Schottky-Barrier Photodiode

1. Theory

Definition: A Schottky-barrier photodiode is a metal-semiconductor rectifying contact used as a photodetector: light passes through a thin, semitransparent metal film, or a transparent conductor, and is absorbed in the semiconductor or in the metal, and the photocurrent is carried by majority carriers.
Barrier: The barrier height $q ϕ_{B}$ (of the order of $q (ϕ_{M} - χ_{s})$ for an n-type semiconductor, but fixed in practice by the interface states) sets the dark current, $I_{s} = A A^{**} T^{2} e^{- q ϕ_{B} / k_{B} T}$ , and the long-wavelength cutoff of the internal photoemission, $λ_{c} = 1.24/ ϕ_{B}$ μm.
Two modes: Photons with $h ν > E_{g}$ are absorbed in the semiconductor, as in a PN photodiode, and photons with $q ϕ_{B} < h ν < E_{g}$ excite carriers in the metal that cross the barrier by internal photoemission, with a yield of the form $Y \propto (h ν - q ϕ_{B})^{2} / h ν$ .
Depletion region: The semiconductor under the metal is depleted to a width $W = 2 ε_{s} (V_{bi} + V_{R}) / q N_{D}$ with a junction capacitance $C_{j} = ε_{s} A / W$ , so that the structure has a role like that of the intrinsic layer of a PIN photodiode.
Speed: Conduction is by majority carriers, so there is no minority-carrier storage and the speed is set by the RC time and the transit time; one source quotes a bandwidth of about 100 GHz for Schottky-barrier photodiodes.
Ultraviolet devices: A semitransparent contact such as 10 nm of Au or 80 nm of ITO on wide-gap GaN gives a visible-blind detector with more than three orders of magnitude of ultraviolet-to-visible contrast.
Infrared devices: PtSi on p-type silicon, with a barrier of about 0.24 eV to 0.27 eV, detects in the 3 μm to 5 μm window by internal photoemission and is described by one source as the most widely used technology for focal-plane arrays; the diodes are cooled (measurements at 55 K to 90 K in one study).
Limits: The metal reflects and absorbs part of the light, which lowers the quantum efficiency of the band-to-band mode, especially at long wavelengths, where one source finds the PIN photodiode better; the internal photoemission is what extends the detection beyond the gap.

2. Working Principle

Illumination: Light is incident through the thin metal film, whose transmittance $T_{f}$ sets the fraction that reaches the semiconductor.
Band-to-band mode: A photon with $h ν > E_{g}$ creates an electron-hole pair in the semiconductor; carriers made in the depletion region drift out and carriers made within a diffusion length $L_{p}$ of it diffuse in, giving $I_{p h} = q Φ_{0} T_{f} [1 - e^{- α W} / (1 + α L_{p})]$ .
Internal photoemission: A photon with $q ϕ_{B} < h ν < E_{g}$ is absorbed by an electron of the metal, which crosses the barrier if it keeps enough energy and has the right direction after the collisions in the film; the yield $Y = M (h ν - q ϕ_{B})^{2} / (8 E_{F} h ν)$ is small and the film is made thin, about 50 Å for PtSi.
Bias: At zero bias the device works in the photovoltaic mode with the dark resistance $R_{0} = n k_{B} T / q I_{s}$ ; a reverse bias widens $W$ , lowers $C_{j}$ and speeds up the collection but raises the dark current through the image-force lowering $Δ ϕ_{B}$ .
Response: The pulse response is set by $t_{10 - 90} = t_{cc}^{2} + t_{d i f f}^{2} + (2.2 R C)^{2}$ ; for a light absorbed within the depletion region the diffusion term is small and the response is limited by the RC time.
Readout: In infrared arrays each diode is read through a CCD transfer gate integrated with it, so that the data of many detectors leave through a few leads.

3. Applications

Visible-blind ultraviolet detection: Schottky photodiodes on GaN and AlGaN (with SiC and AlGaN devices used as reference detectors) detect ultraviolet light without a filter.
Infrared focal-plane arrays: PtSi Schottky-barrier detectors on silicon for the 3 μm to 5 μm window, integrated with CCD readout.
CCD image sensing: The Schottky photodiode is used as the image-sensing element of CCD imagers.
High-speed detection: Short rise times, 13 ps for a GaN diode, for detection of fast ultraviolet pulses.

