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The Role of Surface Roughness in Plasmonic-Assisted Internal Photoemission Schottky Photodetectors Meir Grajower, Uriel Levy, and Jacob B. Khurgin ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00643 • Publication Date (Web): 10 Sep 2018 Downloaded from http://pubs.acs.org on September 10, 2018
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ACS Photonics
The Role of Surface Roughness in Plasmonic-Assisted Internal Photoemission Schottky Photodetectors Meir Grajower1, Uriel Levy1, and Jacob B. Khurgin2* 1
Department of Applied Physics, The Benin School of Engineering and Computer Science, The Center for Nanoscience and
Nanotechnology, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel 2
Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore Maryland 21218, USA
*Corresponding author:
[email protected] Abstract Internal photoemission of charged carriers from metal to semiconductors plays an important role in diverse fields such as sub bandgap photodetectors and catalysis. Typically, the quantum efficiency of this process is relatively low, posing a stringent limitation on its applicability. Here, we show that the efficiency of hot carrier injection from metal into semiconductor across Schottky barrier can be enhanced by as much as an order of magnitude in the presence of surface roughness on the scale of a few atomic layers. Our results are obtained using a simple semi-analytical theory and indicate that properly engineered plasmonic assisted internal photo-emission photodetectors can be a viable alternative in silicon photonics. Other applications, such as plasmonic-enhanced photo catalysis can also benefit from these results. Keywords: plasmonics; photodetectors, Schottky barrier, internal photoemission, surface enhancement, scattering Silicon is the material of choice in the microelectronics industry for several decades1. In recent years, the use of silicon has been expended to new areas. In particular, the field of silicon photonics has been recently emerged, 2–5, involving the ability to guide and manipulating light on a chip which is fabricated in a standard complementary metal oxide silicon (CMOS) technologies and is based on silicon on insulator structures. Above other advantages, silicon is transparent at the telecommunication wavelength bands around 1.3 microns and 1.55 microns. Indeed, a variety of silicon photonics devices have been demonstrated over the years, including e.g.waveguides6, modulators7, splitters8, biosensors9 and more. Yet, while silicon photonics is a promising platform for optical devices, its transparency at the wavelengths of interest implies that it cannot be used as a photodetector at these wavelengths. As so, there is a long lasting effort to integrate other semiconductor materials, with smaller bandgap into the silicon device, such that photodetection becomes possible10–13. In parallel, there is an ongoing effort to find other solutions that are based on CMOS compatible materials and processes. In particular, the approach of detecting infrared light in silicon based on internal photoemission (IPE) is of great interest1,6,14–27. With this approach, one can realize a photodetector by forming a Schottky contact at a metal-semiconductor interface. The Schottky type device is characterized by a potential barrier between the Fermi level of the metal and the conduction band of the semiconductor and shows rectifying electrical characteristics. If the energy of the photon is higher than this potential barrier, there is a probability for the hot electron generated in the metal (via excitation by a photon or by a plasmon) to overcome the barrier and arrive at the semiconductor, where is can be collected as a useful photocurrent. 1 ACS Paragon Plus Environment
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While the IPE based photodetection approach is appealing from its simplicity and material compatibility with CMOS fabrication process6,15, its quantum efficiency is typically low. This is primarily due to the huge momentum mismatch (k-mismatch) between the electronic wavefunction in the metal (typically with energy close to the Fermi level) and in the silicon (typically with energy just above the bottom of the conduction band). Several models studied the process of IPE in details and provided predictions for its efficiency in various configurations18,28–32. These models emerge from the fundamental work of Fowler33, which predicted that the efficiency is proportional to ∝
ℏ ℏ
where ℏ is the energy of
the incident photon. The existing models all predict quantum efficiency well below 1%, which contradicts several experimental results34–36 reported higher responsivities. In some cases the efficiency of hot electrons from Gold nanoparticle in TiO2 films was found to be as high as 25-45%34,36. Several explanations were proposed, including the high confinement of light36,37, excitation of plasmonic resonances22,24,26,38–40, increasing the transmission cross section of the electrons from the metal into the semiconductor 25,41, and others 34–36,42,43. But while all these theories can explicate the enhanced rate of hot carrier excitation in plasmonic nanoparticles, none of them can remotely account for the hot electron injection efficiency exceeding most theoretical prediction by an order of magnitude. While the previous models assumed specular reflection of electrons from a flat interface between metal and semiconductor30,32, a practical ‘smooth’ interface will always show roughness of at least a few atomic layers44–46. Such roughness is expected to play a significant role in altering the probability of electron transfer over the Schottky barrier. Hereby, we analyze the role of surface roughness in enhancing the IPE probability using a perturbative model which shows that even relatively small roughness can lead to a dramatic increase in the transmission probability of electrons into the semiconductor and therefore resulting in a significant increase in the quantum efficiency of IPE based photodetectors. Our work can be used to better understand the experimentally observed enhanced IPE efficiency34-36 and to provide guidelines for achieving better IPE based Schottky photodetectors with higher responsivities, in both free space and guided mode configurations.
