1. Introduction

Surface plasmon polaritons (SPPs) being surface electromagnetic waves propagating along the interface between a metal and an insulator give a unique opportunity to overcome the conventional diffraction limit and thus can be used for light manipulation at the nanoscale [1]. This gives plasmonic components a significant advantage over their photonic counterparts in the integration density and strength of light-matter interaction. However, high mode confinement is paid off by a significant localization of the SPP electromagnetic field in the metal and the resulting propagation length in nanoscale plasmonic waveguides does not exceed a few tens of micrometers due to absorption in the metal [1–3]. Nevertheless, it is possible to overcome this limitation by compensating ohmic losses with optical gain in the adjacent dielectric. Full loss compensation was successfully demonstrated by optical pumping [4–6], which is easily implemented in a laboratory, but is impractical due to its very poor energy efficiency, stray illumination, and necessity of an external high power pump light source. In this regard, an efficient electrical pumping scheme is strongly needed for practical realization of SPP guides and circuits [7–9].

In this paper, we propose a novel SPP amplification scheme based on minority carrier injection in metal-insulator-semiconductor (MIS) structures and demonstrate numerically full loss compensation in an electrically pumped active hybrid plasmonic waveguide. In contrast to the techniques based on heterostructures and quantum wells [10], our approach gives a possibility to bring the active gain medium to the metal surface at a distance of a few nanometers. At the same time, a thin insulator layer can efficiently block the majority carrier current, which is quite high in metal-semiconductor Schottky-barrier-diode SPP amplifiers [11]. This guarantees high mode confinement to the active region and very low threshold currents.

2. Operating principle

Realization of an electrical pumping scheme for loss compensation in SPP waveguides and cavities is a significant challenge, since there are at least two limitations in the development of plasmonic components. Firstly, plasmonics gives a possibility to reduce the mode size well below the diffraction limit, but, in order to use this opportunity, the component size should be comparable with the mode size. Accordingly, one has to use the SPP supporting metal-dielectric or metal-semiconductor interface as an electrical contact. Secondly, a limited choice of low-loss materials with negative real part of permittivity poses a serious challenge for efficient carrier injection, because gold, silver and copper typically form Schottky contacts to direct bandgap semiconductors. As a result, either the modal gain is rather low for full loss compensation, or the threshold current density is quite high [12], which leads to significant power consumption and low energy efficiency of the amplification scheme.

Our technique based on an electrically pumped MIS structure (Fig. 1) is remarkably different from the previously proposed approaches [10, 11, 13–15]. First, SPP propagation losses are reduced by placing a very thin layer of a low refractive index insulator between the metal and semiconductor. In this case, the SPP electromagnetic field is pushed into the insulator reducing the portion of the SPP mode in the metal, which, however, does not impair the mode confinement [3]. Second, and more important, a thin insulator layer can play a dual role: it is semitransparent for electrons moving from the metal to semiconductor, which ensures favourable conditions for efficient minority carrier injection, and blocks the hole current. Such a unique feature gives a possibility to suppress high leakage currents [16], which are unavoidable in metal-semiconductor Schottky contacts [14], and not to form an ohmic contact to the semiconductor at the SPP supporting interface, as opposed to double-heterostructure and quantum-well SPP amplifiers [12]. At the same time, the proposed scheme provides high density of non-equilibrium carriers right near the metal surface, at the distance equal to the thickness of the insulator layer. At a high bias voltage, it becomes possible to create a sufficiently high electron concentration in the semiconductor and to satisfy the condition for population inversion [17], which provides gain for the plasmonic mode propagating in the MIS waveguide.

 

Fig. 1 SPP amplification schemes based on a Schottky barrier diode [11] and tunnel MIS contact. Top panels: Energy band diagrams for the Schottky contact between gold and indium arsenide in equilibrium (a) and at high forward bias (b). The Schottky barrier height for electrons φ_Bn at the Au/InAs contact is negative owing to the large density of surface states and, under high forward bias, electrons (minority carriers in p-type InAs) are freely injected in the bulk of the semiconductor. However, majority carriers (holes) also pass across the Au/InAs contact without any resistance, which results in a high leakage current. Bottom panels: Energy band diagrams for the tunnel Au/HfO₂/InAs MIS structure in equilibrium (c) and at high forward bias (d). If the barrier height for holes χ_VI is substantially greater than that for electrons (φ_MI), the insulator layer can efficiently block majority carriers (holes), but be semi-transparent for minority carriers (electrons) escaping from the metal.

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Previously, an attempt to describe the carrier transport in an electrically pumped hybrid plasmonic waveguide was made in [18]. However, the authors assumed that both electrons and holes pass freely through the insulating layer and used the thermionic emission boundary conditions (literally the same as in the Schottky junction theory [9, 19]). This is erroneous in the presence of the insulating barrier and contradicts the well-established theory of current transfer through thin insulating films [20, 21]. In fact, the insulating barrier strongly suppresses the thermionic current, while the main current component is due to tunneling.

The electron and hole tunnel injection currents J_t,cb and J_t,vb across the MIS contact can be calculated by integrating the flux of carriers incident on the contact timed by the transparency of the insulating barrier (see Appendix A). According to the energy band diagram shown in Fig. 1 (c,d), the ratio of the electron tunneling probability D_cb to the hole tunneling probability D_vb can be easily estimated as

We next explore the carrier transport in the semiconductor. It is important to note that the electric potential is abruptly screened near the interface between the insulator and heavily doped semiconductor, which directly follows from the solution of Poisson’s equation. At a doping density of N_A = 10¹⁸ cm⁻³, the estimated screening length is equal to 2 nm (see Appendix B), while the electron mean free path limited by impurity scattering equals l_fp = 10 nm. This implies that electrons move ballistically through the screening region. Out of the screening region, the driving electric field is weak and the electron transport is governed by diffusion. Accordingly, the density distribution of the injected electrons is given by