Name	Formula	Related Formulae	Variables/Constants
Schottky barrier height (Schottky-Mott rule)	$q ϕ_{B n} = q (ϕ_{M} - χ_{s})$	$V_{bi} = ϕ_{B n} - ϕ_{n}$ $q ϕ_{B p} = E_{g} - q (ϕ_{M} - χ_{s})$	$ϕ_{M}, χ_{s} : w or k f u n c t i o n o f t h e m e t a l an d e l ec t r o n a f f ini t y o f t h e se mi co n d u c t or$ $ϕ_{n} : p o t e n t ia l o f t h e co n d u c t i o n ban d b e l o w t h e F er mi l e v e l in t h e n e u t r a l se mi co n d u c t or$ $(t h e m e a s u r e d ba r r i er h e i g h t s d e p e n d o n t h e in t er f a ce s t a t es an d o n t h e p r ocess in g an d o f t e n d i f f er f r o m t h e r u l e, w hi c h i s w h y t h ey a r e e x t r a c t e d f r o m t h e c u r r e n t - v o l t a g e c ha r a c t er i s t i c, a s o n t h e M e t a l S e mi co n d u c t or C o n t a c t s p a g e)$
Depletion width and capacitance	$W = \frac{2 ε _{s} ( V _{bi} + V _{R} )}{q N _{D}}$	$C_{j} = \frac{ε _{s} A}{W} = A \frac{q ε _{s} N _{D}}{2 ( V _{bi} + V _{R} )}$	$W, C_{j} : w i d t h an d c a p a c i t an ce o f t h e d e pl e t i o n r e g i o n in t h e n - t y p e se mi co n d u c t or u n d er t h e m e t a l, V_{R} b e in g t h e r e v er se bia s (t h e t er m k_{B} T / q i s so m e t im es s u b t r a c t e d f r o m V_{bi} + V_{R})$ $N_{D}, A : d o n or co n ce n t r a t i o n an d a r e a o f t h e co n t a c t (t h e j u n c t i o n i s a o n e - s i d e d o n e, t h e m e t a l b e in g a p er f ec t co n d u c t or)$ $k_{B} : B o l t z mann co n s t an t$
Spectral windows of the two modes	$q ϕ_{B} < h ν < E_{g} : internal photoemission$	$h ν > E_{g} : band to band absorption$ $λ_{c} = \frac{h c}{q ϕ _{B}} \approx \frac{1.24}{ϕ _{B} ( V )} μ m$	$(in t h e f i r s t m o d e a c a r r i er o f t h e m e t a l e x c i t e d b y t h e p h o t o n cr osses t h e ba r r i er in t o t h e se mi co n d u c t or, in t h e seco n d t h e p h o t o n i s ab sor b e d in t h e se mi co n d u c t or a s in a P N p h o t o d i o d e, w hi c h i s t h e m or e e f f i c i e n t m o d e)$ $λ_{c} : l o n g - w a v e l e n g t h c u t o f f o f t h e in t er na l p h o t oe mi ss i o n (4.5 t o 5.2 μ m f or ϕ_{B} = 0.273 t o 0.24 V in a P tS i d i o d e, co m p u t e d), t h e c u t o f f o f t h e ban d - t o - ban d m o d e b e in g 1.24/ E_{g}$
Band-to-band photocurrent and quantum efficiency	$I_{p h} = q Φ_{0} T_{f} [1 - \frac{e ^{- α W}}{1 + α L _{p}}]$	$η = T_{f} [1 - \frac{e ^{- α W}}{1 + α L _{p}}]$ $R = \frac{η λ}{1.24} A/W$	$Φ_{0} : in c i d e n t p h o t o n f l ux (p h o t o n s p er seco n d)$ $T_{f} : t r an s mi tt an ce o f t h e se mi t r an s p a r e n t m e t a l f i l m (1 min u s t h e r e f l ec t an ce an d t h e ab sor pt an ce o f t h e f i l m)$ $W, α, L_{p} : d e pl e t i o n w i d t h, ab sor pt i o n coe f f i c i e n t an d d i f f u s i o n l e n g t h o f t h e min or i t y c a r r i er s in t h e n e u t r a l se mi co n d u c t or (t h e s u m o f t h e d r i f t an d t h e d i f f u s i o n p a r t s, a s o n t h e P N P h o t o d i o d e p a g e)$ $(m e a s u r e d o n G a N : p e ak η = 47% f or an 80 nm I T O co n t a c t an d 27% f or 10 nm o f A u, w i t h R = 0.13 A / W a t 350 nm an d 0.07 A / W a t 310 nm)$
