Effect of anisotropic electron momentum distribution of surface plasmon on internal photoemission of a Schottky hot carrier device

Xin-Hao Li; Jeffrey B. Chou; Wei Lek Kwan; Asma M. Elsharif; Sang-Gook Kim

doi:10.1364/OE.25.00A264

1. Introduction

Metallic-semiconductor nanostructures have been widely used for photocatalytic chemical reactions [1, 2], solar energy harvesting [3, 4] and photo detection [5, 6]. Depending on the band diagram of the Schottky barrier, photo-excited hot electrons or hot holes could be collected to conduction bands of semiconductors through internal photoemission. When barrier height is lower than the bandgap of semiconductor, Schottky hot carrier devices can be a promising solution to harvest sub-bandgap solar energy.

For high efficiency solar to fuel energy conversion, recently we reported an Au/TiO₂ metallic-semiconductor photonic crystal (MSPhC) of 2D nano-cavity arrays with thin Au film as absorber, which showed a broadband absorption below the bandgap of TiO₂ (3.2 eV) [7]. Figure 1(a) shows the focused ion beam (FIB) photo of MSPhC viewed at 30° angle. Thin layers of 60 nm TiO₂ and 20 nm Au are deposited on Al₂O₃ nano-cavity arrays [8, 9] to form Schottky junction, as shown in Fig. 1(b). Hot electrons generated in the Au layer could transfer to the interface and be injected to the conduction band of TiO₂. A layer of 1 nm Al₂O₃ is deposited between TiO₂ and Au to improve the diode performance, which is not shown in the schematic. 30 nm ITO is sputtered as a back contact. Due to the nano-cavity structure, multiple resonance modes result in the device which supports the broadband absorption by the Au layer. Finite-difference time-domain (FDTD) simulation method shows that the structure can support surface plasmon polariton (SPP) at the Au/TiO₂ interface along the side walls of nano-cavities with resonant wavelength near 590 nm, as shown in Fig. 1(c). In Fig. 2(a), both simulated and measured results confirm that MSPhC can achieve an evenly low reflectance, i.e. high absorption, from 400 nm to 800 nm, even though the variations in thickness of the thin gold layer and dimension of cavities across the device fabricated on a 6” silicon wafer may cause frequency shift and broadening effects.

Fig. 1 Schematic of MSPhC device. (a) FIB photo of MSPhC viewed at 30° angle. (b) Cross-section of MSPhC. r and d are the radius (250 nm) and depth (1 µm) of the nano-cavity. (c) Electric filed profile at cross-section of nano-cavity, obtained from FDTD simulation, which shows SPP at the Au/TiO₂ interface along the cavity side wall at 590 nm.

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Fig. 2 (a) Reflectance and normalized photoresponse of MSPhC from 400 nm to 800 nm. The low reflectance from UV-Vis measurement and FDTD simulation indicates high absorption in this range. Value of photoresponse is normalized against the highest value at 590 nm. (b) Normalized IQE of MSPhC (symbols and blue dash line), which is normalized against the value at 2.21 eV (560 nm). The IQE is calculated with the measured photocurrent and absorption by the Au layer. Example of IQE curve (black solid line) based on Fowler’s theory with barrier height of 1.53 eV and arbitrary fitting constant.

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In Fig. 2(a), measurement shows a sub-bandgap photoresponse with only a single peak at 590 nm, which coincides with the SPP resonant wavelength at Au/TiO₂ interface. This indicates SPP plays an important role in the hot electron collection process, which results in a frequency-dependent internal quantum efficiency (IQE) as shown in Fig. 2(b). Therefore, understanding the internal photoemission and the effects of SPP is important for designing Schottky hot carrier devices. The normalized IQE of MSPhC at 1.55 eV (800 nm) is two orders smaller than the IQE at 2.21 eV (560 nm). By fitting I-V curve measurement to the thermionic emission theory [10], the barrier height is estimated as 1.53 ± 0.11 eV. This may be caused by adding the ultrathin layer of Al₂O₃ to improve the diode performance [11]. In the following analysis, we will use 1.53 eV as the barrier height.