The electron current J_0,n at the semiconductor-insulator junction is mainly determined by quantum mechanical tunneling through the insulating layer. In the case of an ideal interface, J_0,n is simply equal to the tunnel current J_t,cb. However, the presence of intrinsic and extrinsic surface states at the semiconductor-insulator interface has a significant impact on the characteristics of the tunnel MIS contact (see Appendix C). First, surface states act as charge storage centers and affect the voltage drop across the insulator layer and band bending in the semiconductor. Second, these states are recombination centers [26] and reduce the efficiency of electron injection. The electron current at the insulator-semiconductor interface J_0,n is smaller than the tunnel injection current J_t,cb by the value of the surface recombination current J_sr,n. Third, direct tunneling from the metal to semiconductor through the surface states is also possible [27, 28]. This process contributes only to the majority carrier current and reduces the energy efficiency of the SPP amplification scheme. Fortunately, intrinsic surface states in InAs have a moderate density ρ_ss ≃ 3 × 10¹² cm⁻²eV⁻¹ [29]. Deposition of metal [30] or insulator [31] layers produces additional defects and ρ_ss can be increased up to 10¹⁴ cm⁻²eV⁻¹ at HfO₂/InAs interfaces [32]. Nevertheless, both intrinsic and extrinsic states can be treated by surface passivation before HfO₂ deposition. The reported values of ρ_ss at the trimethylaluminum-treated HfO₂/InAs interface is about 10¹³ cm⁻²eV⁻¹ [33], while advanced oxygen-termination and surface reconstruction techniques give a possibility to reduce ρ_ss down to 2 × 10¹¹ cm⁻²eV⁻¹ [34], which eliminates the influence of surface states on carrier transport in high quality samples.

3. Results and discussion

3.1. Active hybrid plasmonic waveguide in the passive regime

In order to achieve strong SPP mode localization and efficiently inject both electrons and holes into the active semiconductor region, a modified T-shaped plasmonic waveguide approach [11] has been implemented [Fig. 2(a)]. InAs rib with an acceptor concentration of 10¹⁸ cm⁻³ on InAs substrate is covered by a thin HfO₂ layer and surrounded by a low refractive index dielectric (SiO₂) to confine optical modes in the lateral direction. Finally, a metal layer is placed on top of the semiconductor-insulator structure to form an SPP supporting interface and a tunnel MIS contact for electron injection, while the substrate plays a role of the ohmic contact for majority carrier (hole) injection.

 

Fig. 2 Schematic of a T-shaped hybrid plasmonic waveguide based on the Au/HfO₂/InAs MIS structure, w is the waveguide width, d_i is the thickness of the low refractive index insulator layer between the metal and semiconductor, and H is the waveguide height. (b) Distribution of the normalized energy density per unit length of the waveguide for the fundamental TM₀₀ mode at λ = 3.22 µm, H = 2.5 µm, w = 400 nm and d_i = 3 nm. The dielectric functions of the materials are as follows: ${\epsilon }_{{\mathrm{Si}\mathrm{O}}_2}=2.00$ [35], ${\epsilon }_{{\mathrm{Hf}\mathrm{O}}_2}=3.84$ [36], ε_InAs = 12.38 [37] and ε_Au = −545+38i [11].

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Two-dimensional eigenmode simulations using the finite element method (COMSOL Multiphysics software) reveal that the waveguide depicted in Fig. 2(a) supports a deep-subwavelength TM₀₀ SPP mode [Fig. 2(b)]. For the chosen waveguide dimensions, at a free space wavelength of 3.22 µm, which appears to be optimal for SPP amplification (see the next section for details), the effective index of the TM₀₀ mode is equal to 2.799 and the propagation length in the passive regime assuming the semiconductor to be lossless is about 66 µm corresponding to a modal loss of 2Imβ_psv = 152 cm⁻¹. The other modes supported by the T-shaped waveguide are very leaky photonic modes. In spite of poor field confinement to the lossy metal, propagation lengths of the photonic TM₁₀, TE₀₀, and TE₁₀ modes are 5.8 µm, 12.8 µm and 2.7 µm, respectively, which are much shorter than that of the TM₀₀ mode due to leakage into the high refractive index substrate. The rib height of H = 2.5 µm is close to the optimum, since, as H decreases, the radiation loss of the plasmonic mode becomes substantially greater than the absorption in the metal and the SPP propagation length decreases down to 39 µm at H = 2 µm. On the other hand, greater rib heights do not provide a significant loss reduction, while high-aspect-ratio structures are difficult in fabrication. For the selected optimal geometrical parameters, the SPP mode demonstrates an exceptionally high level of confinement [38] within the InAs region of more than 95% provided by the small mode width and high group index. At the same time, the spontaneous emission enhancement by the Purcell effect does not appreciably decrease the electron lifetime. Thanks to the moderate SPP confinement in the vertical direction, the Purcell factor F_p(d_i) in InAs near the metal surface does not exceed 1.28, which results in a maximum of 7% shortening of τ_R compared to the bulk case.

3.2. Full loss compensation in a deep-subwavelength waveguide

Applying a negative voltage to the top Au electrode of the active plasmonic waveguide shown in Fig. 2(a), one injects electrons into the p-type InAs region. This creates a high density of non-equilibrium electrons in the semiconductor rib, which reduces absorption in InAs [17]. As the bias voltage increases, the quasi-Fermi level for electrons shifts upward towards the conduction band. When the energy difference between quasi-Fermi levels for electrons and holes exceeds the energy $\hslash \omega$ of the SPP quantum, InAs starts to compensate for the SPP propagation losses. The modal gain G is given by the overlap integral of the material gain profile g(y, z) and the electric field distribution of the SPP mode [38]:

Here, E(y, z) and P_z(y, z) are the complex amplitudes of the electric field and the local power density of the SPP mode, respectively, and 2Imβ_psv is attributed to the SPP radiation and ohmic losses. In turn, the material gain coefficient g(y, z) is related to the carrier quasi-Fermi levels F_n(y, z) and F_p(y, z) by the integral over transitions between the energy levels in the conduction and valence bands separated by the energy $\hslash \omega$ (see Appendix D). In heavily doped semiconductors, the material gain strongly depends on the impurity concentration [39], so are the electrical properties (L_D, µ_n, and τ_R). The optimal acceptor concentration is found to be 10¹⁸ cm⁻³. As the doping concentration increases, the Auger recombination rate and the impurity scattering rate rapidly increase greatly reducing the electron diffusion length. On the other hand, at lower doping levels, the number of free electron states in the valence band is insufficient to provide a pronounced gain upon electrical injection. At the given impurity concentration and a reasonably high density of injected electrons of 2×10¹⁶ cm⁻³ (which corresponds to F_n − E_c = k_BT), the material gain spectrum exhibits a maximum of 310 cm⁻¹ at $\hslash \omega =0.385$ eV (λ = 3.22 µm), sufficiently high for the net SPP amplification.