Internal photoemission yield (Fowler)	$Y = \frac{M ( h ν - q ϕ _{B} ) ^{2}}{8 E _{F} h ν}$	$R_{I P E} = \frac{Y λ}{1.24} A/W$ $Y h ν \propto h ν - q ϕ_{B}$	$Y : e l ec t r o n s e mi tt e d o v er t h e ba r r i er p er p h o t o n (t h e so u r ce d oes n o t s a y w h e t h er in c i d e n t or ab sor b e d, an d in c i d e n t i s a ss u m e d in t h e r es p o n s i v i t y)$ $E_{F} : F er mi e n er g y o f t h e m e t a l$ $M : r a t i o o f t h e e f f ec t i v e ma sses o f t h e se mi co n d u c t or an d t h e m e t a l (1 in t h e s im pl e f or m)$ $(t h e pl o t o f Y h ν a g ain s t h ν i s a s t r ai g h t l in e w h ose in t er ce pt g i v es t h e ba r r i er h e i g h t an d w h ose s l o p e g i v es E_{F}, t h e f i t o f o n e P tS i / p - S i d i o d e g i v in g ϕ_{B} = 0.273 V an d E_{F} = 1 e V, t h e e x t e n d e d m o d e l g i v in g 0.24 V an d 5 e V)$ $(f or q ϕ_{B} = 0.24 e V, E_{F} = 5 e V an d h ν = 0.5 e V t h e y i e l d i s 0.34% an d R_{I P E} \approx 6.8 m A / W (co m p u t e d))$
Thermionic-emission dark current and dark resistance	$I_{d} = A A^{**} T^{2} exp (- \frac{q ϕ _{B}}{k _{B} T}) [exp (\frac{q V}{n k _{B} T}) - 1]$	$R_{0} = \frac{n k _{B} T}{q I _{s}}, I_{s} = A A^{**} T^{2} e^{- q ϕ_{B} / k_{B} T}$ $Δ ϕ_{B} = \frac{q E _{ma x}}{4 π ε _{s}}$	$A^{**} : e f f ec t i v e R i c ha r d so n co n s t an t$ $n : i d e a l i t y f a c t or (1.15 an d 1.19 f or t h e I T O an d A u co n t a c t s o n G a N)$ $ϕ_{B} : ba r r i er h e i g h t$ $R_{0} : d a r k (z er o - bia s) r es i s t an ce$ $I_{s} : s a t u r a t i o n c u r r e n t (a ba r r i er hi g h er b y 60 m e V l o w er s i t b y a f a c t or o f 10 a t 300 K, co m p u t e d)$ $Δ ϕ_{B} : ima g e - f or ce l o w er in g o f t h e ba r r i er b y t h e f i e l d E_{ma x} a t t h e in t er f a ce, w hi c h mak es t h e r e v er se c u r r e n t r i se s l o w l y w i t h t h e bia s (t h e t u nn e l l in g an d t h e g e n er a t i o n - r eco mbina t i o n c u r r e n t s a dd t o t h e t h er mi o ni c o n e)$ $k_{B} : B o l t z mann co n s t an t$
Response time and bandwidth	$t_{10 - 90} = t_{cc}^{2} + t_{d i f f}^{2} + (2.2 R C)^{2}$	$t_{cc} = \frac{W}{v _{d}}$ $R C = (R_{s} + R_{L}) C_{j}, f_{R C} = \frac{1}{2 π R C}$	$t_{cc}, t_{d i f f} : t im es o f c ha r g e co l l ec t i o n in t h e d e pl e t i o n r e g i o n a t t h e d r i f t v e l oc i t y v_{d} an d o f d i f f u s i o n f r o m t h e n e u t r a l r e g i o n (s ma l l w h e n t h e l i g h t i s ab sor b e d w i t hin W)$ $R_{s}, R_{L} : ser i es r es i s t an ce o f t h e d i o d e (t h e n e u t r a l r e g i o n an d t h e o hmi c co n t a c t) an d t h e l o a d$ $C_{j} : j u n c t i o n c a p a c i t an ce$ $(an 80 μ m G a N d i o d e a t z er o bia s g a v e a r i se t im e o f 13 p s, a F W H M o f 60 p s, R C = 82 p s (f i tt e d) an d 90 p s (c a l c u l a t e d), an d a 3 d B ban d w i d t h o f 2.6 G H z)$
Johnson-noise-limited specific detectivity at zero bias	$D^{*} = R \frac{R _{0} A}{4 k _{B} T}$	$D^{} = \frac{A}{NEP}$ $D^{} = \frac{η q λ}{h c} \frac{R _{0} A}{4 k _{B} T}$	$D^{*} : s p ec i f i c d e t ec t i v i t y in c m H z^{1/2} / W w h e n t h e n o i se i s t h e J o hn so n n o i se o f t h e d a r k r es i s t an ce R_{0}$ $R_{0} A : p r o d u c t o f t h e d a r k r es i s t an ce an d t h e a r e a (in d e p e n d e n t o f t h e a r e a)$ $NEP : n o i se - e q u i v a l e n t p o w er in W / H z (t h e e x p r ess i o n f o l l o w s f r o m t h e P N P h o t o d i o d e p a g e w i t h R_{s h} r e pl a ce d b y R_{0}, d er i v e d h er e)$ $k_{B} : B o l t z mann co n s t an t$

