$\mathit{GW}+T$ theory of excited electron lifetimes in metals

We develop a first-principle $\mathit{GW}+T$ approach to calculate excited electron lifetimes in metals that includes evaluation of the lowest self-energy term of the many-body perturbation theory in GW approximation and higher terms in the $T$-matrix approximation. The method is applied to studies of the electron lifetimes in Pd, Ta, and Al. We find that the $T$-matrix contribution to the lifetime is more important in Al than in Ta and Pd and relate this to the static screened potential. The inclusion of the $T$-matrix greatly improves agreement between experimental and theoretical results in Ta and Al.

Figure 1

Feynman diagrams for GW and $T$-matrix self-energy of an excited electron. A: GW-term; B: $T$-matrix direct terms with multiple electron-electron scattering; C: $T$-matrix direct terms with electron-hole scattering; D: $T$-matrix exchange terms. The vertical wiggly line represents static screened potential, and lines with the arrows are Green’s functions. The time direction is towards the right. Changing the time direction, one obtains analogous diagrams for the self-energy of an excited hole.

Figure 2

Electron scattering processes included in GW method.

Figure 3

Feynman diagrams for the self-energy of an electron (top figure) and for the BS equation with multiple electron-hole scattering (bottom figure). The positive time direction is to right. For the BS equation with multiple electron-electron scattering the direction of both Green’s functions will be positive.

Figure 4

Electrons scattering processes included in the $T$-matrix theory. In the right column with zigzag line we connect the particles with multiple scattering. In the left column we show Feynman diagrams for the illustrated processes.

Figure 5

Total density of states in Pd, Ta, Al.

Figure 6

Frequency dependence of the main matrix element of Pd spectral function at three points $q=(2\pi ∕a)(x,0,0)$ of the Brillouin zone.

Figure 7

Electron energy band structure in Pd.

Figure 8

Components of the transverse susceptibility in Pd. Solid line: Real part of $R^{-+}$; dashed line: Imaginary part of $R^{-+}$. $q=(2\pi ∕a)(0.0625,0,0)$.

Figure 9

Top panel: Frequency dependence of the main matrix element of the inverse enhancement factor ${(1-W\mathrm{Re}K)}_{11}$ in Pd. Bottom panel: Imaginary part (dashed line) and real part (solid line) of the main matrix element of electron-hole kernel in Pd. $q=(2\pi ∕a)(0.0625,0,0)$.

Figure 10

Momentum-averaged electron and hole lifetimes in Pd. Black diamonds: Lifetimes within GW approach; open diamonds: Lifetimes within $\mathit{GW}+T$ approach, including double-counting corrections; open circles: Lifetimes within $\mathit{GW}+T$ approach without double-counting corrections.

Figure 11

Frequency dependence of the main matrix element $S_{11}$ of Ta spectral function at $q=(2\pi ∕a)(x,0,0)$.

Figure 12

Components of transverse susceptibility in Ta. Solid line: real part of $R^{-+}$; dashed line: Imaginary part of $R^{-+}$. $q=(2\pi ∕a)(0.0625,0,0)$.

Figure 13

Top panel: Frequency dependence of the main matrix element of the inverse enhancement factor ${(1-WK)}_{11}$ in Ta. Bottom panel: Imaginary part (dashed line) and real part (solid line) of the main matrix element of electron-hole kernel in Ta. $q=(2\pi ∕a)(0.0625,0,0)$.

Figure 14

Momentum-averaged electron and hole lifetimes in Ta. Black diamonds: Lifetimes within GW approach; open diamonds: Lifetimes within $\mathit{GW}+T$ approach, including double-counting corrections; small open circles: Lifetimes within $\mathit{GW}+T$ approach without double-counting corrections; open squares: experimental relaxation times in Ta, Ref. 42.

Figure 15

Components of the transverse susceptibility in Al at $q=(2\pi ∕a)(0.05,0.05,-0.05)$. Thick solid line: real part of $R^{-+}$; thick dashed line: Imaginary part of $R^{-+}$. With thin solid (dashed) line in figure for $R_K+R_T$ we show the Fermi liquid results at $r_s=2.0$.

Figure 16

Top panel: Frequency dependence of the inverse enhancement factor ${(1-WK)}_{11}$ for Al. Bottom panel: Imaginary part (dashed line) and real part (solid line) of the main matrix element of kernel in Al. $q=(2\pi ∕a)(0.05,0.05,0.05)$.

Figure 17

Momentum-averaged electron lifetimes in Al. Solid diamonds: ab initio GW calculations; big open diamonds: $\mathit{GW}+T$ calculations with included multiple electron-hole scattering and double counting term D(R); small open diamonds: $\mathit{GW}+T$ calculations with included multiple electron-hole scattering and omitted double counting term; open circles: $\mathit{GW}+T$ calculations with included multiple electron-hole and electron-electron scattering and double counting terms D(R), D(P); solid squares: Experimental data of Ref. 13 not corrected for cascade electrons and transport effects. Thick solid line represents the results of the Fermi-liquid theory, whereas thick dashed line corresponds to Fermi liquid theory with included multiple electron-hole scattering, see text.