Figure 3 shows the simulated gain-current characteristic of the plasmonic TM₀₀ mode in the active hybrid plasmonic waveguide cooled down to 77 K for the HfO₂/InAs interfaces of different quality. All curves exhibit the same trend. At zero bias, the concentration of holes in the semiconductor is many orders of magnitude greater than that of electrons, InAs strongly absorbs the SPP propagating in the waveguide and the modal loss is two-and-a-half times greater than in the case of the lossless (e.g. wide bandgap) semiconductor. As the bias voltage increases, electrons are injected in the InAs rib [Fig. 4(a)], which reduces absorption in the semiconductor near the MIS contact, but still, InAs strongly absorbs at a distance greater than L_D/2 [Fig. 4(b)]. Nevertheless, the SPP modal loss steadily decreases with the injection current (this region is shown in blue in Fig. 3). At some point [green curve in Fig. 4(b)], the overlap between the distributions of the material gain and the SPP field is zero and, at higher bias voltages, the gain in InAs produced by injected electrons partially compensates for the SPP propagation losses.

 

Fig. 3 SPP modal gain versus pump current for different densities of surface states at the HfO₂/InAs interface. Blue, yellow and red regions show three regimes of the active plasmonic waveguide: at low injection currents, the electron concentration in InAs is quite small for population inversion; in the yellow region, the SPP propagation losses are partially compensated by optical gain in InAs; and at high injection currents, the material gain is significantly large for net SPP gain. The black circles highlight the breakdown condition for the HfO₂ insulating layer V_i/d_i = 1 V/nm [40].

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Fig. 4 (a) Contributions of different tunneling processes to the injection current. (b) Material gain profile across the InAs rib at different bias voltages and spatial profile of the SPP electric field averaged over the waveguide width ${|E_{\mathrm{avg}}(z)|}^2={\displaystyle {\int }_{-w/2}^{w/2}{|E_{\mathrm{avg}}(y,z)|}^2}dy/w$. For both panels, the density of surface states equals 10¹³ cm⁻²eV⁻¹.

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Eventually, at an injection current of about 2.7 kA/cm² (Fig. 3), ohmic losses and radiation losses of the plasmonic mode are fully compensated. For comparison, it was recently numerically shown [11] that in active plasmonic waveguides based on the Au/p-InAs Schottky contact under the similar conditions a current density of the order of 20 kA/cm² is needed for full compensation of the SPP propagation losses. Here, we have achieved one order of magnitude reduction in the threshold current density using the tunneling Au/HfO₂/InAs contact instead of the Schottky contact and blocking the majority carrier current with a thin insulator layer.

As evident from Fig. 3, the current density required for full loss compensation is robust against the quality of the HfO₂/InAs interface. It is equal to 2.6 kA/cm² in the case of the ideal interface and increases by only 12% as the density of surface states increases to 2 × 10¹³ cm⁻²eV⁻¹. This small increase is completely attributed to the tunnel current from the metal to surface states [Fig. 4(a)], while the surface recombination current J_sr,n is more than six orders of magnitude lower than the total current. This is explained by the energy barrier for electrons in the semiconductor due to band bending at the HfO₂/InAs interface under high forward bias [41].

Interesting is the possibility to control optical and electrical properties of the active T-shaped hybrid plasmonic waveguide by varying the thickness of the insulator layer. As d_i increases, the portion of the SPP field in the metal decreases, which results in a decrease of ohmic losses. At the same time, thicker insulating layers block the majority carrier current more efficiently as follows from Equation (1). However, it appears that the tunnel current is very sensitive to the thickness of the HfO₂ layer and d_i = 3 nm is the optimal value. As d_i is decreased by 0.5 nm (one atomic layer), the majority carrier current drastically increases due to direct tunneling from the metal to surface states and, at a surface-state density of 10¹³ cm⁻²eV⁻¹, the SPP propagation losses are fully compensated at a very high current density of 7.5 kA/cm² (see Appendix E). On the other hand, the 3.5 nm thick HfO₂ layer efficiently blocks the majority current, but the barrier width for minority carriers is also increased and higher bias voltages are needed for efficient electron injection. As a result, the electric field in HfO₂ rapidly reaches the breakdown value of 1 V/nm [40] as the electron current increases. The minimum achievable SPP modal loss is 240 cm⁻¹ that is much greater than 2Imβ_psv = 155 cm⁻¹ evaluated in the case of the lossless (e.g. wide bandgap) semiconductor. For the optimal HfO₂ layer thickness of 3 nm, the breakdown condition V_i = 3 V corresponds to the net SPP modal gain of 40 cm⁻¹, which can be increased up to 160 cm⁻¹ in high-quality samples, where the breakdown electric field can be as high as 1.2 V/nm [40].

3.3. Low-current loss compensation in copper waveguide

The effective density of states in the valence band of InAs is 75 times greater than in the conduction band. Therefore, in spite of the relatively large ratio ${\mathcal{D}}_{\mathrm{cb}}/{\mathcal{D}}_{\mathrm{vb}}=5$ for the Au/HfO₂/InAs contact, the electron tunnel current does not exceed that of holes [Fig. 4(a)]. In addition, under high forward bias, the effective barrier height for holes, which can be estimated as the energy difference between the hole quasi-Fermi level in InAs near the insulator-semiconductor contact and the valence band edge of HfO₂ at z = d_i, is substantially reduced due to band bending and the ratio J_t,cb/J_t,vb rapidly decreases as the bias voltage increases [Fig. 4(a)]. To improve the efficiency of the amplification scheme, one should increases the difference between barrier heights χ_VI and φ_MI as it can be seen from equation (1). To satisfy this demand, one or more materials in the T-shaped active plasmonic waveguide [Fig. 2(a)] should be replaced.