Important Questions

Ques: Why can a Schottky photodiode detect photons with an energy below the band gap?

A photon with $q ϕ_{B} < h ν < E_{g}$ is absorbed by an electron of the metal, not of the semiconductor, and the excited electron can cross the barrier into the semiconductor if its energy exceeds $q ϕ_{B}$ (internal photoemission). The detection threshold is therefore the barrier height, not the gap: $λ_{c} = h c / q ϕ_{B} \approx 1.24/ ϕ_{B}$ μm, which is 4.5 to 5.2 μm for the 0.273 to 0.24 V of a PtSi/p-Si diode (computed) and covers the 3 to 5 μm atmospheric window. The price is a low yield, because the electron must keep enough energy and direction in the film, so the mode is used where the band-to-band absorption cannot detect the light, as in the infrared with silicon. For $h ν > E_{g}$ the band-to-band mode is more efficient and sets the cutoff near the gap. Reference: Turkish Journal of Physics 24 (2000), study of PtSi/p-Si Schottky-barrier detectors (aj.tubitak.gov.tr/physics/issues/fiz-00-24-4/fiz-24-4-9-9907-8.pdf), Eqs. 1 to 5; University of Pretoria thesis, Chapter 3 ‘Schottky barrier ultraviolet photodetectors’ (repository.up.ac.za), Eqs. 3.5 to 3.22; Electronics Notes, Schottky Photodiode; AZoOptics, What are Schottky Barrier Photodetectors and Diodes?.

Ques: Why does the internal photoemission yield rise as $(h ν - q ϕ_{B})^{2} / h ν$ and how large is it?

In the Fowler model only the electrons of the metal with energy $h ν - q ϕ_{B}$ above the barrier can escape, and the number of excited electrons with sufficient energy and a direction pointing at the barrier grows as $(h ν - q ϕ_{B})^{2}$ , divided by $h ν$ for the absorbed photon and by the Fermi energy. The yield is $Y = M (h ν - q ϕ_{B})^{2} / (8 E_{F} h ν)$ , so $Y h ν$ is linear in $h ν$ and its intercept gives the barrier. For $q ϕ_{B} = 0.24$ eV, $E_{F} = 5$ eV and $h ν = 0.5$ eV, $Y = 0.34%$ and the responsivity is $Y λ /1.24 \approx 6.8$ mA/W (computed), small compared with the 0.5 to 1 A/W of a band-to-band photodiode. The physical argument is textbook, the source giving the formula; the measured spectra deviate from the plot near the threshold, and an extended model with scattering lengths (60 Å and 4000 Å) fitted the data better. Reference: Turkish Journal of Physics 24 (2000), study of PtSi/p-Si Schottky-barrier detectors (aj.tubitak.gov.tr/physics/issues/fiz-00-24-4/fiz-24-4-9-9907-8.pdf), Eqs. 1 to 5.

Ques: What is the barrier height of the PtSi/p-Si diode: 0.273 V or 0.24 V?