Hot electron injection across the smooth interface We start our analysis by calculating the probability of hot electrons with energy E above the Fermi level to be injected over the effective (i.e. including the effects of image force and reverse bias) potential barrier φb in the absence of roughness (i.e. flat interface). For that, we find the relation between the lateral wavevector k ||m (continuous across the boundary) and the normal to the interface wavevectors
k z , m and k z , s (the wavevectors in metal and semiconductor, respectively)
k zm = k F2 − k||2m 2 k zs = kmax − k||2m
2 (E) = kmax
(1)
m E − φb 2 2ms E − φb ) = s kF 2 ( h m0 EF
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where kF is the Fermi wave vector, E F = h 2 k F2 / 2 m0 is the Fermi energy, and m0 , ms are the mass of the electron in the metal and the effective mass of the electron in the semiconductor respectively. The wavevectors are all shown in Fig. 1.
Figure 1 - An illustration of the wavefunctions and wavevectors involved in hot electron injection from metal into semiconductor. Orange – metal, blue – semiconductor. a) k-space diagram of the plane representing the metal-semiconductor interface. - the maximal lateral k-vector that will still allow transport of electrons from the metal to the semiconductor. || - the lateral silicon k-vector, which can take any value between 0 and . - the Fermi wavevector in the metal, ||, the lateral k-vector in the metal and – the momentum added by the roughness. b) Illustration of the interface roughness, including the electron wavefunctions in the semiconductor and the metal. H is the Hamiltonian which couples the high momentum hot electrons into allowed states in the silicon c) vitalization of the escape cone of hot electrons in the specific example of Au-Si interface. The k-vectors are normalized to . As can be seen, most of the hot electrons are outside of the escape cone and are thus reflected from the metal interface and cannot be transmitted into the silicon.
Only the hot electrons with lateral wave vectors less than have a finite probability to be inject over the Schottky barrier, i.e. ||, . This defines a cone of allowed wave vectors of hot electrons which can be injected into the semiconductor14,23,32,47. The cone angle is given by sin
!
"#$% . "&
The
escape cone can be seen (purple solid cone) in Fig. 1c.