It would be naive to expect better characteristics in plasmonic structures based on metals different from gold and silver, however, in electrically pumped active plasmonic waveguides, both electrical and optical processes are involved, and the optimal configuration is typically the result of the interplay between them. Copper is characterized by higher absorption at optical frequencies than gold [42, 43] and the modal loss of the plasmonic TM₀₀ mode in the T-shaped waveguide increases from 152 cm⁻¹ to 189 cm⁻¹. On the other hand, the electron work function of copper is equal to 4.7 eV, which is 0.4 eV lower than that of gold. Such a small difference is sufficient to produce an appreciable effect on carrier tunneling through the insulating barrier and increase the efficiency of minority carrier injection in InAs. The electron-to-hole transparency ratio ${\mathcal{D}}_{\mathrm{cb}}/{\mathcal{D}}_{\mathrm{vb}}$ of the Au/HfO₂/InAs contact is as large as 20, and the electron tunnel current J_t,cb substantially exceeds the hole tunnel current J_t,vb (Fig. 5).

 

Fig. 5 Gain-current characteristic for the SPP mode of the T-shaped active hybrid plasmonic waveguide based on the Cu/HfO₂/InAs MIS structure and relative contributions of different tunnelling processes to the total current. The density of surface states is equal to 5×10¹² cm⁻²eV⁻¹ and the dimensions of the waveguide are the same as in Fig. 2(b).

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At a surface-state density of 5 × 10¹² cm⁻²eV⁻¹, the propagation losses of the plasmonic mode are fully compensated at a current density of 0.95 kA/cm², which is significantly smaller than in the case of the gold active plasmonic waveguide despite higher absorption losses of the SPP mode. Passivation of the InAs surface reduces the density of surface states by orders of magnitude [34], and the regime of full loss compensation is reached at a very low current J = 0.82 kA/cm², which is comparable with the threshold currents in double-heterostructure InAs lasers [44]. In addition, the copper waveguide operates at smaller bias voltages than the gold one. First, this results in lower energy consumption per unit waveguide length, which decreases from 30 mW/mm to 8.2 mW/mm in the regime of full loss compensation. Second, and more importantly, smaller bias voltages create lower electric fields in the HfO₂ layer. The breakdown condition (V_i/d_i = 1 V/nm) corresponds to the net SPP modal gain of 220 cm⁻¹ that is much higher than in the similar gold waveguide. This allows one to use the proposed amplification scheme in nanolasers [45], where not only ohmic, but also high radiation losses have to be compensated.

4. Conclusion

In summary, we have proposed a novel approach for SPP amplification in deep-subwavelength structures under electrical pumping and demonstrated full loss compensation in an active hybrid plasmonic waveguide. The amplification scheme is based on efficient minority carrier injection in metal-insulator-semiconductor contacts at large forward bias, which creates a population inversion in the semiconductor and provides optical gain for the plasmonic mode propagating along the metal interface. At the same time, a thin insulator layer blocks majority carriers giving a possibility to reduce the leakage current to that in double-heterostructure lasers. Comprehensive electronic/photonic simulations show that the SPP propagation losses can be fully compensated in the electrically driven active hybrid plasmonic waveguide based on the Au/HfO₂/InAs structure at a current density of only 2.6 kA/cm². Moreover, by replacing gold with copper, one significantly improves the efficiency of minority carrier injection, and, in spite of higher ohmic losses, the current density in the regime of lossless SPP propagation is reduced down to 0.8 kA/cm². Such an exceptionally low value demonstrates the potential of electrically pumped active plasmonic waveguides and plasmonic nanolasers for future high-density photonic integrated circuits.

Appendix A. Calculation of the tunnel current

The tunnel injection current across the metal-insulator-semiconductor (MIS) contact from the metal to the k-th band of the semiconductor [k = cb stands for the conduction and k = vb stands for the valence band] can be expressed as an integral over the transverse momenta p_⊥ and energies E of the incident electrons [20]:

(4)$$J_{\mathrm{t},k}=\frac{4\pi e}{{\left(2\pi \hslash \right)}^3}{\displaystyle \underset{E_{\mathrm{c}}}{\overset{+\infty }{\int }}dE\left[f_m(E)-f_k(E)\right]}{\displaystyle \underset{0}{\overset{p_{\perp ,\max }}{\int }}{p}_{\perp }d{p}_{\perp }}{\mathcal{D}}_k(E,p_{\perp }).$$

Here, D_k (E, p_⊥) is the probability of carrier tunneling to the k-th band of the semiconductor, p_⊥,max is the maximum transverse momentum of the carrier in the semiconductor at a given energy E and f_m(E), f_cb(E), and f_vb(E) are the electron distribution functions in the metal and respective bands of the semiconductor. For electron tunneling to the conduction band, $p_{\perp ,\max }=\sqrt{2m_{\mathrm{e}}(E-E_{\mathrm{c}})}$, where m_e = 0.026m₀ is the effective electron mass [51] and E_c is the conduction band edge in InAs. In the case of tunneling to the heavy (light) hole band, $p_{\perp ,\max }=\sqrt{2m_{\mathrm{hh}(\mathrm{lh})}(E_{\mathrm{v}}{|}_{z=d_{\mathrm{i}}}-E)}$, where m_lh = 0.024m₀ and m_hh = 0.36m₀ are the light and heavy hole effective masses, respectively, and $E_{\mathrm{v}}{|}_z{}_{=d_{\mathrm{i}}}$ is the valence band edge in InAs at the interface. The contributions from light and heavy holes are summed over when calculating the net hole current.

