ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First - PowerPoint PPT Presentation

Nambu-Gor’kov Formalism A generalized 2 × 2 matrix Green’s functions ˆ G n k ( τ ) is used to describe electron quasiparticles and Cooper pairs on an equal footing. imaginary time Wick’s time-ordering operator G n k ( τ ) = −� T τ Ψ n k ( τ ) Ψ † ˆ n k (0) � � � two-component ˆ c n k ↑ Ψ n k = c † field operator ˆ − n k ↓ Margine, Lecture Fri.2 12/36

Nambu-Gor’kov Formalism A generalized 2 × 2 matrix Green’s functions ˆ G n k ( τ ) is used to describe electron quasiparticles and Cooper pairs on an equal footing. imaginary time Wick’s time-ordering operator G n k ( τ ) = −� T τ Ψ n k ( τ ) Ψ † ˆ n k (0) � � � two-component c n k ↑ ˆ Ψ n k = c † field operator ˆ − n k ↓ � c † � � T τ ˆ c n k ↑ ( τ )ˆ n k ↑ (0) � � T τ ˆ c n k ↑ ( τ )ˆ c − n k ↓ (0) � ˆ G n k ( τ ) = − c † c † c † � T τ ˆ − n k ↓ ( τ )ˆ n k ↑ (0) � � T τ ˆ − n k ↓ ( τ )ˆ c − n k ↓ (0) � Margine, Lecture Fri.2 12/36

Nambu-Gor’kov Formalism ˆ G n k ( τ ) is periodic in the imaginary time τ and can be expanded in a Fourier series: e − iω j τ ˆ ˆ � G n k ( τ ) = T G n k ( iω j ) iω j where iω j = i (2 j + 1) πT ( j integer) are electronic Matsubara frequencies and T is the temperature. � G n k ( iω j ) F n k ( iω j ) � ˆ G n k ( iω j ) = F ∗ n k ( iω j ) − G − n k ( − iω j ) Margine, Lecture Fri.2 13/36

Nambu-Gor’kov Formalism ˆ G n k ( τ ) is periodic in the imaginary time τ and can be expanded in a Fourier series: e − iω j τ ˆ ˆ � G n k ( τ ) = T G n k ( iω j ) iω j where iω j = i (2 j + 1) πT ( j integer) are electronic Matsubara frequencies and T is the temperature. � G n k ( iω j ) F n k ( iω j ) � ˆ G n k ( iω j ) = F ∗ n k ( iω j ) − G − n k ( − iω j ) • Diagonal elements are the normal state Green’s functions and describe single-particle electronic excitations. Margine, Lecture Fri.2 13/36

Nambu-Gor’kov Formalism ˆ G n k ( τ ) is periodic in the imaginary time τ and can be expanded in a Fourier series: e − iω j τ ˆ ˆ � G n k ( τ ) = T G n k ( iω j ) iω j where iω j = i (2 j + 1) πT ( j integer) are electronic Matsubara frequencies and T is the temperature. � G n k ( iω j ) F n k ( iω j ) � ˆ G n k ( iω j ) = F ∗ n k ( iω j ) − G − n k ( − iω j ) • Diagonal elements are the normal state Green’s functions and describe single-particle electronic excitations. • Off-diagonal elements are the anomalous Green’s functions and describe Cooper pairs amplitudes (become non-zero below T c , marking the transition to the superconducting state). Margine, Lecture Fri.2 13/36

Nambu-Gor’kov Formalism ˆ G n k ( iω j ) can be evaluated by solving Dyson’s equation: ˆ ˆ ˆ G − 1 G − 1 n k ( iω j ) = 0 ,n k ( iω j ) − Σ n k ( iω j ) Margine, Lecture Fri.2 14/36

Nambu-Gor’kov Formalism ˆ G n k ( iω j ) can be evaluated by solving Dyson’s equation: non-interacting Green’s function ˆ ˆ ˆ G − 1 G − 1 n k ( iω j ) = 0 ,n k ( iω j ) − Σ n k ( iω j ) Margine, Lecture Fri.2 14/36

Nambu-Gor’kov Formalism ˆ G n k ( iω j ) can be evaluated by solving Dyson’s equation: non-interacting Green’s function ˆ ˆ ˆ G − 1 G − 1 n k ( iω j ) = 0 ,n k ( iω j ) − Σ n k ( iω j ) G − 1 ˆ 0 ,n k ( iω j ) = iω j ˆ τ 0 − ( ǫ n k − ǫ F )ˆ τ 3 Margine, Lecture Fri.2 14/36

