Impact of Nanostructure Configuration on the Photovoltaic Performance of Quantum-Dot Arrays

In this paper, a mesoscopic model based on the nonequilibrium Green’s function formalism for a tight-binding-like effective Hamiltonian is used to investigate a selectively contacted quantum-dot array designed for operation as a single-junction quantum-dot solar cell. By establishing a direct relation between nanostructure configuration and optoelectronic properties, the investigation reveals the influence of interdot and dot-contact coupling strengths on the rates of charge-carrier photogeneration, radiative recombination, and extraction at contacts, and, consequently, on the ultimate performance of photovoltaic devices with finite quantum-dot arrays as the active medium. For long carrier lifetimes, the dominant configuration effects originate in the dependence of the joint density of states on the interdot coupling in terms of band width and effective band gap. In the low-carrier-lifetime regime, where recombination competes with carrier extraction, the extraction efficiency shows a critical dependence on the dot-contact coupling.

Figure 1

Schematic band diagram and phenomenological parameters characterizing the selectively contacted QD array. Isolated QD states at energies ${ϵ}_{c,v}$ are connected through interdot coupling terms $t_{c,v}$. To enable photocarrier extraction and injection of dark current under applied bias voltage $V_{\mathrm{bias}}$, the QD states are connected to contacts via coupling elements $V_B$. The carrier selectivity of the contacts, i.e., $V_{cL}=V_{vR}=0$, is required for photocurrent rectification at flat-band conditions in the absence of a doping-induced junction. While left and right contacts are assumed to be equilibrated with charge-carrier populations characterized by chemical potentials ${\mu }_L$ and ${\mu }_R={\mu }_L+V_{\mathrm{bias}}$, respectively, the carrier population inside the absorber emerges entirely from the quantum-statistical formalism applied.

Figure 2

Local (left) and total (right) density of electron and hole states for a selectively contacted 20-QD array with $V_{cR}=V_{vL}=0.1\mathrm{eV}$, $t_c=-0.03\mathrm{eV}$, and $t_v=-0.02\mathrm{eV}$. At the closed contacts, the band width approaches the infinite single-orbital tight-binding chain value of $4|t_b|$, $b=c,v$ (dotted lines). On the contrary, at the open contacts, hybridization with contact states induces a pronounced widening of the bands and a shift of the band edge towards the electrode edges at ${ϵ}_{cR}^{\prime }=0.5\mathrm{eV}$ and ${ϵ}_{vL}^{\prime }=-0.6\mathrm{eV}$, according to the local density of contact states at the surface of the semi-infinite electrode (dashed lines).

Figure 3

Convergence of the average density of electronic states per QD to the infinite 1D TB chain limit with increasing number of QDs between the contacts, assuming symmetric contact coupling $V_{cL}=V_{cR}=0.1\mathrm{eV}$, $t_c=-0.03\mathrm{eV}$, and a homogeneous broadening ${\eta }_c=0.5\mathrm{meV}$.

Figure 4

Site-integrated density of electron states for a 20-QD array ($V_{cL}=V_{cR}=0.1\mathrm{eV}$, $t_c=-0.03\mathrm{eV}$) with coupling-induced broadening only (“coh”), constant homogeneous broadening $\eta =0.5\mathrm{meV}$ (“hom”), and for quasielastic coupling to optical phonons of energy $\hslash \mathrm{\Omega }=0.2\mathrm{meV}$ (“phon”).

Figure 5

Convergence of the JDOS of a 20-QD system with increasing nonlocality and to the infinite 1D TB chain for large system size (inset).

Figure 6

(a) AC and SR of the 20-QD cell as obtained from the NEGF, in comparison with the AC of the infinite 1D TB model. (b) Spectral emission rate for the 20-QD cell at a terminal bias voltage $V_{\mathrm{bias}}\equiv {\mu }_R-{\mu }_L=0.6\mathrm{V}$ as derived from the NEGF formalism (empty dots) as well as from the generalized Planck law using the NEGF absorption (solid line) or the infinite 1D TB model (dotted line). The inset shows the integrated emission rate as a function of $V_{\mathrm{bias}}$, with open symbols for the values given by the current expression (30) and filled squares for the values obtained from the global emission rate (33).

