far as we know - but the data isn't sufficiently dense to completely answer this question.
Anisotropy. DB turns to the question of anisotropy in the optical conductivity. What happens to the conductivity in the in-plane direction? In the cuprates, the c-axis response often looks like an insulator. But in the iron-based, there is no qualitative difference between inplane and interlayer transport. In the cuprates, you can still pass current along the insulating c-axis direction. Remarkably you can still see the plasma edge in the c-axis response - this derives from c-axis Josephson coupling. A moderate field H~ 8T eliminates the plasma edge, yet sc still survives. So its clear that interplane coupling does not matter for the superconductivity.
In the c-axis optical properties of the iron-based sc, one sees almost no difference between normal and sc state. This has recently been resolved for122 samples. It turns out to be an artifact that is connected with "cleaning the surface". Newer measurements show a marked drop in low frequency optical conductivity.
Infrared nano-scopy
A new innovation in IR spectroscopy, actually a hybrid of STM and IR spectroscopy. The tip has a radius of 8-10nm, allowing a 3 or 4 order of magnitude increase in the resolution from direct optical methods. It also gives you access to much shorter wavelength than available with light.
It also gives you access to much shorter wavelength than available
with light. Dimitri showed images of monolayer graphene using this method.
Addendum. In the discussion on Thursday, Dimitri was asked to show the optical conductivity of graphene. He talked about the work of Z. Q. Li, Nature Physics 2008 obtained with the above method. Its flat, undoped, but doped, as an interband threshold E_F, a Drude peak that is very sharp, and an incoherent part that is not yet understood. He said that higher mobility samples (best are now above 100,000) need to be measured. He finds also that sigma_1/sigma_2 has oscillations in the Friedel oscillations.
An infrared probe of electronic correlations and many body effects in solids: a case study of high-Tc pnictides and graphene
Blogged by Piers Coleman
Good morning folks. Andre Marie is introducing Dmitri Basov. He remembers Dimitri from his Toronto days, when, with Tom Timusk, he discovered the optical signatures of the pseudogap in the cuprates. Andre Marie mentioned how much new insight into high temperature superconductivity have come from Dimitri's work.
Dimitri mentions how many of his collaborations began here at the ICTP. He has recently written a review on optics in RMP, cited below
D. N. Basov, R. D. Averitt, D. van der Marel, M. Dressel, K. Haule “Electrodynamics of correlated electron materials” Rev. Mod. Phys. 85(2), 471 (2011).
Dimitri begins with a review of frequencies, emphasizing the breadth of energies that govern cuprates and graphene, from 10 to 30,000cm^-1. (1cm^-1 = 0.124 meV = 1.4K). Dimitri reminded us that j = sigma E, that the dielectric constant is related to the optical conductivity
epsilon = 1 + (4 pi i) sigma/omega
Sigma obeys the remarkable sum rule
This sum rule, he said, derives from the fact sigma is a response function (blogger: its really a statement of impulse - this integral is the instantaneous current that responds to the impulse from a pulse of electric field - at short times, the acceleration is given by Newtons law).
Dimitri discusses the Drude model - which he says plays the same roll to electrodynamics as Shakespeare does to literature! Drude's model came out just three years after the discovery of the electron. Here is a summary of the main points:
DB shows the optical conductivity of a weakly interacting metal. The area under the Drude peak is just the electron density (see above). The relaxation rate tau(\omega) ~ omega^2 is frequency dependent. In an electron phonon system, the scattering rate is determined by the phonon spectrum, alpha^2 F(omega), and one can actually invert the expression. C-60 is an example of an electron-phonon superconductivity with a relatively high Tc. But what about strongly interacting systems - here the Drude weight is dramatically reduced below the band-structure value. The frequency dependence is substantially different to a simple Lorentzian - the coherent part drops away more rapidly, there is also a large incoherent background.
Question - what is the connection of sigma(omega) to the reflection of IR light?