A widely used model of Schottky internal photoemission has been Fowler’s theory [10, 12]. It assumes that the generated hot electrons in metal are distributed isotopically in the momentum space. Based on this assumption, the IQE of a Schottky device could be fitted to Fowler’s equation [10, 12]:

η_{IQE} = C \frac{{(h ν - q ϕ_{B})}^{2}}{h ν \cdot q ϕ_{B}}

where $q ϕ_{B}$ is Schottky barrier height, $h ν$ is photon energy and C is fitting constant. A typical IQE curve calculated by Fowler’s equation with $q ϕ_{B}$ of 1.53 eV and arbitrary fitting constant is shown as the black solid line in Fig. 2(b). It is obvious that Fowler’s theory cannot explain the internal photoemission in MSPhC, especially the peak centered at 2.21 eV, which suggests its assumption is not valid when there is SPP at the interface. Previous experimental results have also shown that the applicability of Fowler’s theory is limited [3, 13, 14] when plasmon resonance exists.

Not only Fowler’s theory, other previous models estimating IQE of Schottky hot carrier device have not incorporated the possible anisotropic momentum distribution of hot electrons at the presence of SPP [15–17]. In the following sections, we analyze the effect of anisotropic hot electron momentum distribution caused by SPP on the internal photoemission of MSPhC. Considering the energy, momentum and spatial distribution of generated hot electrons, we developed an effective theoretical model for IQE estimation incorporating this effect, which matches the experimental results of MSPhC better than the old theories. It indicates that the anisotropic hot electron momentum distribution should not be ignored when designing Schottky hot carrier devices. Combining our model and electromagnetic design, photoresponse and IQE of Schottky hot carrier devices could be tuned with surface plasmon resonance.

2. Results and discussion

The internal photoemission at the metal-semiconductor Schottky junction could be divided into three steps: (1) hot electrons excitation, (2) transport to the interface and (3) injection across the junction. After generated with initial energy and momentum at different location inside the metal, hot electrons travel with scattering against electrons, phonons and defects, and a part of them can reach the metal-semiconductor interface. At the interface, hot electron injection can happen if the normal component of hot electron’s kinetic energy is higher than the Schottky barrier. It is important to understand how hot electrons are generated and how their initial states are for predicting IQE of metal-semiconductor Schottky devices.

Electron excitation mechanisms in metal could be divided into interband transition (D to S band) and intraband transition (S to S band) [18, 19]. Nanoscale geometries provide required momentum for intraband transition, which makes it preferable for intraband transition to happen in metallic nanostructures. First principle calculations [20–23] and experimental work [24] have shown that the intraband transition dominates light absorption below the interband transition threshold and generates highly energetic hot electrons [20, 21, 24]. Above the threshold, interband transition dominates light absorption and generates low energy hot electrons close to the Fermi level. Since D band locates about 2.3 eV below the Fermi level in Au [4, 15], hot electrons excited through interband transition by absorbing a photon with energy from 1.55 eV to 3.1 eV (from 400 nm to 800 nm) could not overcome the Schottky barrier between Au and TiO₂, which is typically 1.0~1.2 eV [4, 25], and about 1.53 eV in our case. On the other hand, hot electrons generated through intraband transition can have energy level higher than the Schottky barrier, which makes it the only source for internal photoemission [15, 16]. In the following steps, the energy, momentum and spatial distribution of hot electrons generated through intraband transition are modeled with a free-electron-like Jellium model at the Fermi level of 5.5 eV.

Firstly, the energy distribution of hot electrons are investigated. With the free-electron-like band structure, electrons could be excited from an energy level E₀, below the Fermi level E_F in the range $E_{F} - h ν \leq E_{0} \leq E_{F}$ , to an energy E₁ level in the range $E_{F} \leq E_{1} \leq E_{F} + h ν$ by absorbing a photon with energy $h ν$ . The distribution of generated hot electrons’ energy depends on the probability of each transition between two different energy levels. Here we use electron distribution joint density of states (EDJDOS) D [15, 16] to evaluate the number of possible electron transitions between two specific states:

