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^{1,a)}and Kaustav Banerjee

^{1,a)}

### Abstract

Metallic nanoparticle assisted band-to-band tunneling is proposed, and the impact of such nanoparticle induced states on the tunneling probability and current is modeled and analyzed. An analytical formula for tunneling probability is derived for the case of constant force, and it is shown that the incorporation of these particles in the forbidden gap can lead to a substantial increase in the tunneling current. The effect of the Fermi-level pinning position on the tunneling current is studied, and the pinning value for obtaining maximum improvement in current is discussed depending on the force conditions. It is also shown that an asymmetric pinning is required to leverage maximum benefits from the insertion of metallic nanoparticles.

This work was supported by the National Science Foundation, Grant No. CCF-0811880. Authors would like to thank A. C. Gossard and C. J. Palmstrøm at UCSB for their useful discussions.

### Key Topics

- Tunneling
- 41.0
- Nanoparticles
- 15.0
- Semiconductors
- 6.0
- Conduction bands
- 4.0
- Valence bands
- 4.0

## Figures

(Color online) (a) Schematic band diagram of a tunnel junction where metallic nanoparticles have been incorporated leading to zero bandgap between segment-1 and segment-2. Due to the presence of the nanoparticles, tunneling process can now be divided into two parts: through segment-1 and segment-2 (denoted by horizontal arrows). *V* is the voltage applied across the junction. () denote the energy differences between the valence (conduction) band in segment-1 (segment-2) and the Fermi level. The Fermi level pinning position at the metal-semiconductor interface is denoted by *p* which is defined from the conduction band as a fraction of the bandgap (*E _{G} *). Segment-1 is shown separately in (b), highlighting the energy dependent (0 to

*E*) tunneling length (

_{max}*d*

_{0}), while for (c) the standard case without any nanoparticles,

*d*

_{0}is independent of energy (

*E*). Note that the constant force (linear band bending) is assumed to derive analytical expressions. In real devices, Poisson-Schrodinger equations should be solved simultaneously for calculation of actual band-bending.

(Color online) (a) Schematic band diagram of a tunnel junction where metallic nanoparticles have been incorporated leading to zero bandgap between segment-1 and segment-2. Due to the presence of the nanoparticles, tunneling process can now be divided into two parts: through segment-1 and segment-2 (denoted by horizontal arrows). *V* is the voltage applied across the junction. () denote the energy differences between the valence (conduction) band in segment-1 (segment-2) and the Fermi level. The Fermi level pinning position at the metal-semiconductor interface is denoted by *p* which is defined from the conduction band as a fraction of the bandgap (*E _{G} *). Segment-1 is shown separately in (b), highlighting the energy dependent (0 to

*E*) tunneling length (

_{max}*d*

_{0}), while for (c) the standard case without any nanoparticles,

*d*

_{0}is independent of energy (

*E*). Note that the constant force (linear band bending) is assumed to derive analytical expressions. In real devices, Poisson-Schrodinger equations should be solved simultaneously for calculation of actual band-bending.

(Color online) Tunneling probability as a function of energy level *E* (shown in Fig. 1(b)) highlighting the energy dependence of MN-tunneling probability. For all figures in this paper, *E _{G} * = 0.93 eV and

*m*

^{*}^{ }= 0.064

*m*

_{0}.

(Color online) Tunneling probability as a function of energy level *E* (shown in Fig. 1(b)) highlighting the energy dependence of MN-tunneling probability. For all figures in this paper, *E _{G} * = 0.93 eV and

*m*

^{*}^{ }= 0.064

*m*

_{0}.

(Color online) Current as a function of Fermi level pinning position *p* (defined in Fig. 1(a)) for similar (*F* _{1} = *F* _{2}) as well as dissimilar (*F* _{1} ≠ *F* _{2}) forces in the two segments. For *F* _{1} = *F* _{2}, the peak current occurs at *p* = 0.5, while for *F* _{1} ≠ *F* _{2,} the peak position shifts depending on the forces. The fraction of voltage drop and hence the depletion region width in each segment (in Fig.1(a)) change so that the current continuity is satisfied.

(Color online) Current as a function of Fermi level pinning position *p* (defined in Fig. 1(a)) for similar (*F* _{1} = *F* _{2}) as well as dissimilar (*F* _{1} ≠ *F* _{2}) forces in the two segments. For *F* _{1} = *F* _{2}, the peak current occurs at *p* = 0.5, while for *F* _{1} ≠ *F* _{2,} the peak position shifts depending on the forces. The fraction of voltage drop and hence the depletion region width in each segment (in Fig.1(a)) change so that the current continuity is satisfied.

(Color online) Current as a function of the force (*F* _{1} = *F* _{2}) for two different values of *V* and *p* = 0.5. It is observed that MN-tunneling leads to a substantial increase in current compared to those without MN.

(Color online) Current as a function of the force (*F* _{1} = *F* _{2}) for two different values of *V* and *p* = 0.5. It is observed that MN-tunneling leads to a substantial increase in current compared to those without MN.

(Color online) (a) Current density as a function of the energy level *E* plotted from 0 to Δ*E.* It is seen that the current initially increases with increase in *E* due to increase in the tunneling probability, but then decreases due to decrease in (*f* _{i}(*E*) − *f* _{f}(*E*)). (b) Schematic band diagram showing different pinning positions defined by *p* _{1} and *p* _{2} at the two segments where *p* _{1} > 0.5 while *p* _{2} < 0.5. (c) The difference in Fermi function between the left segment and the metal given by (*f* _{1}(*E*) − *f* _{m}(*E*)) as a function of the tunneling length for different *p* _{1} with *p* _{2} = 1 − *p* _{1}. We observe that as *p* _{1} increases, the window of tunneling length over which current flows (which also corresponds to the energy window Δ*E*) shifts towards lower values of tunneling lengths. (d) Effective increase in the tunneling current with increase in *p* _{1}.

(Color online) (a) Current density as a function of the energy level *E* plotted from 0 to Δ*E.* It is seen that the current initially increases with increase in *E* due to increase in the tunneling probability, but then decreases due to decrease in (*f* _{i}(*E*) − *f* _{f}(*E*)). (b) Schematic band diagram showing different pinning positions defined by *p* _{1} and *p* _{2} at the two segments where *p* _{1} > 0.5 while *p* _{2} < 0.5. (c) The difference in Fermi function between the left segment and the metal given by (*f* _{1}(*E*) − *f* _{m}(*E*)) as a function of the tunneling length for different *p* _{1} with *p* _{2} = 1 − *p* _{1}. We observe that as *p* _{1} increases, the window of tunneling length over which current flows (which also corresponds to the energy window Δ*E*) shifts towards lower values of tunneling lengths. (d) Effective increase in the tunneling current with increase in *p* _{1}.

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