The modified Fowler fit of one paper gives $q ϕ_{B} = 0.273$ eV with $E_{F} = 1$ eV, and the extended model with scattering gives 0.24 eV with $E_{F} = 5$ eV. The author notes that the first barrier is higher than the typical values from current-voltage measurements and that the Fermi energy of the second is in the suggested range for PtSi, and the extended model fitted better, so 0.24 eV is the more likely barrier. The cutoff is $λ_{c} = 1.24/ ϕ_{B}$ , 5.2 μm for 0.24 V against 4.5 μm for 0.273 V (computed), which shows how sensitive the long-wavelength limit is to the model used for the fit. Reference: Turkish Journal of Physics 24 (2000), study of PtSi/p-Si Schottky-barrier detectors (aj.tubitak.gov.tr/physics/issues/fiz-00-24-4/fiz-24-4-9-9907-8.pdf), Eqs. 1 to 5.

Ques: Why does a semitransparent metal contact trade quantum efficiency against dark current?

The photocurrent is proportional to the transmittance $T_{f}$ of the film: on GaN an 80 nm ITO contact gave a peak $η = 47%$ and 10 nm of Au gave 27%, a gain of about two. The barrier heights from the forward characteristics were 0.59 eV for ITO and 0.75 eV for Au, and the dark current $I_{s} \propto e^{- q ϕ_{B} / k_{B} T}$ differs by $e^{0.16/0.0259} \approx 490$ for this difference (computed). The measured leakage at 1 V reverse was above about 1 μA for ITO (100 μm) and below 1 nA for Au (150 μm), a ratio above 1000, the same order. The transparent contact gives the efficiency and the high-barrier metal the low dark current, so each choice improves one figure at the expense of the other. One source gives the ITO leakage as 1 mA, which is probably a typo for μA. Reference: E. Ozbay et al., Appl. Phys. Lett. (2001), GaN Schottky photodiodes with ITO and Au contacts (Bilkent University); University of Pretoria thesis, Chapter 3 ‘Schottky barrier ultraviolet photodetectors’ (repository.up.ac.za), Eqs. 3.5 to 3.22.

Ques: Why does a Schottky photodiode show a 13 ps rise time but only a 2.6 GHz bandwidth?

The rise of the pulse response is set by the excitation and the fast collection of the carriers, which are made within the depletion region of a majority-carrier device with no diffusion tail, (an interpretation, since the source gives the RC value but no explanation of the rise), whereas the fall follows the RC time of the diode and its load, 82 ps fitted and 90 ps calculated for the 80 μm GaN device. A single-pole response with $f_{3 d B} = 1/ (2 π R C)$ would give 1.9 GHz for 82 ps (computed), the same order as the 2.6 GHz from the Fourier transform of the pulse, whereas $0.35/ t_{r}$ would give 27 GHz (computed) for the 13 ps rise time, which is not a bandwidth of the device. The response is not single-pole, so the two numbers are not related by $0.35/ t_{r}$ , and the bandwidth is limited by the RC time. Reference: E. Ozbay et al., Appl. Phys. Lett. (2001), GaN Schottky photodiodes with ITO and Au contacts (Bilkent University); University of Pretoria thesis, Chapter 3 ‘Schottky barrier ultraviolet photodetectors’ (repository.up.ac.za), Eqs. 3.5 to 3.22.

Photons with $h ν < E_{g}$ do not create pairs by band-to-band absorption in GaN, and the gap of 3.4 eV gives a band-edge wavelength of $1.24/3.4 \approx 365$ nm (computed), in agreement with the highest sensitivity at 350 to 360 nm in the sources. Internal photoemission over a barrier of 0.6 to 1 eV would be expected to extend the response into the visible with a small yield, and the measured ultraviolet-to-visible contrast was more than three orders of magnitude for both ITO and Au contacts. The cutoff is set by the gap, so no filter is required; the transmission of the contact sets the short-wavelength end, and the ITO film absorbs less than 10% near the band edge. Reference: E. Ozbay et al., Appl. Phys. Lett. (2001), GaN Schottky photodiodes with ITO and Au contacts (Bilkent University); Spbstu, GaN MSM photodetector with Ni/Au contacts, Peter the Great St. Petersburg Polytechnic University (2022).

Ques: Does a Schottky junction have a depletion region?