Nambu-Gor’kov Formalism ˆ G n k ( iω j ) can be evaluated by solving Dyson’s equation: non-interacting Green’s function ˆ ˆ ˆ G − 1 G − 1 n k ( iω j ) = 0 ,n k ( iω j ) − Σ n k ( iω j ) G − 1 ˆ 0 ,n k ( iω j ) = iω j ˆ τ 0 − ( ǫ n k − ǫ F )ˆ τ 3 ˆ Σ n k ( iω j ) = iω j [1 − Z n k ( iω j )] ˆ τ 0 + χ n k ( iω j )ˆ τ 3 + ∆ n k ( iω j ) Z n k ( iω j )ˆ τ 1 mass renormalization energy superconducting function shift gap function Margine, Lecture Fri.2 14/36

Nambu-Gor’kov Formalism ˆ G n k ( iω j ) can be evaluated by solving Dyson’s equation: non-interacting Green’s function ˆ ˆ ˆ G − 1 G − 1 n k ( iω j ) = 0 ,n k ( iω j ) − Σ n k ( iω j ) G − 1 ˆ 0 ,n k ( iω j ) = iω j ˆ τ 0 − ( ǫ n k − ǫ F )ˆ τ 3 ˆ Σ n k ( iω j ) = iω j [1 − Z n k ( iω j )] ˆ τ 0 + χ n k ( iω j )ˆ τ 3 + ∆ n k ( iω j ) Z n k ( iω j )ˆ τ 1 mass renormalization energy superconducting function shift gap function 1 ˆ G n k ( iω j ) = − Θ n k ( iω j ) { iω j Z n k ( iω j )ˆ τ 0 + [( ǫ n k − ǫ F ) + χ n k ( iω j )] ˆ τ 3 + ∆ n k ( iω j ) Z n k ( iω j )ˆ τ 1 } Margine, Lecture Fri.2 14/36

Migdal-Eliashberg Approximation � d q ˆ τ 3 ˆ � Σ n k ( iω j ) = − T ˆ G m k + q ( iω j ′ )ˆ τ 3 Ω BZ mj ′ �� � | g mnν ( k , q ) | 2 D q ν ( iω j − iω j ′ ) + V n k ,m k + q ( iω j − iω j ′ ) × ν Margine, Lecture Fri.2 15/36

Migdal-Eliashberg Approximation � d q ˆ τ 3 ˆ � Σ n k ( iω j ) = − T ˆ G m k + q ( iω j ′ )ˆ τ 3 Ω BZ mj ′ �� � | g mnν ( k , q ) | 2 D q ν ( iω j − iω j ′ ) + V n k ,m k + q ( iω j − iω j ′ ) × ν � ∞ 2 ω D q ν ( iω j ) = dω ( iω j ) 2 − ω 2 δ ( ω − ω q ν ) 0 Margine, Lecture Fri.2 15/36

Migdal-Eliashberg Approximation � d q ˆ τ 3 ˆ � Σ n k ( iω j ) = − T ˆ G m k + q ( iω j ′ )ˆ τ 3 Ω BZ mj ′ �� � | g mnν ( k , q ) | 2 D q ν ( iω j − iω j ′ ) + V n k ,m k + q ( iω j − iω j ′ ) × ν � ∞ 2 ω D q ν ( iω j ) = dω ( iω j ) 2 − ω 2 δ ( ω − ω q ν ) 0 � ∞ 2 ω � j + ω 2 | g mnν ( k , q ) | 2 δ ( ω − ω q ν ) λ n k ,m k + q ( ω j ) = N F dω ω 2 0 ν Margine, Lecture Fri.2 15/36

Migdal-Eliashberg Approximation � d q ˆ τ 3 ˆ � Σ n k ( iω j ) = − T ˆ G m k + q ( iω j ′ )ˆ τ 3 Ω BZ mj ′ �� � | g mnν ( k , q ) | 2 D q ν ( iω j − iω j ′ ) + V n k ,m k + q ( iω j − iω j ′ ) × ν � ∞ 2 ω D q ν ( iω j ) = dω ( iω j ) 2 − ω 2 δ ( ω − ω q ν ) 0 � ∞ 2 ω � j + ω 2 | g mnν ( k , q ) | 2 δ ( ω − ω q ν ) λ n k ,m k + q ( ω j ) = N F dω ω 2 0 ν Migdal’s theorem Only the leading terms in Feynman diagram of the self-energy are included. The neglected terms are of the order of ( m e /M ) 1 / 2 ∝ ω D /ǫ F . Margine, Lecture Fri.2 15/36