D (E_{0} + h ν, h ν) = ρ (E_{0}) f (E_{0}) ρ (E_{0} + h ν) f (E_{0} + h ν)

where $D (E_{0} + h ν, h ν)$ is the EDJDOS of exciting a hot electron from energy level E₀ by absorbing a photon with energy $h ν$ . $ρ (E)$ is the density of states (DOS) of the two energy levels, which is proportional to E^1/2 given the free-electron-like band structure. $f (E)$ is the Fermi-Dirac distribution function. This kind of simplification has been applied to estimate the photoresponse of a range of noble metals [26]. Then energy distribution of generated hot electrons could be calculated with the probability G of generating a hot electron from the energy level E₀ by absorbing a photon with energy $h ν$ :

G (E_{0} + h ν, h ν) = \frac{D (E_{0} + h ν, h ν) d E}{\int_{E_{F} - h υ}^{E_{F}} D (E + h ν, h ν) d E}

Secondly, the momentum distribution of hot electrons is analyzed. Recent theoretical [23, 27, 28] and experiment works [29, 30] have shown that photoresponse is proportional to the electric field normal to the metal-semiconductor interfaces, which suggests that the momentum distribution of generated hot electrons is not isotropic but depends on the electric field inside metallic structure. Based on these previous results and the requirement of energy conservation, a model describing the anisotropic momentum distribution of hot electrons is developed, which has been neglected in previous theories of Schottky internal photoemission. In momentum space, the probability of electron transition between two energy levels by changing a specific wave number is proportional to the square of electric field along that direction. Then, the integration of probability P of generating a hot electron with momentum (k_x, k_y, k_z) on a surface of constant energy needs to satisfy:

\underset{{| k |}^{2} = constant}{∭} P (k_{x}, k_{y}, k_{z}) \cdot \frac{k_{z}^{2}}{k_{x}^{2} + k_{y}^{2} + k_{z}^{2}} d k_{x} d k_{y} d k_{z} = \frac{{| E_{z} |}^{2}}{{| E |}^{2}}

where z is the direction normal to the interface, E_z is internal electric field normal to the interface and E is the total internal electric field. One possible momentum distribution that can satisfy the above restriction of energy conservation is:

P (θ) = \frac{1}{2} \sin (θ) + \frac{12 \cdot K - 4}{3 π} \cos (2 θ)

where $θ$ is the angle against the k_z direction, and $K = {| E_{z} |}^{2} / {| E |}^{2}$ . We did not treat k_x and k_y differently because they are tangential to the interface and do not affect the momentum distribution regarding the normal direction. Thus the distribution could be simplified as a function of $θ$ . In Eq. (5), the second term on the right hand side represents the effect of internal electric field on the momentum distribution. Need to notice that Eq. (5) is only a first order model satisfying the restriction of Eq. (4). The value of K is limited ( $1 / 3 \leq K \leq (3 π + 8) / 24$ ) to satisfy that $P (θ) \geq 0$ for $0 \leq θ \leq π$ . A more realistic momentum distribution in nano scale system could be calculated with the random phase approximation method used in previous theoretical works [23, 27, 28]. It relies on knowing the exact electron’s wave equation and has only been applied to simple small geometries such as nanoparticles or bulk materials. Our model, on the other hand, could be effectively applied to more complex and larger scale systems, like MSPhC. Figure 3(a) show the FDTD simulation result of averaged K inside the Au layer. The electric field in each mesh cell is collected. Then K is calculated based on the geometry of the nano-cavity. The red line shows the analytical result of K for interface electromagnetic wave between semi-infinite Au and TiO₂, which is calculated with $K = {| \sqrt{ε_{T i O_{2}} / ε_{A u}} |}^{2} / ({| \sqrt{ε_{T i O_{2}} / ε_{A u}} |}^{2} + 1)$ [31]. The permittivity $ε_{A u}$ of Au and $ε_{T i O_{2}}$ of TiO₂ is adopted from Johnson & Christy [32] and Palik [33]. Electric field inside the Au layer is dominated by SPP at the range near the resonant wavelength. As shown in Fig. 3(a), from 1.9 eV to 2.3 eV, the factor K matches well with the analytical result. Meanwhile the decay of SPP along the direction normal to the interface causes the deviation between simulation and analytical result. Based on Eq. (5) Fig. 3(b) shows the anisotropic momentum distribution of hot electrons on a surface of constant energy when incident light is at 510 nm, 580 nm and 620 nm. z is the direction normal to the interface. It clearly shows that the momentum distribution of hot electrons is not isotropic due to the effect of internal electric field. A higher K will cause the concentration of hot electron distributed along the direction normal to the interface. This indicates that the anisotropic momentum distribution should not be neglect when modeling internal photoemission process.

