1. Introduction

Operation frequency of modern microprocessors does not exceed a few gigahertz due to high heat generation and interconnect delays. SPPs, which are surface electromagnetic waves propagating at the interface between a metal and an insulator, are considered as very promising information carriers that can replace electrons in integrated circuits [1–3]. An exceedingly short wavelength and a very high spatial localization of the electromagnetic field near the interface allow to get over the usual diffraction limit and design ultracompact interconnects with the transverse size of the order of 100 nm [1,4] that is comparable with electronic components. Unfortunately, high propagation losses due to Joule heating restrict the application of SPPs. Thus, one should increase the SPP propagation length, i.e. partially or fully compensate Joule losses. This can be done by using an active media placed near a metal surface [5]. In recent years, a number of paper devoted to the SPP amplification have been published [6–14] and several methods have been proposed. Despite the advantages of these methods, the necessity of an external high power pump laser prevents us to use them in nanoscale circuits.

In this paper, we propose a different technique that is based on a minority carrier injection effect in metal-semiconductor contacts that gives one a possibility to use compact electrical pumping instead of a bulky optical approach.

2. Schottky barrier diode

Usually, Schottky diodes are treated as majority carrier devices. For instance, if one has an n-type semiconductor-metal contact, the electron concentration is much greater than the concentration of holes all over the semiconductor at all bias voltages (here and below, only the case of forward bias is considered) and the hole current is much less than the electron one. However, the situation changes drastically when the metal work function ${\psi }_{\text{M}}$ exceeds ${\chi }_{\text{e}}+E_{\text{g}}/2$, where ${\chi }_{\text{e}}$ and $E_{\text{g}}$ are the electron affinity and the band gap of the semiconductor, respectively. In this case, the concentration of holes (minority carriers) near the metal-semiconductor contact becomes greater than the concentration of electrons (majority carriers) and it is said that an inversion layer is formed. Under forward bias, holes are injected into the bulk of the semiconductor and recombine with electrons that results in light emission [15]. So, Schottky barriers can be used to design efficient and compact light- [15] and plasmon-emitting diodes [16], but what about lasers and amplifiers? To design a laser, one should satisfy the condition for net stimulated emission or gain [17,18]

How can we satisfy inequality (1)? Firstly, if we use a degenerate semiconductor, $F_{\text{e}}-E_{\text{c}}$ is positive nearly everywhere inside the semiconductor under sufficient forward bias. Hence, one should only maximize the difference $E_{\text{v}}-F_{\text{h}}.$ It is obvious that, near the metal-semiconductor contact, $F_{\text{h}}$ is very close to the metal Fermi level $F_{\text{m}}=0$ and one can increase $E_{\text{v}}-F_{\text{h}}$ by increasing the metal work function or decreasing the electron affinity of the semiconductor. Inside the semiconductor, $E_{\text{v}}-F_{\text{h}}$ will decrease but in the region near the Schottky contact condition (1) is still satisfied. Thus, one should maximize ${\psi }_{\text{M}}-{\chi }_{\text{e}}-E_{\text{g}}$ and make it positive.

To demonstrate the principle of operation of the proposed device, consider a structure depicted in Fig. 1(a) . For simplicity, assume the back contact to be an ideal ohmic contact, i.e.

 

Fig. 1 (a) Sketch of the ideal Schottky barrier diode and schematic band diagram under zero bias, L = 400 nm, ψ _M = 5.4 eV [34,35], ε _st = 13.94 [36], χ _e = 4.5 eV [36], E _g = 0.75 eV that correspond to Ga_0.47In_0.53As at T = 300 K and donor concentration N _d = 9.3 × 10¹⁷ cm⁻³. (b) Carrier density distribution and (c) band diagram under zero bias.

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Thus, we have four boundary conditions (Eqs. (1) and (2)) and six nonlinear first order differential equations that describe the carrier behavior within the semiconductor:

The carrier recombination rate consists of three components $U=U_{\text{spont}}+U_{\text{stim}}+U_{\text{nr}}$, as long as there are three recombination processes: spontaneous emission $(U_{\text{spont}}),$ stimulated emission $(U_{\text{stim}})$ and non-radiative Schockley-Read-Hall and Auger recombination $(U_{\text{nr}}).$ In direct-band-gap semiconductors, $U_{\text{nr}}$ is usually much less than $U_{\text{spont}},$ therefore it is not discussed in the present work.