Migdal-Eliashberg Approximation � d q ˆ � τ 3 ˆ Σ n k ( iω j ) = − T ˆ G m k + q ( iω j ′ )ˆ τ 3 Ω BZ mj ′ � � × λ n k ,m k + q ( ω j − ω j ′ ) + V n k ,m k + q ( iω j − iω j ′ ) Margine, Lecture Fri.2 16/36

Migdal-Eliashberg Approximation � d q ˆ � τ 3 ˆ Σ n k ( iω j ) = − T ˆ G m k + q ( iω j ′ )ˆ τ 3 Ω BZ mj ′ � � × λ n k ,m k + q ( ω j − ω j ′ ) + V n k ,m k + q ( iω j − iω j ′ ) 1 ˆ G n k ( iω j ) = − Θ n k ( iω j ) { iω j Z n k ( iω j )ˆ τ 0 + [( ǫ n k − ǫ F ) + χ n k ( iω j )] ˆ τ 3 + ∆ n k ( iω j ) Z n k ( iω j )ˆ τ 1 } Margine, Lecture Fri.2 16/36

Migdal-Eliashberg Approximation � d q ˆ � τ 3 ˆ Σ n k ( iω j ) = − T ˆ G m k + q ( iω j ′ )ˆ τ 3 Ω BZ mj ′ � � × λ n k ,m k + q ( ω j − ω j ′ ) + V n k ,m k + q ( iω j − iω j ′ ) 1 ˆ G n k ( iω j ) = − Θ n k ( iω j ) { iω j Z n k ( iω j )ˆ τ 0 + [( ǫ n k − ǫ F ) + χ n k ( iω j )] ˆ τ 3 + ∆ n k ( iω j ) Z n k ( iω j )ˆ τ 1 } � 1 � 0 � 1 � � � 0 1 0 τ 0 = ˆ τ 1 = ˆ ˆ τ 3 = 0 1 1 0 0 − 1 τ 3 ˆ ˆ τ 0 ˆ τ 3 = ˆ τ 0 and ˆ τ 3 ˆ τ 1 ˆ τ 3 = − ˆ τ 1 Margine, Lecture Fri.2 16/36

Migdal-Eliashberg Approximation � d q λ n k ,m k + q ( ω j − ω j ′ ) − N F V n k ,m k + q ( iω j − iω j ′ ) ˆ � Σ n k ( iω j ) = − T Ω BZ Θ m k + q ( iω j ′ ) mj ′ � � � × iω j ′ Z m k + q ( iω j ′ )ˆ τ 0 + ( ǫ m k + q − ǫ F ) + χ m k + q ( iω j ′ ) τ 3 ˆ � − ∆ m k + q ( iω j ′ ) Z m k + q ( iω j ′ )ˆ τ 1 Margine, Lecture Fri.2 17/36

Migdal-Eliashberg Approximation � d q λ n k ,m k + q ( ω j − ω j ′ ) − N F V n k ,m k + q ( iω j − iω j ′ ) ˆ � Σ n k ( iω j ) = − T Ω BZ Θ m k + q ( iω j ′ ) mj ′ � � � × iω j ′ Z m k + q ( iω j ′ )ˆ τ 0 + ( ǫ m k + q − ǫ F ) + χ m k + q ( iω j ′ ) τ 3 ˆ � − ∆ m k + q ( iω j ′ ) Z m k + q ( iω j ′ )ˆ τ 1 ˆ Σ n k ( iω j ) = iω j [1 − Z n k ( iω j )] ˆ τ 0 + χ n k ( iω j )ˆ τ 3 + ∆ n k ( iω j ) Z n k ( iω j )ˆ τ 1 Margine, Lecture Fri.2 17/36

