First-principles nonequilibrium dynamical cluster theory for quantum transport simulations of disordered nanoelectronic devices

As important progress for simulating realistic device materials, we report the first-principles nonequilibrium dynamical cluster theory for simulating the quantum transport properties of nanoelectronics with inevitable disordered defects or dopants. In this method, we formulate the nonequilibrium dynamical cluster theory in Keldysh's Green's function representation, and implement it with the exact muffin-tin orbital based density functional theory. With this method, the important correlation effects of disordered scattering and short-range order effects can be effectively treated for the nonequilibrium electronic structure and quantum transport calculations of devices under finite bias. Moreover, a double-energy-contour technique is devised to considerably improve the numerical convergence in the nonequilibrium electron structure calculation. As the demonstration, the first-principles nonequilibrium dynamical cluster theory is applied to calculate Cu/Co junction with disordered interface and Fe/vacuum/Fe magnetic tunnel junction with surface roughness. We find that a sizable transmission decrease can be induced by including the correlation effects of disorders of few layers in the Cu/Co junction, presenting the important transport channel closing due to disordered quantum interference. For Fe/vacuum/Fe junction, we find that short-range order of surface roughness, with the important clustering and anticlustering tendencies, can dramatically change the transmission properties compared to the case of (or close to) complete randomness. The development of first-principles nonequilibrium dynamical cluster theory provides an important approach for analyzing the process-dependent device performance, extending the capability of first-principles quantum transport simulation.

Figure 1

(a) The Schwinger-Keldysh closed-time contour: starts from $-\infty$, passes through $\tau$ and ${\tau }^{\prime }$, and finally returns to $-\infty$. There are four real-time quantities $Q^{t,\overline{t},>,<}(\tau ,{\tau }^{\prime })$ with $\tau$ and ${\tau }^{\prime }$ on the two branches $C_{+}$ or $C_{-}$. (b) Schematic illustration of a disordered two-probe device. The device is divided into different principal layers labeled by $ip$ and contains clean left electrode $ip\le 0$, disordered central region $1\le ip\le N_p$, and clean right electrode $ip\ge N_p+1$. The disordered sites are included in different DCA clusters. (c) Schematic illustration of disorder-averaged two-probe device with the DCA effective medium.

Figure 2

(a) Schematic illustration of DCA cluster with Born–von Karman periodic boundary condition. (b) Schematic illustration of BZ partition for 2D BZ of fcc(111) plane when $N_{xy}=4$. The 4 ${\vec{K}}_{\parallel }$ are marked in the 2D BZ, the points of ${\vec{k}}_{\parallel }$ within the same coarse-grained region of ${\vec{K}}_{\parallel }$ are denoted with the same color.

Figure 3

Flow chart of the EMTO-DFT-NEGF-DCA self-consistent loop for the quantum transport simulation of device materials.

Figure 4

Schematic illustration of the energy contour for the energy integration in nonequilibrium electron density calculation. (a) The conventional contour is composed of $C_{T,1}$ in the upper complex plane from $E_{\text{BVal}}$ to $E_{f,R}$ and $C_{T,2}$ on the real energy axis from $E_{f,R}$ to $E_{f,L}$. (b) Double-energy contours are composed of four parts: $C_1, C_2$, and $C_4$ in the upper complex energy, and $C_3$ on the real energy axis from $E_{f,R}$ to $E_{f,L}$. (c) Structure of $\mathrm{Fe}/{\mathrm{Fe}}_{0.5}{\mathrm{Va}}_{0.5}\left(1\mathrm{ML}\right)/\mathrm{Va}\left(6\mathrm{MLs}\right)/{\mathrm{Fe}}_{0.5}{\mathrm{Va}}_{0.5}$(1 ML)/Fe MTJ with disordered interface roughness. (d), (e) For the integrated valence charge versus nonequilibrium self-consistent iterations using the conventional contour (e) and double-energy contour [for the 11th atomic layer in disordered Fe/Va/Fe MTJ as shown in (c) under the bias 1.0 V (starting from the equilibrium potential)]: blue, orange, and green lines are for the calculations with the respective $k_{\parallel }$ meshes of $24\times 24, 48\times 48$, and $96\times 96$.

Figure 5

(a)-(d): the four possible SRO maps for the DCA cluster of $N_{xy}=4$ and $N_z=1$. (e) Schematic illustration of SROs with clustering and anticlustering tendencies in comparison with complete randomness.

Figure 6

(a) Schematic illustration of 2D square-lattice model with seven disordered atomic layers along the $z$ direction in the central scattering region. (b), (c) Present the averaged transmission coefficients $\overline{\mathcal{T}}$ with different methods, for the case ${\epsilon }_{\text{im}}=10.0$ (${\epsilon }_{\text{host}}=1.0$). (b) For $\overline{\mathcal{T}}$ versus energy for device with seven disordered layers. (c) For $\overline{\mathcal{T}}$ versus disordered layers $N_z$ at energy is equal to 1.0. The pink dashed and black dotted-dashed lines with the respective functions of $A+B/N_z$ and $Aexp(-B{N}_z)+C$ are fitting to the results of CPA-NVC and SC methods (for understanding the scaling behaviors of different methods).

Figure 7

The total transmission coefficients versus interfacial disorder concentration x in $\mathrm{Cu}/{\mathrm{Cu}}_{1-x}{\mathrm{Co}}_x/\mathrm{Co}$ (111) junction with two ((a), (c)) and three ((b), (d)) disordered monolayers. Plots (a) and (b) are for the spin-up channel, (c) and (d) are for spin-down channel. Blue circles are for CPA-NVC results, orange squares, green up triangles, red down triangles, and purple stars are for the DCA calculations with the respective $N_{xy}^{\text{DCA}}=1,4,9$, and 16.

Figure 8

The total transmission coefficient of different spin channels (left axis) and TMR (right axis) with the CPA-NVC and DCA methods (with various SROs) for Fe/Va/Fe MTJ with surface roughness of ${\mathrm{Fe}}_{0.5}{\mathrm{Va}}_{0.5}$.

Figure 9

The ${\vec{k}}_{\parallel }$-resolved transmission (coherent and diffusive parts) of different spin channels in PC and APC with the CPA-NVC and DCA methods (with various SROs) for Fe/Va/Fe MTJ with surface roughness of ${\mathrm{Fe}}_{0.5}{\mathrm{Va}}_{0.5}$.

Figure 10

The nonequilibrium properties of Fe/Va/Fe MTJ with surface roughness of ${\mathrm{Fe}}_{0.5}{\mathrm{Va}}_{0.5}$: (a) nonequilibrium distribution function versus sites in the central region for the energy point $E=0.5+E_{f,R}$ under the bias voltage 1 V. (b) The potential difference from the equilibrium versus sites in central region: blue circle, orange square, and green up triangle are for the respective applied bias voltages 0.2, 0.6, and 1.0 V. The transmission coefficients versus energy for different spin channels for the bias voltage 1.0 V: (c)–(f) are for the spin-up and -down channels in PC, spin-up, and spin-down channels in APC, respectively.