Fig. 3 (a) Ratio $K = {| E_{z} |}^{2} / {| E |}^{2}$ in the thin Au layer of MSPhC, obtained from FDTD simulation (black symbols and dash line). Analytical result of ratio K for interface electromagnetic wave between semi-infinite Au and TiO₂ (red solid line). (b) Calculated momentum distribution of hot electrons at 510 nm, 580 nm and 620 nm on a surface of energy constant with Eq. (5). The scale bar shows the distribution probability normalized against uniform distribution in natural log scale. As shown in the left upper corner of (b), only hot electrons with enough normal energy component could be injected, i.e. the “escape cone” model.

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Thirdly, we consider the spatial distribution. SPP could increase the Au layer absorption by generating strong electric filed oscillation within the decay length (usually less than 10 nm [18]) near the metal-semiconductor interface [31]. Compared with hot electrons excited through other “bulk” paths of intraband transition, plasmonic hot electrons are generated near the interface [18], which minimizes the possible scattering loss. To account the effect of SPP, we calculated the absorption contribution by the Au layer as a function of thickness to the interface by FDTD simulation. As shown in Fig. 4, the contribution by Au layer within 3 nm next to the interface increases from 20% to over 50% when photon energy decreases from 2.4 eV to 2.0 eV, which indicates the presence of SPP enhances the absorption by the Au layer near the interface.

Fig. 4 The percentage of absorption by Au layer as a function of thickness to the Au/TiO₂ interface in MSPhC by FDTD simulation. The presence of SPP traps light at the interface and increases the absorption contribution of the Au layer near the interface.

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The total probability for hot electrons to reach the metal-semiconductor interface can be estimated with an exponential attenuation model [23, 34, 35].

P_{transfer} = \int_{0}^{H} η (h) \exp (- h / l_{mfp}) d h

where h is the normal distance to the interface, H is the thickness of Au layer, $η (h) d h$ is the percentage of absorption by the thin layer dh with distance h to the interface. $l_{mfp}$ is the mean free path (MFP) of hot electrons, which equals to:

l_{m f p}^{- 1} = l_{e e}^{- 1} + l_{e p}^{- 1}

where l_ee is electron-electron scattering MFP and l_ep is electron-phonon scattering MFP. Experimental measurement have shown that the electron MFP depends on its energy [34, 36]. l_ee is in the range of 10~100 nm, while l_ep is in the range of 10~100 nm of hot electrons within 6 eV above the Fermi level. For simplicity, here we fix the mean free path as l_mfp = 10 nm for hot electrons from Fermi level to about 3 eV above [36].

To model the injection probability when hot electrons reach the Schottky interface, we use the so-called “escape cone” model while incorporating the effect of anisotropic momentum distribution. Hot electrons with normal kinetic energy greater than the Schottky barrier have injection probability of 1 while the others have injection probability of 0. This criteria allows us to obtain an analytical expression of the injection probability, while more delicate models based on calculating the transmission coefficient of electrons through a rectangular potential step and considering the effects of electrons effective mass jump across the interface could also be found in literature [37]. The injection probability P_inject of a hot electron at energy level $E_{h} \geq (E_{F} + q ϕ_{B})$ is:

\begin{array}{l} P_{inject} (E_{h}) = \int_{0}^{Ω_{escape}} P (θ) d θ \\ \approx \frac{1}{2} (1 - \sqrt{\frac{q ϕ_{B} + E_{F}}{E_{h}}}) + \frac{6 K - 2}{3 π} [\sqrt{\frac{q ϕ_{B} + E_{F}}{E_{h}}} \sqrt{1 - \frac{q ϕ_{B} + E_{F}}{E_{h}}}] \end{array}

where the escape angle $Ω_{escape}$ is defined as $\cos (Ω_{escape}) = \sqrt{(q ϕ_{B} + E {}_{F}) / E_{h}}$ . The first term on the right hand of Eq. (8) is equivalent to the injection probability in Fowler’s theory [17], while the second term reflects the effects of anisotropic momentum distribution, which has not been considered by previous theories. When K is larger than 1/3, the concentration of hot electron along the direction normal to the metal-semiconductor interface will enhance the injection probability.