Migdal-Eliashberg Approximation � d q λ n k ,m k + q ( ω j − ω j ′ ) − N F V n k ,m k + q ( iω j − iω j ′ ) ˆ � Σ n k ( iω j ) = − T Ω BZ Θ m k + q ( iω j ′ ) mj ′ � � � × iω j ′ Z m k + q ( iω j ′ )ˆ τ 0 + ( ǫ m k + q − ǫ F ) + χ m k + q ( iω j ′ ) τ 3 ˆ � − ∆ m k + q ( iω j ′ ) Z m k + q ( iω j ′ )ˆ τ 1 ˆ Σ n k ( iω j ) = iω j [1 − Z n k ( iω j )] ˆ τ 0 + χ n k ( iω j )ˆ τ 3 + ∆ n k ( iω j ) Z n k ( iω j )ˆ τ 1 Θ n k ( iω j ) = [ ω j Z n k ( iω j )] 2 +[( ǫ n k − ǫ F ) + χ n k ( iω j )] 2 +[ Z n k ( iω j )∆ n k ( iω j )] 2 Margine, Lecture Fri.2 17/36

Migdal-Eliashberg Approximation � d q λ n k ,m k + q ( ω j − ω j ′ ) − N F V n k ,m k + q ( iω j − iω j ′ ) ˆ � Σ n k ( iω j ) = − T Ω BZ Θ m k + q ( iω j ′ ) mj ′ � � � × iω j ′ Z m k + q ( iω j ′ )ˆ τ 0 + ( ǫ m k + q − ǫ F ) + χ m k + q ( iω j ′ ) τ 3 ˆ � − ∆ m k + q ( iω j ′ ) Z m k + q ( iω j ′ )ˆ τ 1 ˆ Σ n k ( iω j ) = iω j [1 − Z n k ( iω j )] ˆ τ 0 + χ n k ( iω j )ˆ τ 3 + ∆ n k ( iω j ) Z n k ( iω j )ˆ τ 1 Θ n k ( iω j ) = [ ω j Z n k ( iω j )] 2 +[( ǫ n k − ǫ F ) + χ n k ( iω j )] 2 +[ Z n k ( iω j )∆ n k ( iω j )] 2 Equating the scalar coefficients of the Pauli matrices leads to the anisotropic Migdal-Eliashberg equations. Margine, Lecture Fri.2 17/36

Anisotropic Migdal-Eliashberg Equations � d q T ω j ′ Z m k + q ( iω j ′ ) � Z n k ( iω j ) = 1 + ω j N F Ω BZ Θ m k + q ( iω j ′ ) mj ′ � � × λ n k ,m k + q ( ω j − ω j ′ ) − N F V n k ,m k + q ( iω j − iω j ′ ) � d q χ n k ( iω j ) = − T ( ǫ m k + q − ǫ F ) + χ m k + q ( iω j ′ ) � N F Ω BZ Θ m k + q ( iω j ′ ) mj ′ � � × λ n k ,m k + q ( ω j − ω j ′ ) − N F V n k ,m k + q ( iω j − iω j ′ ) � d q T Z m k + q ( iω j ′ )∆ m k + q ( iω j ′ ) � Z n k ( iω j )∆ n k ( iω j ) = N F Ω BZ Θ m k + q ( iω j ′ ) mj ′ � � × λ n k ,m k + q ( ω j − ω j ′ ) − N F V n k ,m k + q ( iω j − iω j ′ ) Margine, Lecture Fri.2 18/36

Anisotropic Migdal-Eliashberg Equations Standard approximations • only the off-diagonal contributions of the Coulomb self-energy are retained in order to avoid double counting of Coulomb effects Margine, Lecture Fri.2 19/36

Anisotropic Migdal-Eliashberg Equations Standard approximations • only the off-diagonal contributions of the Coulomb self-energy are retained in order to avoid double counting of Coulomb effects • static screening approximation → the Coulomb contribution to the self-energy is given by the ˆ τ 1 component of G n k ( iω j ) which is off-diagonal → the Coulomb contribution to Z n k ( iω j ) vanishes � d q T ω j ′ Z m k + q ( iω j ′ ) � Z n k ( iω j ) = 1 + ω j N F Ω BZ Θ m k + q ( iω j ′ ) mj ′ � � × λ n k ,m k + q ( ω j − ω j ′ ) − N F V n k ,m k + q ( iω j − iω j ′ ) Margine, Lecture Fri.2 19/36