One source states that the Schottky junction has no depletion layer, unlike the PN diode; the other sources show that it has one: the space charge of the ionized donors under the metal gives $W = 2 ε_{s} (V_{bi} + V_{R}) / q N_{D}$ and the capacitance $C_{j} = ε_{s} A / W$ , which is used for capacitance-voltage profiling, and the photocurrent expression contains $W$ . The depletion region is present, which makes the first statement wrong; what a Schottky diode lacks is the injection and storage of minority carriers, which may be what that source meant, though it does not say so. The depleted layer is the active region of the photodiode, so the analogy of another source with the intrinsic layer of a PIN photodiode is useful. Reference: AZoOptics, What are Schottky Barrier Photodetectors and Diodes?; Wikipedia, Schottky barrier (accessed October 2026); University of Pretoria thesis, Chapter 3 ‘Schottky barrier ultraviolet photodetectors’ (repository.up.ac.za), Eqs. 3.5 to 3.22; Electronics Notes, Schottky Photodiode.

Ques: Why is the barrier height of Ni/Au on GaN given as 0.85 to 1.16 eV when the literature value is 1.04 eV?

The barrier height extracted from the current-voltage characteristic depends on the processing: the MSM devices of one study gave 0.85 eV (ideality 2.8) as deposited, 0.88 eV at 400 °C, 1.12 eV at 500 °C and 1.16 eV at 600 °C, while the cited literature value is 1.04 eV, and the 10 nm Au contact on GaN gave 0.75 eV (ideality 1.19) in another study. Ideality factors of 2.1 to 2.8 mean the current is not ideal thermionic emission, so the extracted barriers are unreliable. A rise from 0.85 to 1.12 eV would lower an ideal thermionic current by $e^{0.27/0.0259} \approx 3 \times 1 0^{4}$ (computed), yet the dark current fell by at most 30 times, showing that leakage and not thermionic emission dominated. Annealed values of 1.1 to 1.2 eV are close to the literature value, so about 1.0 to 1.1 eV is the more likely barrier of a good Ni/Au contact, and the as-deposited 0.85 eV points to a defective interface. Reference: Spbstu, GaN MSM photodetector with Ni/Au contacts, Peter the Great St. Petersburg Polytechnic University (2022); E. Ozbay et al., Appl. Phys. Lett. (2001), GaN Schottky photodiodes with ITO and Au contacts (Bilkent University).

Quartz 5

Explorer

Schottky-Barrier Photodiode

1. Theory

2. Working Principle

3. Applications

Important Questions

Ques: Why can a Schottky photodiode detect photons with an energy below the band gap?

Ques: Why does the internal photoemission yield rise as $(h ν - q ϕ_{B})^{2} / h ν$ and how large is it?

Ques: What is the barrier height of the PtSi/p-Si diode: 0.273 V or 0.24 V?

Ques: Why does a semitransparent metal contact trade quantum efficiency against dark current?

Ques: Why does a Schottky photodiode show a 13 ps rise time but only a 2.6 GHz bandwidth?

Ques: Why is a Schottky photodiode on GaN visible-blind without a filter?

Ques: Does a Schottky junction have a depletion region?

Ques: Why is the barrier height of Ni/Au on GaN given as 0.85 to 1.16 eV when the literature value is 1.04 eV?

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Table of Contents

Backlinks

Quartz 5

Explorer

Schottky-Barrier Photodiode

1. Theory

2. Working Principle

3. Applications

Important Questions

Ques: Why can a Schottky photodiode detect photons with an energy below the band gap?

Ques: Why does the internal photoemission yield rise as (hν−qϕB​)2/hν and how large is it?

Ques: What is the barrier height of the PtSi/p-Si diode: 0.273 V or 0.24 V?

Ques: Why does a semitransparent metal contact trade quantum efficiency against dark current?

Ques: Why does a Schottky photodiode show a 13 ps rise time but only a 2.6 GHz bandwidth?

Ques: Why is a Schottky photodiode on GaN visible-blind without a filter?

Ques: Does a Schottky junction have a depletion region?

Ques: Why is the barrier height of Ni/Au on GaN given as 0.85 to 1.16 eV when the literature value is 1.04 eV?

Graph View

Table of Contents

Backlinks

Ques: Why does the internal photoemission yield rise as $(h ν - q ϕ_{B})^{2} / h ν$ and how large is it?