Combing all of the above analysis of the internal photoemission process together, the IQE of a Schottky device as a function of photon energy $h ν$ can be calculated with:

η_{IQE} (h ν) = \frac{A_{intra}}{A_{intra} + A_{inter}} P_{transfer} \cdot \int_{E_{F} + q ϕ_{B}}^{E_{F} + h ν} G (E_{h}) P_{inject} (E_{h}) d E_{h}

where A_intra and A_inter are the absorption by intra and interband transition of Au. Even though interband transition does not contribute to photocurrent, it still contributes to light absorption and IQE. Thus it is necessary to consider its effect at range above the interband transition threshold. The ratio between A_intra and total absorption by Au is estimated as $Im (ε_{intra}) / (Im (ε_{inter}) + Im (ε_{intra}))$ . $ε_{intra}$ is the interband contribution to permittivity, and $ε_{intra}$ is the intraband contribution estimated by Drude model. The values of them in optical range are adopted from Johnson & Christy [32].

Figure 5 shows the normalized IQE calculated with our model and Fowler’s theory incorporating the effects of interband transition. From Fig. 5 we can see that the IQE is affected by both the intrinsic properties of Au and external effects of nanostructure and illumination condition. The black symbols of IQE based on the assumptions of Fowler’s theory only considers the effects of inter- and intraband transition, i.e. the band structure of Au. It shows the platform shape of IQE due to interband transition. On the other hand, the anisotropic momentum distribution caused by SPP largely enhances the IQE with a peak near 2.2 eV, which is shown with our modified model. It matches with the IQE of MSPhC better than the widely used Fowler’s theory. This results indicates that the effects of anisotropic electron momentum distribution by surface plasmon should be considered when designing Schottky hot carrier devices. A device that can generate hot electrons with momentum preferentially normal to the Schottky interface can largely enhance the device’s IQE. Our theoretical modeling could provide design guidance to tune and to enhance the photoresponse and IQE of Schottky hot carrier devices.

Fig. 5 Normalized IQE calculated based on FDTD simulation and the modified model (red round symbol, red dash line is a 4th order polynomial fitting), Fowler’s theory (black diamond symbol) and normalized IQE of MSPhC (blue square symbol). The calculated value is normalized against the IQE at 2.21 eV with Eq. (9). Through considering the factor of anisotropic momentum distribution of hot electrons and SPP effects, the modified model fits the pattern of measured IQE of MSPhC better than the Fowler’s theory.

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3. Conclusion

To summarize, we analyze the effect of anisotropic hot electron momentum distribution caused by SPP on the internal photoemission process in the MSPhC, and developed a theoretical model of IQE for Schottky hot carrier devices incorporating this effect. We find only intraband transition contributes to photoresponse for our device with incident light from 400 nm to 800 nm wavelength. It allows us to estimate the energy distribution of hot electrons with a simplified free-electron-like band structure. In the momentum space, SPP can cause hot electrons concentrated along the direction normal to the Au/TiO₂ interface by modifying the electric field inside the thin Au layer with direction preferentially normal to the interface, increasing the IQE resultantly. Meanwhile the presence of SPP enhances the absorption near the interface and reduces the hot electron scattering loss. By incorporating the factor of anisotropic momentum distribution into the “escape cone” injection model and the factor of SPP into electron transport efficiency estimation, we developed a model to estimate the IQE of plasmonic devices. Compared with the widely used Fowler’s theory, our model can better and effectively predict internal photoemission in Schottky devices. This theoretical work shows that the anisotropic hot electron momentum distribution is one major factor in designing Schottky hot carrier devices.

Acknowledgments

This work was supported under the Cooperative Agreement between Masdar Institute of Science and Technology, UAE and Massachusetts Institute of Technology, Cambridge, USA, reference number 02/MI/MIT/CP/11/07633/GEN/G/00.

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Effect of anisotropic electron momentum distribution of surface plasmon on internal photoemission of a Schottky hot carrier device

Abstract

1. Introduction

2. Results and discussion

3. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (9)

Optics Express