Anisotropic Migdal-Eliashberg Equations Standard approximations • only the off-diagonal contributions of the Coulomb self-energy are retained in order to avoid double counting of Coulomb effects • static screening approximation → the Coulomb contribution to the self-energy is given by the ˆ τ 1 component of G n k ( iω j ) which is off-diagonal → the Coulomb contribution to Z n k ( iω j ) vanishes • all quantities are evaluated around the Fermi surface → χ n k ( iω j ) vanishes when integrated on the Fermi surface because it is an odd function of ω j Margine, Lecture Fri.2 19/36

Anisotropic Migdal-Eliashberg Equations Standard approximations • only the off-diagonal contributions of the Coulomb self-energy are retained in order to avoid double counting of Coulomb effects • static screening approximation → the Coulomb contribution to the self-energy is given by the ˆ τ 1 component of G n k ( iω j ) which is off-diagonal → the Coulomb contribution to Z n k ( iω j ) vanishes • all quantities are evaluated around the Fermi surface → χ n k ( iω j ) vanishes when integrated on the Fermi surface because it is an odd function of ω j • the electron density of states is assumed to be constant Margine, Lecture Fri.2 19/36

Anisotropic Migdal-Eliashberg Equations Standard approximations • only the off-diagonal contributions of the Coulomb self-energy are retained in order to avoid double counting of Coulomb effects • static screening approximation → the Coulomb contribution to the self-energy is given by the ˆ τ 1 component of G n k ( iω j ) which is off-diagonal → the Coulomb contribution to Z n k ( iω j ) vanishes • all quantities are evaluated around the Fermi surface → χ n k ( iω j ) vanishes when integrated on the Fermi surface because it is an odd function of ω j • the electron density of states is assumed to be constant • the dynamically screened Coulomb interaction N F V n k ,m k ′ is embedded into the semiempirical pseudopotential µ ∗ c Margine, Lecture Fri.2 19/36

Anisotropic Migdal-Eliashberg Equations � d q πT ω j ′ � Z n k ( iω j ) = 1 + ω j N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ mass renormalization function × λ n k ,m k + q ( ω j − ω j ′ ) δ ( ǫ m k + q − ǫ F ) � d q Z n k ( iω j )∆ n k ( iω j ) = πT ∆ m k + q ( iω j ′ ) � N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ superconducting gap function λ n k ,m k + q ( ω j − ω j ′ ) − µ ∗ � � × δ ( ǫ m k + q − ǫ F ) c Margine, Lecture Fri.2 20/36

Anisotropic Migdal-Eliashberg Equations � d q πT ω j ′ � Z n k ( iω j ) = 1 + ω j N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ mass renormalization function × λ n k ,m k + q ( ω j − ω j ′ ) δ ( ǫ m k + q − ǫ F ) � d q Z n k ( iω j )∆ n k ( iω j ) = πT ∆ m k + q ( iω j ′ ) � N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ superconducting gap function λ n k ,m k + q ( ω j − ω j ′ ) − µ ∗ � � × δ ( ǫ m k + q − ǫ F ) c Margine, Lecture Fri.2 20/36

Anisotropic Migdal-Eliashberg Equations � d q πT ω j ′ � Z n k ( iω j ) = 1 + ω j N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ mass renormalization function × λ n k ,m k + q ( ω j − ω j ′ ) δ ( ǫ m k + q − ǫ F ) � d q Z n k ( iω j )∆ n k ( iω j ) = πT ∆ m k + q ( iω j ′ ) � N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ superconducting gap function λ n k ,m k + q ( ω j − ω j ′ ) − µ ∗ � � × δ ( ǫ m k + q − ǫ F ) c � ∞ 2 ω � j + ω 2 | g mnν ( k , q ) | 2 δ ( ω − ω q ν ) λ n k ,m k + q ( ω j ) = N F dω ω 2 0 ν anisotropic e-ph coupling strength Margine, Lecture Fri.2 20/36

Anisotropic Migdal-Eliashberg Equations � d q πT ω j ′ � Z n k ( iω j ) = 1 + ω j N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ mass renormalization function × λ n k ,m k + q ( ω j − ω j ′ ) δ ( ǫ m k + q − ǫ F ) � d q Z n k ( iω j )∆ n k ( iω j ) = πT ∆ m k + q ( iω j ′ ) � N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ superconducting gap function λ n k ,m k + q ( ω j − ω j ′ ) − µ ∗ � � × δ ( ǫ m k + q − ǫ F ) c � ∞ 2 ω � j + ω 2 | g mnν ( k , q ) | 2 δ ( ω − ω q ν ) λ n k ,m k + q ( ω j ) = N F dω ω 2 0 ν � ∞ 2 ω anisotropic e-ph dωα 2 F n k ,m k + q ( ω ) = ω 2 j + ω 2 coupling strength 0 anisotropic Eliashberg spectral function Margine, Lecture Fri.2 20/36