To begin with, we demonstrate that it is possible to satisfy inequality (1). For this purpose, let $U_{\text{stim}}$ be zero, while $U_{\text{spont}}=B(np-n_{\text{eq}}{p}_{\text{eq}})$, where $n_{\text{eq}}$ and $p_{\text{eq}}$ are the equilibrium concentrations and $B=1.43\times {10}^{-10}{\text{cm}}^3{\text{s}}^{-1}$ for In_0.53Ga_0.47As [23]. In the presence of degeneracy and in the case of high minority carrier injection, system of Eqs. (4) cannot be solved analytically and we have to implement the Newton-Raphson method. Under zero bias, namely at thermal equilibrium, $F_{\text{e}}=F_{\text{h}}=F_{\text{m}}=0$ (Figs. 1(b) and 1(c)). As the bias increases, holes are injected into the bulk of the semiconductor and $F_{\text{h}}$ shifts downward. At the same time, the concentration of electrons changes slightly and $F_{\text{e}}$ remains constant (Fig. 2 ). Under high forward bias (Fig. 2(b)), the difference between quasi-Fermi levels exceeds $E_{\text{g}}$ and we satisfy inequality (1). This clearly demonstrates that it is possible to realize a SPP amplifier based on a Schottky barrier diode [24].

 

Fig. 2 Quasi-Fermi levels at V = 0.8 V (a) and V = 0.85 V (b).

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3. SPP dispersion

The dispersion relation for SPPs propagating along the planar interface between a metal and a semiconductor (Fig. 3(a) ) with permittivities ${\epsilon }_1$ and ${\epsilon }_2$, respectively, has the form ${\kappa }_2{\epsilon }_1={\kappa }_1{\epsilon }_2,$ where ${\kappa }_i=\sqrt{{\beta }^2-{(\omega /c)}^2{\epsilon }_i}$ is the penetration constant ($i=1,2$ and β is the SPP wavevector). Assuming the semiconductor to be lossless and taking into account losses in the metal [25], we calculate the propagation length of the SPP at the interface between Au and Ga_0.47In_0.53As. At a light wavelength of 1.7 µm (0.73 eV), $\beta =(143926+806i){\text{cm}}^{-1}$ that corresponds to a wavelength of 436 nm and a propagation length of 6.2 µm. The imaginary part of β is much less than the real one and losses almost do not affect the field distribution and SPP wavelength, therefore we will use the power flow approach. The essence of the method is that only the real parts of permittivities are used to determine the real part of the wavevector $(\mathrm{Re}\beta =144018{\text{cm}}^{-\text{1}}\text{),}$ while $\mathrm{Im}\beta$ is found from the power flow equation [26,27]. This method provides a simple treatment and gives a clear physical interpretation of the SPP attenuation or amplification. Let us denote by P the power flow per unit guide width ($P={\displaystyle {\int }_{-\infty }^{+\infty }{S}_{x}dz}$, where $S_x$ is the x-component of the complex Poynting vector) and by R the Joule loss power per unit guide length per unit guide width $(R=\omega /8\pi {\displaystyle {\int }_{-\infty }^0\mathrm{Im}{\epsilon }_1}{\left|E\right|}^{2}dz).$ Then $dP/dx=-2\mathrm{Im}\beta P=-R$ [27] and consequently $\mathrm{Im}\beta =R/2P=815{\text{ cm}}^{-1}.$ The power flow in the metal is much less than in the semiconductor that is due to the great difference in penetration depths of the SPP field inside the metal (22 nm) and semiconductor (223 nm). Despite that, the absorption in the metal is high enough and the material gain of the order of $2\mathrm{Im}\beta ={\text{1630 cm}}^{-1}$ is required in the semiconductor medium to compensate losses.

 

Fig. 3 (a) Schematic view of the SPP propagation along the metal-semiconductor interface. (b) Dependence of the material gain on the minority carrier density in In_0.53Ga_0.47As at a light wavelength of 1.7 µm. (c) Gain spectra of In_0.53Ga_0.47As.

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4. SPP amplification by stimulated emission of radiation

Stimulated emission recombination rate is given as

Taking into account stimulated emission, we solve Eq. (4) numerically in the same way as it was done in section 2 (Fig. 4 ). For a small signal of 5 mW/µm or less (Fig. 4(a)), the stimulated emission recombination rate is greater than the spontaneous but its absolute value is nevertheless quite small $(eU<<(J_{\text{n}}+J_{\text{p}})/L)$ and does not affect the carrier distribution (Fig. 4(a)). In this case, the net SPP gain G at $V=$1.07 V is positive and equals $1/P\times \left({\displaystyle {\int }_0^{L}g(z)S_x(z)}dz-R\right)=1780-1630=150{\text{cm}}^{-1}.$ At a high signal power (Fig. 4(b)), $eU>(J_{\text{n}}+J_{\text{p}})/L$ in the region near the metal-semiconductor interface that affects the carrier density distribution. The material gain is smaller (Fig. 4(b)) and the net SPP gain becomes negative ($G=-620{\text{cm}}^{-1}$ at $V=$1.07 V and $P=$50 mW/μm).

 

Fig. 4 Gain, current density (dashed lines correspond to the electron current and solid lines to the hole current) and carrier density distribution across the active region of the plasmonic waveguide at four different biases for two different values of the power per unit guide width.

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5. Conclusion

To conclude, we have proposed a novel SPP amplification scheme that utilizes a compact electrical pumping and gives an ability to design really nanoscale amplifiers and spasers [33]. For the analysis of the scheme, we have developed a fully self-consistent one-dimensional steady-state model of the amplifier and presented an accurate numerical solution.