DB - what we actually measure is the reflectance R(omega). From this we can extract both the real and imaginary part of the conductivity using Kramers-Kronig. If you have a transparent system, you can directly pull out the real and imaginary parts of sigma.
Now he turns to the optical conductivity as a probe of correlations in cuprate and iron-based superconductors. These pose many interesting questions:
(1) Exotic superconductors - are they all alike - at least within one family? (cf Tolstoy "Happy Families are all alike")
(2) Unusual normal state properties - are they a pre-requisite for Hi Tc?
(See D. N. Basov and A. V. Chubukov, Nature Phys 7, 272, (2011) ).
Dmitri contrasts the phase diagram of the two systems. Dimitri points out that both have a pseudo-gap. Doping has a different meaning in the two types of system. The Co and P doped Ba(Fe2As2) show a nice Drude peak, with a spectral weight that does not change a lot with doping. (See below, rhs - note that the log scale exaggerates the difference between doped and undoped).
By contrast, the doped cuprates show a strong growth of the Drude peak with doping - because you are doping an insulator. He notes that the scattering rates must be driven by interactions.
Q: How does one know where the interband processes begin?
DB: I know that most of the low energy weight must come from intraband weight (in the iron-based). This is the only way to get consistency with other probes.
Q: Can the changes in the optical conductivity be connected to electron electron interactions
DB: Yes - the spectral weight is much smaller than non interacting theories predict. Secondly the form of the conductivity is quite different that predicted by non-interacting theories.
DB replots the data on a linear scale to contrast the cuprates and iron sc. The low frequency part is coherent, the high frequency part, with a large excess over and above the Drude, is the "incoherent"
part
Intraband = coherent + incoherent
Now we turn to SC dynamics. DB notes that Michael Tinkham measured the energy gap before BCS theory (1956), with an onset of conductivity (absorption) at twice the energy gap. In the early days, it was not known how to connect the absence of absorption at low energies, with the unchanged spectrum at high energies. Tinkham showed there is a threshold. Now what about the cuprates and iron sc? In
Q: The shape of the curve - does it depend on the symmetry of the gap?
DB: Yes, the symmetry of the gap can impact the form of the optical conductivity. d-wave sc have certain peculiarities - where there is a node - but the gross features are not that different from an s-wave sc. The dominant effect is the suppression of absorption below the gap.
Q: Optical conductivity averages all k, so what gap is measured?
DB: The strongest absorption measures the maximum gap.
However the situation is more complicated than in lead. There is a lot of extra low energy absorption
associated with multi-band behavior.
DB shows the optical spectrum of pseudo-gap materials, where there are signs of the pseudo-gap even in the normal state, above Tc. This gap-like behavior is called the "pseudogap". One is gapping part of the Fermi surface, suppressing some scattering, which leads to a narrowing of the remaining Drude peak.
Its particularly interesting to look at the frequency-dependent scattering rate. The scattering just above Tc has all the features of the fully developed sc. Amazingly, there is a similar, though slightly less marked pseudo-gap feature in the iron optical conductivity. The pseudogap scale determines a crossover from weak low frequency scattering to strong high-frequency scattering rate. In the iron-sc, the pseudogap is related to the SDW.
Q: How is this related to transport?
DB: Transport is the dc limit. But we get more information from the frequency dependence - it shows us how we are deviating from Drude. These spectra highlight what is significant in the response.
Q: Can one probe stripe order or phase coexistence using optical c?
DB: Yes, we tried to probe stripe order in La 214, and you can look at the anisotropy of the conductivity parallel and perpendicular to the stripes using polarized light. This has also been done in the pnictides, where recently Leo De Georgi is investigating anisotropies associated with nematicity.
R: You associated the pgap in the pnictides with the SDW, whereas AM Tremblay associated it with Mott physics. Could you contrast the two in the second lecture?
DB: Yes I will.
Lecture 2.
In pnictides, pseudogap physics disappears beyond optimal doping.
DB then turned to the discussion of the coupling to the bosonic spectrum.
Tutorial 3: the pairing glue.