What about the Coulomb Interaction? Screened Coulomb interaction V n k ,m k + q = � n k , − n k | W | m k + q , − m k + q � Giustino, Cohen, Louie, PRB 81, 115105 (2010); Lambert and Giustino, PRB 88, 075117 (2013) Margine, Lecture Fri.2 21/36

What about the Coulomb Interaction? Screened Coulomb interaction V n k ,m k + q = � n k , − n k | W | m k + q , − m k + q � W can be calculated within the random phase approximation in Giustino, Cohen, Louie, PRB 81, 115105 (2010); Lambert and Giustino, PRB 88, 075117 (2013) Margine, Lecture Fri.2 21/36

What about the Coulomb Interaction? Screened Coulomb interaction V n k ,m k + q = � n k , − n k | W | m k + q , − m k + q � W can be calculated within the random phase approximation in µ c = N F �� V n k ,m k + q �� FS Giustino, Cohen, Louie, PRB 81, 115105 (2010); Lambert and Giustino, PRB 88, 075117 (2013) Margine, Lecture Fri.2 21/36

What about the Coulomb Interaction? Screened Coulomb interaction V n k ,m k + q = � n k , − n k | W | m k + q , − m k + q � W can be calculated within the random phase approximation in µ c = N F �� V n k ,m k + q �� FS Morel-Anderson semiempirical pseudopotential µ c µ ∗ c = 1 + µ c log( ω el /ω ph ) Giustino, Cohen, Louie, PRB 81, 115105 (2010); Lambert and Giustino, PRB 88, 075117 (2013) Margine, Lecture Fri.2 21/36

Anisotropic Migdal-Eliashberg Equations � d q πT ω j ′ � Z n k ( iω j ) = 1 + ω j N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ × λ n k ,m k + q ( ω j − ω j ′ ) δ ( ǫ m k + q − ǫ F ) � d q ∆ m k + q ( iω j ′ ) Z n k ( iω j )∆ n k ( iω j ) = πT � N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ λ n k ,m k + q ( ω j − ω j ′ ) − µ ∗ � � × δ ( ǫ m k + q − ǫ F ) c Margine, Lecture Fri.2 22/36

Anisotropic Migdal-Eliashberg Equations � d q πT ω j ′ � Z n k ( iω j ) = 1 + ω j N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ × λ n k ,m k + q ( ω j − ω j ′ ) δ ( ǫ m k + q − ǫ F ) � d q ∆ m k + q ( iω j ′ ) Z n k ( iω j )∆ n k ( iω j ) = πT � N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ λ n k ,m k + q ( ω j − ω j ′ ) − µ ∗ � � × δ ( ǫ m k + q − ǫ F ) c • The coupled nonlinear equations need to be solved self-consistently at each temperature T Margine, Lecture Fri.2 22/36

Anisotropic Migdal-Eliashberg Equations � d q πT ω j ′ � Z n k ( iω j ) = 1 + ω j N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ × λ n k ,m k + q ( ω j − ω j ′ ) δ ( ǫ m k + q − ǫ F ) � d q ∆ m k + q ( iω j ′ ) Z n k ( iω j )∆ n k ( iω j ) = πT � N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ λ n k ,m k + q ( ω j − ω j ′ ) − µ ∗ � � × δ ( ǫ m k + q − ǫ F ) c • The coupled nonlinear equations need to be solved self-consistently at each temperature T • The equations must be evaluated on dense electron k - and phonon k ′ -meshes to properly describe anisotropic effects Margine, Lecture Fri.2 22/36

Anisotropic Migdal-Eliashberg Equations � d q πT ω j ′ � Z n k ( iω j ) = 1 + ω j N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ × λ n k ,m k + q ( ω j − ω j ′ ) δ ( ǫ m k + q − ǫ F ) � d q ∆ m k + q ( iω j ′ ) Z n k ( iω j )∆ n k ( iω j ) = πT � N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ λ n k ,m k + q ( ω j − ω j ′ ) − µ ∗ � � × δ ( ǫ m k + q − ǫ F ) c • The coupled nonlinear equations need to be solved self-consistently at each temperature T • The equations must be evaluated on dense electron k - and phonon k ′ -meshes to properly describe anisotropic effects • The sum over Matsubara frequencies must be truncated (typically set to four to ten times the largest phonon energy) Margine, Lecture Fri.2 22/36