The tunneling probability can be directly obtained in the WKB-approximation

(5)$${\mathcal{D}}_k(E,p_{\perp })=\exp \left[-\frac{2\sqrt{2m_{\mathrm{i}}}}{h}{\displaystyle \underset{z_1}{\overset{z_2}{\int }}\sqrt{E-\frac{p_{\perp }^2}{2m_{\mathrm{i}}}-U_k(z)dz}}\right].$$

Here, z_1,2 are the classical turning points, m_i is the effective mass in the insulator [22], and U_cb(z) and U_vb(z) are the conduction and valence band edges in the insulator. To obtain closed-form expressions for the tunneling probability, we introduce the following notations [see Fig. 6(a)]: φ_MI is the barrier height at the metal-insulator interface; E_c and E_c z_=d are the conduction band edges in the bulk and at the interface, respectively; F_i = V_i/d_i is the the electric field in the insulator, and F_s is the electric field in the semiconductor at the semiconductor-insulator interface. The height and width of the barrier and, consequently, its transparency strongly depend on the electron energy E which we reckon from the Fermi level in the metal.

1. Electrons with energies $E-p_{\perp }^{2}2m_{\mathrm{i}}>{\phi }_{\mathrm{MI}}-e{V}_{\mathrm{i}}$ pass through the triangular barrier in the insulator [red arrow in Fig. 6(a)]. The corresponding transparency, D_cb,I, is given by

   (6)$$\ln D_{\mathrm{cb},\mathrm{I}}(E,p_{\perp })=-\frac{4\sqrt{2m_{\mathrm{i}}}}{3e\hslash F_{\mathrm{i}}}{\left[{\phi }_{\mathrm{MI}}+\frac{p_{\perp }^2}{2m_{\mathrm{i}}}-E\right]}^{3/2}.$$
2. Electrons with energies ${E_{\mathrm{c}}|}_{z=d_{\mathrm{i}}}<E-p_{\perp }^2/2m_{\mathrm{i}}<{\phi }_{\mathrm{MI}}-e{V}_{\mathrm{i}}$ tunnel through the trapezoid barrier in the insulator [green arrow in Fig. 6(a)]. The transparency D_cb,II of the trapezoid barrier is

   (7)$$\ln D_{\mathrm{cb},\mathrm{II}}(E,p_{\perp })=-\frac{4\sqrt{2m_{\mathrm{i}}}}{3e\hslash F_{\mathrm{i}}}\left\{{\left[{\phi }_{\mathrm{MI}}+\frac{p_{\perp }^2}{2m_{\mathrm{i}}}-E\right]}^{3/2}-{\left[{\phi }_{\mathrm{MI}}-e{V}_{\mathrm{i}}+\frac{p_{\perp }^2}{2m_{\mathrm{i}}}-E\right]}^{3/2}\right\}.$$
3. Finally, electrons with energies E_c < E − p² ⊥/2me < Ec z=d tunnel through the trapezoid barrier in the insulator followed by a short section of the bandgap in the semiconductor [blue arrow in Fig. 6(a)]. It should be noted here that the electric field in heavily doped semiconductors is abruptly screened near the surface. At a doping density of N_A = 10¹⁸ cm⁻³, the screening length in InAs is about l_s=2 nm [see Fig. 6(a)]. Hence, the contribution of these electrons to the tunnel current is non-negligible. The transparency of such a compound barrier is given by

   (8)$$\ln D_{\mathrm{cb},\mathrm{III}}(E,p_{\perp })=\ln D_{\mathrm{II}}(E,p_{\perp })-\frac{4}{3}\frac{\sqrt{2m_{\mathrm{e}}}}{e\hslash F_{\mathrm{s}}}{\left[{E_{\mathrm{c}}|}_{z=d_{\mathrm{i}}}-E+\frac{p_{\perp }^2}{2m_{\mathrm{e}}}\right]}^{3/2}.$$

 

Fig. 6 (a) Illustration of three different components of the electron tunnel current: tunneling through the triangular barrier (I), tunneling through the trapezoid barrier (II), and tunneling through the trapezoid barrier followed by a short (~ 2 nm) section of the bandgap in the semiconductor (III). (b) Schematic illustration of carrier transport in the MIS structure under forward bias. Black arrows indicate the tunnel currents between the metal and the conduction band (J_t,cb) and between the metal and the valence band (J_t,vb). Color arrows show the processes related to the population and depopulation of surface states: capture and thermal emission of electrons (blue arrows), capture and thermal emission of holes (red arrows), direct tunneling from the metal to surface states (green arrows).

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The main contribution to the tunnel current emerges from the low transverse momenta p_⊥. Thus, one can expand the right-hand sides of Eqs. (6–8) over p_⊥ and integrate over the transverse momenta in Eq. (4). The tunnel current density is then written as

(9)$$J_{\mathrm{t},\mathrm{cb}}=\frac{4\pi e{m}_{\mathrm{i}}}{{\left(2\pi \hslash \right)}^3}{\displaystyle \underset{E_{\mathrm{c}}}{\overset{+\infty }{\int }}\left[f_{\mathrm{m}}(E)-f_{\mathrm{cb}}(E)\right]}S(E)dE,$$

where the functional form of S(E) depends on the energy of the incident electron. For E > φ_MI − eV_i (triangular barrier),

(10)$$S(E)=E_{\mathrm{t},\mathrm{I}}{D}_{\mathrm{cb},\mathrm{I}}(E,0)\left[1-\exp \left(-\frac{m_{\mathrm{e}}}{m_{\mathrm{i}}}\frac{E-E_{\mathrm{c}}}{E_{\mathrm{t},\mathrm{I}}}\right)\right],$$

where the characteristic tunneling energy E_t,I is

(11)$$E_{\mathrm{t},\mathrm{I}}=\frac{e\hslash F_{\mathrm{i}}}{2{\left[2m_{\mathrm{i}}({\phi }_{\mathrm{MI}}-E)\right]}^{1/2}}.$$