Discussion of the Allen formula and how to extract \alpha^2 F(w) from the second derivative of 1/tau.
The discussion first focused on the cuprates, but then was extended to pnictides.
Back to the pseudogap behavior of 1/tau in cuprates and pnictides. In pnictides, the structure associated with the pseudogap is quite different from the SC gap. New aspect of the pseudogap story -- presence of SDW. The pseudogap scale in the pnictides == the same SDW scale as extracted from ARPES.
Q. How \alpha^2 F was extracted from the data?
A. By using the theoretical formula with phenomenological pseudogap parameter and a coupling to "some" mode.
Tutorial 5 (Tutorial 4 was probably eliminated) -- condensate formation below T_c and from what frequencies the condensate is formed.
Q: Does the ratio of Delta/T_c remain constant.
DB: No - there is quite a bit of spread of this ratio, but within the log-log scaling, this doesn't matter.
Q: Wouldn't this also work for say lead.
DB: Yes it would, but the resistivity would be driven by dirt, not inelastic scattering.
Tutorial 6 Electronic Kinetic Energy and Correlations.
At the two extremes, we have Mott insulators - localized, Coulomb arrests metallic transport, at the other fully itinerant simple metals. What we are dealing with is "correlated metals" that lie between these two extremes. One way to characterize this is using Kinetic energy, in ref to the KE from band calculations.
Exptal KE = integral of sigma(omega)domega
Kband <---- band theory
The ratio of the two is a measure of correlation strength in the material.
Q: What cut-off do you use in obtaining these plots?
DB: We integrate up to the bottom of the inter-band transitions, trying to catch the coherent component of the sigma.
KE is suppressed in all exotic superconductors. Iron, cuprate, ruthenate, heavy fermion and organics all experience this. This implies that there are changes in the spectrum at high energies - appears to be a condition for correlated superconductors. High Tc systems are not in general, super-correlated, because of course, strong e correlations kills the Drude peak, which also implies a reduced superfluid density and thus suppresses the superconductor. This appears to be going on in the under-doped side of the cuprate phase diagram.
Q: Is there a sharp transition from correlated to conventional metals?
DB: Not as far as we know - but the data isn't sufficiently dense to completely answer this question.
Anisotropy. DB turns to the question of anisotropy in the optical conductivity. What happens to the conductivity in the in-plane direction? In the cuprates, the c-axis response often looks like an insulator. But in the iron-based, there is no qualitative difference between inplane and interlayer transport. In the cuprates, you can still pass current along the insulating c-axis direction. Remarkably you can still see the plasma edge in the c-axis response - this derives from c-axis Josephson coupling. A moderate field H~ 8T eliminates the plasma edge, yet sc still survives. So its clear that interplane coupling does not matter for the superconductivity.
Infrared nano-scopy
Addendum. In the discussion on Thursday, Dimitri was asked to show the optical conductivity of graphene. He talked about the work of Z. Q. Li, Nature Physics 2008 obtained with the above method. Its flat, undoped, but doped, as an interband threshold E_F, a Drude peak that is very sharp, and an incoherent part that is not yet understood. He said that higher mobility samples (best are now above 100,000) need to be measured. He finds also that sigma_1/sigma_2 has oscillations in the Friedel oscillations.
Perhaps this will be addressed in the afternoon, but I'd like to know some of the experimental limits of infrared spectroscopy:
ReplyDelete- How destructive are the measurements? Is there heating?
- How much are you limited to the surface?
- Can't you go below 2K?
- Are there conflicts with other spectroscopies?
Our venerable chief organizer warns in his lecture notes that one should be careful using the Kramers-Kronig relations when interpreting spectroscopic data, and I'm wondering how this is done in practice.
ReplyDeleteThe question concerning excitation of plasma waves on graphene with a help of a microscope tip ("scanning tunneling nanoscopy"): how to extract real and imaginary parts of the conductivity from measured quantities? And what are these quantities: intensity of scattered light? Its phase relative to the incident beam? Angular distribution?
ReplyDelete