Anisotropic Migdal-Eliashberg Equations � d q πT ω j ′ � Z n k ( iω j ) = 1 + ω j N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ × λ n k ,m k + q ( ω j − ω j ′ ) δ ( ǫ m k + q − ǫ F ) � d q ∆ m k + q ( iω j ′ ) Z n k ( iω j )∆ n k ( iω j ) = πT � N F Ω BZ � ω 2 j ′ +∆ 2 m k + q ( iω j ′ ) mj ′ λ n k ,m k + q ( ω j − ω j ′ ) − µ ∗ � � × δ ( ǫ m k + q − ǫ F ) c • The coupled nonlinear equations need to be solved self-consistently at each temperature T • The equations must be evaluated on dense electron k - and phonon k ′ -meshes to properly describe anisotropic effects • The sum over Matsubara frequencies must be truncated (typically set to four to ten times the largest phonon energy) • Z n k and ∆ n k are only meaningful for n k at or near the Fermi surface Margine, Lecture Fri.2 22/36

Isotropic Migdal-Eliashberg Equations Z ( iω j ) = 1 + πT ω j ′ � λ ( ω j − ω j ′ ) ω j � ω 2 j ′ +∆( iω j ) j ′ ∆( iω j ′ ) � λ ( ω j − ω j ′ ) − µ ∗ � � Z ( iω j )∆( iω j ) = πT c � ω 2 j ′ +∆ 2 ( iω j ′ ) j ′ Margine, Lecture Fri.2 23/36

Isotropic Migdal-Eliashberg Equations Z ( iω j ) = 1 + πT ω j ′ � λ ( ω j − ω j ′ ) ω j � ω 2 j ′ +∆( iω j ) j ′ ∆( iω j ′ ) � λ ( ω j − ω j ′ ) − µ ∗ � � Z ( iω j )∆( iω j ) = πT c � ω 2 j ′ +∆ 2 ( iω j ′ ) j ′ Isotropic e-ph coupling strength � ∞ 2 ω dωα 2 F ( ω ) λ ( ω j ) = ω 2 j + ω 2 0 Isotropic Eliashberg spectral function � d k � d q 1 � | g mnν ( k , q ) | 2 α 2 F ( ω ) = N F Ω BZ Ω BZ nmν × δ ( ω − ω q ν ) δ ( ǫ n k − ǫ F ) δ ( ǫ m k + q − ǫ F ) Margine, Lecture Fri.2 23/36

Examples from calculations and experiments Margine, Lecture Fri.2 24/36

Supeconductivity in Pb • Isotropic Migdal-Eliasbergh formalism (EPW) Figures adapted from Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 25/36

Supeconductivity in Pb • Isotropic Migdal-Eliasbergh formalism (EPW) superconducting gap edge ∆ 0 is defined as ∆ 0 = ∆( iω = 0) Figures adapted from Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 25/36

Supeconductivity in Pb • Isotropic Migdal-Eliasbergh formalism (EPW) superconducting gap edge ∆ 0 is defined as ∆ 0 = ∆( iω = 0) Figures adapted from Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 25/36

Supeconductivity in Pb • Isotropic Migdal-Eliasbergh formalism (EPW) superconducting gap edge ∆ 0 T c is defined as the temperature is defined as ∆ 0 = ∆( iω = 0) at which ∆ 0 = 0 Figures adapted from Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 25/36

Supeconductivity in Pb • Comparison between Migdal-Eliashberg and SCDFT formalism Right top and bottom figures from Marques et al, Phys. Rev. B 72, 024546 (2005) and Floris et al, Phys. Rev. B 75, 054508 (2007) Margine, Lecture Fri.2 26/36

Supeconductivity in Pb • Comparison between Migdal-Eliashberg and SCDFT formalism Right top and bottom figures from Marques et al, Phys. Rev. B 72, 024546 (2005) and Floris et al, Phys. Rev. B 75, 054508 (2007) Margine, Lecture Fri.2 26/36

Supeconductivity in Pb • Comparison between Migdal-Eliashberg and SCDFT formalism Right top and bottom figures from Marques et al, Phys. Rev. B 72, 024546 (2005) and Floris et al, Phys. Rev. B 75, 054508 (2007) Margine, Lecture Fri.2 26/36