For $E_{\mathrm{c}}{|}_z{}_{=d_{\mathrm{i}}}<E<{\phi }_{\mathrm{MI}}-e{V}_{\mathrm{i}}$ (trapezoid barrier),

(12)$$S(E)=E_{\mathrm{t},\mathrm{II}}{D}_{\mathrm{cb},\mathrm{II}}(E,0)\left[1-\exp \left(-\frac{m_{\mathrm{e}}}{m_{\mathrm{i}}}\frac{E-E_{\mathrm{c}}}{E_{\mathrm{t},\mathrm{II}}}\right)\right],$$

where

(13)$$E_{\mathrm{t},\mathrm{II}}=\frac{e\hslash F_{\mathrm{i}}}{2\sqrt{2m_{\mathrm{i}}}\left({\left[{\phi }_{\mathrm{MI}}-E\right]}^{1/2}-{\left[{\phi }_{\mathrm{MI}}-E-e{V}_{\mathrm{i}}\right]}^{1/2}\right)}.$$

Finally, for low-energy electrons $\left(E_{\mathrm{c}}<E<{E_{\mathrm{c}}|}_z{}_{=d_{\mathrm{i}}}\right)$,

(14)$$S(E)=E_{\mathrm{t},\mathrm{III}}{D}_{\mathrm{cb},\mathrm{III}}(E,0)\left[1-\exp \left(-\frac{m_{\mathrm{e}}}{m_{\mathrm{i}}}\frac{E-E_{\mathrm{c}}}{E_{\mathrm{t},\mathrm{III}}}\right)\right],$$

(15)$$E_{\mathrm{t},\mathrm{III}}={\left\{\frac{2\sqrt{2m_{\mathrm{i}}}\left({|{\phi }_{\mathrm{MI}}-E|}^{1/2}-{|{\phi }_{\mathrm{MI}}-E-e{V}_{\mathrm{i}}|}^{1/2}\right)}{e\hslash F_{\mathrm{i}}}+\frac{m_{\mathrm{i}}}{m_{\mathrm{e}}}\frac{2{\left[2m_{\mathrm{e}}\left(E-E_{\mathrm{c}}\right)\right]}^{1/2}}{e\hslash F_{\mathrm{s}}}\right\}}^{-1}.$$

The probability D_vb(E, p_⊥) of electron tunneling from the metal to the valence band of the semiconductor or, equivalently, the hole tunneling probability is obtained in a similar way. Namely, holes near the valence band edge $\left({E_{\mathrm{v}}|}_z{}_{=d_{\mathrm{i}}}-E-p_{\perp }^2/2m_{\mathrm{i}}<{\chi }_{\mathrm{VI}}-e{V}_{\mathrm{i}}\right)$ tunnel through the trapezoid barrier, whose transparency is given by

(16)$$\begin{array}{l}\ln D_{\mathrm{vb},\mathrm{I}}(E,p_{\perp })=-\frac{4\sqrt{2m_{\mathrm{i}}}}{3e\hslash F_{\mathrm{i}}}\{{\left[{\chi }_{\mathrm{VI}}+\frac{p_{\perp }^2}{2m_{\mathrm{i}}}-{E_{\mathrm{v}}|}_{z=d_{\mathrm{i}}}+E\right]}^{3/2}-\\ {\left[{\chi }_{\mathrm{VI}}-e{V}_{\mathrm{i}}+\frac{p_{\perp }^2}{2m_{\mathrm{i}}}-{E_{\mathrm{v}}|}_{z=d_{\mathrm{i}}}+E\right]\}}^{3/2},\end{array}$$

while holes with sufficiently high energies $\left({E_{\mathrm{v}}|}_{z=d_{\mathrm{i}}}-E-p_{\perp }^2/2m_{\mathrm{i}}>{\chi }_{\mathrm{VI}}-e{V}_{\mathrm{i}}\right)$ tunnel through the triangular barrier with transparency

(17)$$\ln D_{\mathrm{vb},\mathrm{II}}(E,p_{\perp })=-\frac{4\sqrt{2m_{\mathrm{i}}}}{3e\hslash F_{\mathrm{i}}}{\left[{\chi }_{\mathrm{VI}}+\frac{p_{\perp }^2}{2m_{\mathrm{i}}}-{E_{\mathrm{v}}|}_{z=d_{\mathrm{i}}}+E\right]}^{3/2}.$$

Appendix B. Distribution of electric potential

As in most semiconductor tunnel structures [20], the injection current density in the MIS structure [Eq. (9)] depends exponentially on the voltage drop $V_{\mathrm{i}}={\mathrm{E}}_{\mathrm{i}}{d}_{\mathrm{i}}$ across the thin insulator layer [see Fig. 6(a)]. Hence, the accurate determination of V_i becomes critical. It can be found from the boundary condition for the electric displacement at the semiconductor-insulator interface:

(18)$${\epsilon }_{\mathrm{i}}{V}_{\mathrm{i}}/d_{\mathrm{i}}-{\epsilon }_{\mathrm{s}}{\mathrm{E}}_{\mathrm{s}}=Q_{\mathrm{ss}}/{\epsilon }_0.$$

Here, ε_i = 25, [46] and ε_s = 15, [47] are the static dielectric constants of the insulator and semiconductor, respectively, and ${\mathrm{E}}_{\mathrm{s}}$ is the electric field in the semiconductor at the insulator-semiconductor interface. This field is found from Poisson’s equation for the electric potential φ in the semiconductor

(19)$$\frac{d^2\phi }{d{x}^2}=\frac{e(N_A-p)}{{\epsilon }_s{\epsilon }_0},$$

where p is the density of holes. The contribution of electrons to the charge density can be neglected, since the population inversion in p-type InAs is achieved at an electron density of the order of 10¹⁶ cm⁻³, which is far below the hole density and the the acceptor concentration at the chosen doping level.