Inside the cone, the transmission coefficient of electrons from metal to semiconductor is:
T (km ) = 1 −
| k zm − k zs |2 | k zm + k zs |2
(2)
and as we only consider the in plane wavevectors up to k max (see Fig. 1a), all the wavevectors are real and:
T (km ) =
4kzm kzs (kzm + kzs )2
(3)
The transmission probability is calculated by integrating Eq. 3 over all the allowed wave vectors. Before integration, let us make an approximation by assuming the constant density of states in the metal in the
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vicinity of Fermi level. Normalizing all the wavevectors to kF we find the transmission probability via averaging over the lateral wave-vector k ||m kmax
T ( E ) ≡ T (kmax ) =
∫ T (k
m
)k m dk m
1
∫k
m
2 1 − k||2m kmax − k||2m
kmax
=8
0
∫ 0
dk m
(
1− k + k 2 ||m
2 max
−k
2 ||m
)
2
k m dk m
(4)
0
To ascertain the importance of finite reflection we first find out the transmission probability in the absence of reflection within the allowed cone of transmission in the momentum space. This is achieved assuming perfect transmission, i.e. T ( km ) = 1 in Eq. 3. Substituting in Eq. 4, one obtains: 2 T0 ( E ) = kmax (E) =
ms E − φ m0 EF
(5)
shown as dotted line in Fig. 2(a). Next, we calculate the exact result from Eq. 4, including the effect of the finite transmission within the cone of angles in the momentum space. The result is shown as a solid line in Fig. 2(a) for the case of Au
(E
F
= 5.5eV , k F = 12 nm −1 ) - Si ( ms = 0.3m0 ) interface with the barrier φ = 0.5eV . As one can see,
the finite reflection reduces the probability of hot electron injection by as much as a factor of two. We now turn to describe the probability of internal photoemission of a hot electron which is generated by a photon with energy hω . In the absence of reflection, this is achieved by integrating Eq. 5 over all the energies from φb to hω . Doing so, we get:
1 ms T0 (hω ) = hω m0
hω
∫
φb
E − φb 1 EF ms hω − φb dE = EF 2 hω m0 EF
2
(6)
which is of course the Fowler formula33. This result is shown in Fig.2 (b) as a dotted line. Also shown (as solid line) the accurate result of internal photoemission which takes into account the reflection within the cone. It is calculated according to:
T ( hω ) =
1 hω
hω
∫ T ( E )dE
(7)
φb
Where T ( E ) is taken from Eq. 4. Clearly, including finite reflectivity of the smooth interface is important to evaluation of the hot electron injection probability.
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Figure 2 - (a) Hot electron injection probability from Au into Si as a function of hot electron energy including (solid line, Eq. 4) and neglecting (dotted line, Eq. 5) finite reflectivity at the interface. (b) Hot electron injection probability from Au into Si as a function of the incident photon (plasmon) energy including (solid line, Eq. 7) and neglecting (dotted line, Eq. 6) finite reflectivity at the interface. In both cases, the finite reflection reduces the probability of hot electron injection by as much as a factor of two.
Interface roughness scattering as a pathway to increased efficiency of hot electron injection For a flat interface between the metal and the semiconductor, an electron having a parallel momentum k||m > kmax (outside of the transmission cone) will be reflected and thus cannot be transported into the semiconductor. However, surface roughness can assist in relaxing the momentum mismatch and provide a path for the electron to be transmitted into the semiconductor. In the limit of the perturbation theory (small roughness), the probability of surface roughness scattering of the electron incident onto the interface with a lateral wavevector k||m > kmax into the state in the semiconductor with the lateral wavevector k||s can be evaluated according to Fermi Golden rule as:
P ( E , k||m , k||s ) =
2π Ψ m ( E , k||m ) H R Ψ s ( E , k||s ) h
2
ρ s ( E , k||s )
(8)
Here the wavefunction of the incident electron includes contributions of the incident, reflected and evanescent waves,
jk zm z k zm − jqs − jkzm z + e e k zm + jqs 1 Ψ m ( E , k||m ) = 2k zm vzm e − jqs z k zm + jqs
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z≤0 (9)
z>0
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2 , qs is the wavevector in the semiconductor and the wavefunction has been where k zs = k||2m − kmax
normalized to the electron velocity in the metal vzm = hk zm / m0 as it is commonly done in current probability calculations. The wavefunction in the semiconductor involves incident and reflected waves as well as the wave transmitted into the metal and propagating with the wavevector k zm1 = k F2 − k||2s ,
− jk zs z k zs − k zm1 jkzs z + e e k zs + k zm1 1 Ψ s ( E , k||s ) = 2k zs La e− jkzm1z k zs + k zm1
z>0 (10)
z