Figure 7

Evolution during the NEGF self-consistency iteration of the energy-integrated rates for charge-carrier generation ($R_{\mathrm{gen}}$), radiative recombination ($R_{\mathrm{em}}$), and extraction ($I^{\mathrm{tot}}/q$) normalized to the absorption rate given by the incident photon flux and the absorptance, for different regimes of operation. (a) Optical extraction only ($\to {\eta }_{\mathrm{ext}}=0$), i.e., photoluminescence at closed contacts ($V_c=V_v\equiv V_B=0$) and a recombination lifetime ${\tau }_r\approx 1\mathrm{ps}$. (b) Unit carrier extraction at electrodes ($\to {\eta }_{\mathrm{ext}}=1$) due to strong dot-contact coupling $V_{cR}=V_{vL}\equiv V_B=0.15\mathrm{eV}$ and long recombination lifetime ${\tau }_r\approx 1\mathrm{s}$. (c) Incomplete carrier extraction ($\to 0<{\eta }_{\mathrm{ext}}<1$) for recombination lifetimes (${\tau }_r\approx 10\mathrm{ps}$) comparable to the escape time at finite but low contact coupling $V_{cR}=V_{vL}\equiv V_B=0.02\mathrm{eV}$. In all cases, an interdot coupling of $|t_{c,v}|=0.03,0.02\mathrm{eV}$ is used.

Figure 8

Local density of electron states of a right-contacted three-QD system for varying values (a) of the contact coupling $V_{cR}$ at fixed interdot coupling $|t_c|=0.02\mathrm{eV}$ and (b) of the interdot coupling $|t_c|$ at fixed contact coupling $V_{cR}=0.1\mathrm{eV}$. The LDOS far from the contact acquires a regular form with peak multiplicity determined by the number of QDs. The spectral shape of the LDOS in the direct vicinity of the contact exhibits the asymmetric Fano-type signatures of the coupling of a discrete level system to the continuum of states in the bulk electrode. The effects of the contact coupling are a level broadening and a redshift that both increase with the coupling strength $V_{cR}$. The energy separation of the resonances, on the other hand, is directly proportional to the interdot coupling $|t_c|$.

Figure 9

LDOS of a selectively contacted 20-QD array resolved on the individual dot sites and displayed for different values of the contact coupling $V_{cR}=0.05,0.15,0.3\mathrm{eV}$, at a fixed interdot coupling of $|t_c|=0.03\mathrm{eV}$. While for the weakest coupling, the DOS is almost symmetric both along the array and with respect to the central level energy at ${ϵ}_c=0.6\mathrm{eV}$, the asymmetry induced to the real part of the contact self-energy is already visible at intermediate coupling and results in the formation of a bound interface state below the low-energy edge of the miniband at the contact layer. In addition, there is the broadening of the DOS increasing with coupling strength, which originates in the imaginary part of the contact self-energy.

Figure 10

The absorption cross section for the 20-QD cell is presented in (a) for various values of dot-contact couplings $V_{cR}=V_{vL}\equiv V_B$ at $|t_{c,v}|=0.03,0.02\mathrm{eV}$ and in (b) for different interdot coupling parameters $t=|t_c|=|t_v|$ at $V_{cR}=V_{vL}\equiv V_B=0.05\mathrm{eV}$ (dashed lines indicate band width of 1D TB chain).

Figure 11

Spectral emission rate for the 20 QDs at a terminal bias voltage $V_{\mathrm{bias}}=0.6\mathrm{V}$ in (a) for different dot-contact couplings and in (b) for different interdot couplings. The inset displays the corresponding variation of the energy-integrated emission rate.

Figure 12

Convergence of the terminal current in the NEGF self-consistency iteration for different values of contact coupling $V_B$, fixed interdot coupling $|t_{c,v}|=0.03,0.02\mathrm{eV}$ and recombination lifetimes of (a) ${\tau }_r\approx 100\mathrm{ps}$, (b) ${\tau }_r\approx 10\mathrm{ps}$, and (c) ${\tau }_r\approx 1\mathrm{ps}$.

Figure 13

Convergence of the local current resolved on individual dot position and for different values of the contact coupling $V_B$, revealing a pronounced dependence of photocurrent rectification on the size of the coupling, which is related to the extraction rate as well as to reflection of photogenerated charge carriers.

Figure 14

Convergence of the terminal current in the NEGF self-consistency iteration for different values of interdot coupling $|t_{c,v}|$, fixed contact coupling $V_B=0.05\mathrm{eV}$, and recombination lifetimes of (a) ${\tau }_r\approx 100\mathrm{ps}$, (b) ${\tau }_r\approx 10\mathrm{ps}$, and (c) ${\tau }_r\approx 1\mathrm{ps}$.

Figure 15

Carrier extraction efficiency for configurations of interdot coupling $|t_{c,v}|\in [5,50]\mathrm{meV}$ and dot-contact coupling $V_B\in [0.01,0.15]\mathrm{eV}$ and different recombination lifetimes. At all lifetimes, the predominant effect on the extraction efficiency originates in the dot-contact coupling.