One-step approach to ARPES from strongly correlated solids: A Mott-Hubbard system

An expression is derived for angle-resolved photocurrent from a semi-infinite correlated system. Within the sudden approximation, the photocurrent is proportional to the spectral function of a one-particle two-time retarded Green's function $\mathcal{G}$ of an operator that creates an electron in a special quantum state $\chi$ localized at the surface. For a system described by a many-body single-band model, we present an analytical expression that relates the Green's function $\mathcal{G}$ with the Green's function of an infinite crystal $G_{b,\mathbf{k}}\left(\omega \right)$ in Wannier representation. The role of final states and of the crystal surface is analyzed for a model Green's function of the infinite crystal with a three-peak spectral function typical of a Mott-Hubbard metal. The momentum dependences of both the quasiparticle pole position and the spectral weight of the incoherent band manifest themselves in the shape of the photocurrent energy distribution curve.

Figure 1

The spectral density of the bulk Green's function (40) $A_{\mathrm{b}}(\mathbf{k},\omega +i\eta )=-\mathrm{Im}{G}_{b,\mathbf{k}}(\omega +i\eta )/\pi$ (33) for ${\varepsilon }_{{\mathbf{k}}_{\parallel }}=-2.2T,k_z^{\prime }c=0,0.3\pi ,0.5\pi ,0.7\pi ,\pi$ (from bottom to the top), and the imaginary part of its self-energy $\mathrm{Im}{\mathrm{\Sigma }}_{{\mathbf{k}}_{\parallel },\omega +i\eta }/\pi b^2$ (70); $b_1=5T,b_2=b=T$. Here and below a small imaginary constant $i\eta ,\eta =0.001$ is added to the energy argument in order to visualize the coherent $\delta$-function peaks in the spectral densities. Dark red dotted line shows the quasiparticle dispersion $k_z=arccos\left[({\sigma }_{{\mathbf{k}}_{\parallel },\omega }-\omega )/2T\right]$.

Figure 2

The density of states $A_l$ (73) projected on a Bloch sum of Wannier functions located in the layer $z=lc$, Eq. (26). ${\varepsilon }_{{\mathbf{k}}_{\parallel }}=-2.2T$.

Figure 3

The spectral density $A(\omega +i\eta )$ (67) compared with the spectral density $A_{\mathrm{b}}(\mathbf{k},\omega +i\eta )$ (33) of the bulk Green's function (40) shown in Fig. 1 for (a) $V_{\mathrm{i}}=0.1T$ and (b) $V_{\mathrm{i}}=T$. The parameters are $\epsilon \left({\mathbf{k}}_0\right)=\hslash \mathrm{\Omega }-2.86T,{\varepsilon }_{{\mathbf{k}}_{\parallel }}=-2.2T{\epsilon }_z^{\prime }=10T/c,k_{0}c=0.36\pi$. The contributions coming from the bulk $[A_1(\omega +i\eta )$, Eq. (68)], and the surface $[A_2(\omega +i\eta )$, Eq. (69)] terms of the Green's function (46) are also shown. The thin blue line in the upper panel shows the spectral density behavior near the $(\delta$-functional) coherent peak of the bulk Green's function Lorentzian (61).

Figure 4

The spectral density $A(\omega +i\eta )$ (67) for different photon energies $\hslash \mathrm{\Omega }$. The lowest curve corresponds to $k_{0}c=0$, and the topmost one to $k_{0}c=\pi$. The energy step between the curves is $\hslash \mathrm{\Delta }\mathrm{\Omega }=0.89T$ and $V_{\mathrm{i}}=T$. The notation is the same as in the previous figures.

Figure 5

(a) $k_z^{\prime }$ (79) - thin lines, and $k_z^{\prime \prime }$ (80) - thick lines, for various values of “optic potential” $V_i$; $E_G=\hslash \mathrm{\Omega }-5T,{\varepsilon }_{{\mathbf{k}}_{\parallel }}=-2.2T,W=0.5T,{\epsilon }_z^{\prime }=10T/c$. The spectral density $A(\omega +i\eta )$ compared with the spectral density of the bulk Green's function (40) shown in Fig. 1 for (b) $V_i=0.1T$ and (c) $V_i=T$. The energy $E_0=\hslash \mathrm{\Omega }$ is used for the normalization in Eqs. (67, 68, 69).