Supeconductivity in Pb • Comparison between Migdal-Eliashberg and SCDFT formalism Right top and bottom figures from Marques et al, Phys. Rev. B 72, 024546 (2005) and Floris et al, Phys. Rev. B 75, 054508 (2007) Margine, Lecture Fri.2 26/36

Supeconductivity in Pb • Comparison between Migdal-Eliashberg and SCDFT formalism Right top and bottom figures from Marques et al, Phys. Rev. B 72, 024546 (2005) and Floris et al, Phys. Rev. B 75, 054508 (2007) Margine, Lecture Fri.2 26/36

Supeconductivity in MgB 2 Bottom left and right figures from Kortus et al, Phys. Rev. Lett. 86, 4656 (2001) and Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 27/36

Supeconductivity in MgB 2 Bottom left and right figures from Kortus et al, Phys. Rev. Lett. 86, 4656 (2001) and Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 27/36

Supeconductivity in MgB 2 FS Bottom left and right figures from Kortus et al, Phys. Rev. Lett. 86, 4656 (2001) and Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 27/36

Supeconductivity in MgB 2 FS Bottom left and right figures from Kortus et al, Phys. Rev. Lett. 86, 4656 (2001) and Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 27/36

Supeconductivity in MgB 2 e-ph coupling strength FS Bottom left and right figures from Kortus et al, Phys. Rev. Lett. 86, 4656 (2001) and Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 27/36

Supeconductivity in MgB 2 • Anisotropic Migdal-Eliasbergh formalism (EPW) superconducting gap on FS Left and right figures from Ponc´ e et al, Comp. Phys. Commun. 209, 116 (2016) and Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 28/36

Supeconductivity in MgB 2 • Anisotropic Migdal-Eliasbergh formalism (EPW) superconducting gap on FS Left and right figures from Ponc´ e et al, Comp. Phys. Commun. 209, 116 (2016) and Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 28/36

Supeconductivity in MgB 2 • Anisotropic Migdal-Eliasbergh formalism (EPW) superconducting gap on FS Left and right figures from Ponc´ e et al, Comp. Phys. Commun. 209, 116 (2016) and Margine and Giustino, Phys. Rev. B 87, 024505 (2013) Margine, Lecture Fri.2 28/36

Supeconductivity in MgB 2 • SCDFT formalism Figure and table from Floris et al, Physics C 456, 45 (2007) Margine, Lecture Fri.2 29/36

Supeconductivity in MgB 2 • SCDFT formalism A fully anisotropic calculation gave T c = 22 K. Figure and table from Floris et al, Physics C 456, 45 (2007) Margine, Lecture Fri.2 29/36

Supeconductivity in C 6 CaC 6 resistance vs temperature in C 6 CaC 6 Left and right figures from Ichinokura et al, ACS Nano 10, 2761 (2016) and Chapman et al, Sci. Rep. 6, 23254 (2016) Margine, Lecture Fri.2 30/36

Supeconductivity in C 6 CaC 6 resistance vs temperature magnetisation vs temperature in C 6 CaC 6 in Ca-doped graphite laminates Left and right figures from Ichinokura et al, ACS Nano 10, 2761 (2016) and Chapman et al, Sci. Rep. 6, 23254 (2016) Margine, Lecture Fri.2 30/36

Supeconductivity in C 6 CaC 6 • Anisotropic Migdal-Eliasbergh formalism with ab initio Coulomb pseudopotential µ ∗ c = 0 . 155 (EPW and SternheimerGW) Figures adapted from Margine et al, Sci. Rep. 6, 21414 (2016) Margine, Lecture Fri.2 31/36

Supeconductivity in C 6 CaC 6 • Anisotropic Migdal-Eliasbergh formalism with ab initio Coulomb pseudopotential µ ∗ c = 0 . 155 (EPW and SternheimerGW) superconducting gap on FS Figures adapted from Margine et al, Sci. Rep. 6, 21414 (2016) Margine, Lecture Fri.2 31/36

Supeconductivity in C 6 CaC 6 • Anisotropic Migdal-Eliasbergh formalism with ab initio Coulomb pseudopotential µ ∗ c = 0 . 155 (EPW and SternheimerGW) superconducting gap on FS Figures adapted from Margine et al, Sci. Rep. 6, 21414 (2016) Margine, Lecture Fri.2 31/36