By multiplying Eq. (19) by dφ/dx and integrating over dx, one easily obtains the expression for the electric field in the semiconductor at the semiconductor-insulator interface

(20)$${\mathrm{E}}_{\mathrm{s}}={\left\{\frac{2}{{\epsilon }_{\mathrm{s}}{\epsilon }_0}{\displaystyle {\int }_{{\zeta }_{\mathrm{p}}}^{{\zeta }_{\mathrm{p}}+e{V}_{\mathrm{s}}}\left[\left({{\epsilon }^{\prime }}_F\right)-N_A\right]}d{{\epsilon }^{\prime }}_F\right\}}^{1/2},$$

where ζ_p = E_v − F_p is the hole Fermi energy in the bulk of the semiconductor and V_s = V − V_i + (χ_CI − E_G − ζ_p)/e is the voltage drop across the semiconductor [see Fig. 6(b)]. Here, we have assumed that the distribution of holes is equilibrium, which is justified by a large barrier height for holes at the semiconductor-insulator interface.

Appendix C. Surface states at the semiconductor-insulator interface: charge density, surface recombination and tunnel current through the surface states

The presence of the surface states at the semiconductor-insulator interface has a significant impact on the characteristics of the tunnel MIS contact [21]. First, surface states accumulate electric charge and affect the band bending and voltage drop V_i across the insulator layer. Second, the direct tunnel current from the metal to the surface states contributes to the net current across the junction [27]. Finally, surface states act as recombination centers [26] and reduce the electron injection current:

(21)$$J_{0,\mathrm{n}}=J_{\mathrm{t},\mathrm{cb}}-J_{\mathrm{sr},\mathrm{n}},$$

where J_0,n is the electron current density at the insulator-semiconductor interface, J_t,cb is the electron tunnel current density across the MIS contact, and J_sr,n is the surface recombination current density.

Following [48, 49], we assume that surface states ’communicate’ with the electron states in metal and semiconductor bands, but not with each other. Thus, the occupancy of each state f_ss(E) with energy E is independent from the other states. f_ss(E) is found from the detailed balance principle between three population/depopulation processes schematically shown in Fig. 6(b):

1. electron capture from the conduction band of the semiconductor to the surface states and their thermal emission to the band;
2. hole capture and thermal emission;
3. direct tunnelling from the metal to surface states.

In a steady state, f_ss is found to be given by

(22)$$f_{\mathrm{ss}}(E)=\frac{v_{\mathrm{n}}+v_{\mathrm{p}}{e}^{(F_{\mathrm{p}}-E)/k_{\mathrm{B}}T}+v_{\mathrm{t}}(E)f_{\mathrm{m}}(E)}{v_{\mathrm{n}}\left[e^{(E-F_{\mathrm{n}})/k_{\mathrm{B}}T}+1\right]+v_{\mathrm{p}}\left[e^{(F_{\mathrm{p}}-E)/k_{\mathrm{B}}T}+1\right]+v_{\mathrm{t}}(E)},$$

where we have introduced the rates of electron capture $v_{\mathrm{n}}={n|}_{z=d_{\mathrm{i}}}{\sigma }_{\mathrm{n}}{v}_{\mathrm{n}}/4$, hole capture $v_{\mathrm{p}}={n|}_{z=d_{\mathrm{i}}}{\sigma }_{\mathrm{p}}{v}_{\mathrm{p}}/4$, and electron tunnel escape from the surface state to the metal ν_t (E). In the above equations, σ_n and σ_p are the electron and hole capture cross-sections, v_n and v_p are the carrier thermal velocities.

The capture-cross section depends on whether the surface trap is donor type or acceptor type. Experimental studies have revealed the donor-type behavior of surface states in InAs [31]. The hole capture cross-sections by a neutral trap is approximately equal to 10⁻¹⁵ cm⁻² [26], while the electron capture cross-section σ_n by a positively charged center is calculated using the theory of carrier capture mediated by cascade phonon emission [50]:

(23)$${\sigma }_{\mathrm{n}}=\frac{4\pi }{3}{r}_{\mathrm{T}}^3{l}_{\epsilon }^{-1}.$$

Here, r_T = e²/4πε₀ε_sk_BT is the Bohr radius for an electron with an energy equal to the thermal energy, and l_ε is the energy relaxation length. The latter is smaller than the momentum relaxation length l_p by a factor of kT/2m_es² due to the quasi-elastic character of electron-phonon relaxation (here s is the sound velocity in the semiconductor). The momentum relaxation length l_p = v_nτ_p is estimated using the electron mobility in intrinsic InAs µ_n,i = eτ_p/m_e. Substituting the values appropriate to InAs (effective mass m_e = 0.026m₀, average sound velocity s = 3 × 10⁵ m/s, mobility µ_n,i, = 10⁵ cm² V⁻¹ s⁻¹ [51]), we obtain l_p = 500 nm, kT/2m_es² ≈ 2350, r_T = 14 nm, and σ_n = 10⁻¹⁶ cm⁻². The above derivation shows that such a small capture cross section is due to the large energy relaxation length in InAs.

Following [28], we can write the tunnel escape rate as

(24)$$v_{\mathrm{t}}(E)={\tau }_0^{-1}D(E,0),$$

where D(E,0) is the tunneling probability evaluated at zero transverse momentum and τ₀ is the characteristic escape time:

(25)$${\tau }_0=\frac{m_{\mathrm{i}}^2{d}_{\mathrm{i}}}{2\hslash m_0}\frac{{\kappa }^2+k_{\mathrm{F}}^2}{{\kappa }^2{k}_{\mathrm{F}}}{\left[1-\sqrt{\frac{\kappa }{d_{\mathrm{i}}{k}_F}}ℐ\left(\sqrt{\frac{d_{\mathrm{i}}{k}_{\mathrm{F}}^2}{\kappa }}\right)\right]}^{-1}.$$

In the above expression, $\kappa =\sqrt{2m_{\mathrm{i}}{\chi }_{\mathrm{CI}}}/\hslash$ is the electron wave function decay constant in the insulator, k_F is the Fermi wave vector in the metal, m₀ is the electron effective mass in metal assumed to be equal to the free electron mass, and $\mathrm{I}\left(x\right)$ stands for the Dawson integral:

(26)$$ℐ(x)=e^{-x^2}{\displaystyle \underset{0}{\overset{x}{\int }}{e}^{{\mathrm{t}}^2}}dt.$$

In fact, the characteristic escape time weakly depends on the barrier width d_i and the Fermi energy of the metal. For the 3-nm-thick HfO₂ layer and Au contact (Fermi energy 5.5 eV), τ₀ = 0.25 fs, while for Cu contact (Fermi energy 7.0 eV) τ₀ = 0.29 fs.

Assuming the density of surface states ρ_ss to be uniform over the band gap, we readily express the surface charge density

(27)$$Q_{\mathrm{ss}}=e{\rho }_{\mathrm{ss}}{\displaystyle \underset{E_{\mathrm{v}}}{\overset{E_{\mathrm{c}}}{\int }}[1-f_{\mathrm{ss}}(E)]}dE,$$

the surface recombination current

(28)$$J_{\mathrm{sr},\mathrm{n}}=\frac{1}{4}e{v}_{\mathrm{n}}{n|}_{z=d_{\mathrm{i}}}{\sigma }_{\mathrm{n}}{\rho }_{\mathrm{ss}}{\displaystyle \underset{E_{\mathrm{v}}}{\overset{E_{\mathrm{c}}}{\int }}\left\{[1-f_{\mathrm{ss}}(E)]-f_{\mathrm{ss}}(E)e^{(E-F_{\mathrm{n}})/k_{\mathrm{B}}T}\right\}dE,}$$

and the direct tunnel current through these states

(29)$$J_{\mathrm{t},\mathrm{ss}}=e{\rho }_{\mathrm{ss}}{\displaystyle \underset{E_{\mathrm{v}}}{\overset{E_{\mathrm{c}}}{\int }}{v}_{\mathrm{t}}(E)[f_{\mathrm{m}}(E)-f_{\mathrm{ss}}(E)]}dE.$$

We finally note that the electron density at the semiconductor-insulator interface ${n|}_{z=d_{\mathrm{i}}}$ entering the expressions for f_ss (E) and J_sr,n is considerably smaller than the electron density n_bulk in the bulk of the semiconductor (i.e. at z > l_s + d_i, where l_s is the screening length) at large forward bias. The reason for this is that the energy barrier prevents the electrons from reaching the surface [41] and ${n|}_{z=d_{\mathrm{i}}}\approx n_{\mathrm{bulk}}{e}^{-e{V}_{\mathrm{s}}/*k_{\mathrm{B}}T}$, where V_s is the voltage drop across the semiconductor.

Appendix D. Calculation of the material gain

Optical gain in the semiconductor is obtained by integrating the transition probabilities between electron states in the conduction and valence bands over the states’ energies:

(30)$$\begin{array}{c}g(F_{\mathrm{n}},F_{\mathrm{p}})=\frac{\pi e^2\hslash }{n_{\mathrm{In}\mathrm{As}}{m}_0^2{\epsilon }_0}\frac{{|M_b|}^2}{6\hslash \omega }{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\rho }_{\mathrm{cb}}}(E){\rho }_{\mathrm{vb}}(E-\hslash \omega )\\ {|M_{\mathrm{env}}(E,E-\hslash \omega )|}^2\left[f_{\mathrm{cb}}(E,E_{\mathrm{n}})-f_{\mathrm{vb}}(E-\hslash \omega ,F_{\mathrm{p}})\right]dE.\end{array}$$

Here, ρ_cb(E) and ρ_vb(E) are the densities of states in the conduction and valence bands of the semiconductor, respectively, f_cb(E, F_n) and f_vb(E, F_p) are the Fermi-Dirac distribution functions for the conduction and valence bands, M_b is the average matrix element connecting Bloch states near the band edge [52], and M_env is the envelope matrix element calculated using Stern’s model [39]. In heavily doped semiconductors, ρ_cb, ρ_vb, M_env and consequently the material gain, strongly depend on the impurity concentration, while at low doping levels M_env is strongly peaked at the energy corresponding to the exact momentum conservation for the electron-photon system [17, 39], and equation (30) is reduced to the that in the band-to-band transition model with the k-selection rule.

The calculated gain spectra at an acceptor density of 10¹⁸ cm⁻³ are shown in Fig. 7. The maximum of the gain spectrum shifts slightly towards shorter wavelengths as the electron density increases. At an electron concentration of 2 × 10¹⁶ cm⁻³ near the MIS contact, which corresponds to the full loss compensation regime, the maximum is achieved at λ = 3.22 µm.

 

Fig. 7 Gain spectra of InAs calculated using Stern’s model [see Eq. (30)] at different electron densities, the acceptor density is equal to 10¹⁸ cm⁻³ for all curves.

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Appendix E. Gain characteristics at different thicknesses of the insulator layer

We have varied the thickness of the insulator layer in order to achieve the regime of full loss compensation at the lowest possible injection current. The simulated gain-current and current-voltage characteristics are presented in Fig. 8. One readily notes that the insulator thickness of d_i = 3 nm is optimal. At higher thicknesses, full loss compensation cannot be achieved before the breakdown of the HfO₂ layer at V_i/d_i ≈ 1 V [40]. At lower oxide thicknesses, the current density in the regime of full loss compensation is greater than 5 kA/cm² due to the large tunnel current J_t,ss through the surface states [Fig. 8(c)].

 

Fig. 8 (a) SPP modal gain versus injection current density for three different thicknesses of the HfO₂ layer of the Au/HfO₂/InAs active hybrid plasmonic waveguide. Dashed curves correspond to the voltage range above the breakdown voltage of HfO₂. (b) Current-voltage characteristics (color coding is the same as in panel (a)). (c, d) Contribution of different tunneling processes to the injection current for the 2.5-nm-thick (c) and 3.5-nm-thick (d) HfO₂ layers. For all panels, the density of surface states is equal to 10¹³ cm⁻²eV⁻¹.

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Acknowledgments

The work was supported by the Russian Science Foundation, grant no. 14-19-01788.

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