This is the submitted version of the article: Antidormi A., Cummings A.W.. Optimizing the Photothermoelectric Effect in Graphene. Physical Review Applied, (2021). 15. 054049: - . 10.1103/PhysRevApplied.15.054049. Available at: https://dx.doi.org/10.1103/PhysRevApplied.15.054049 Optimizing the photothermoelectric effect in graphene Aleandro Antidormi∗ and Aron W. Cummings† Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain (Dated: May 18, 2021) Among its many uses, graphene shows significant promise for optical and optoelectronic applications. In particular, devices based on the photothermoelectric effect (PTE) in graphene can offer a strong and fast photoresponse with high signal-to-noise ratio while consuming minimal power. In this work we discuss how to optimize the performance of graphene PTE photodetectors by tuning the light confinement, device geometry, and material quality. This study should prove useful for the design of devices using the PTE in graphene, with applications including optical sensing, data communications, multi-gas sensing, and others. I. INTRODUCTION telecommunications wavelengths that can reach switch- ing frequencies up to and beyond 65 GHz [5, 24, 25]; Owing to its unique properties, graphene is a promis- and plasmonically-enhanced detectors reaching photore- ing material for a wide range of applications [1]. Par- sponses over 12 V/W that can also be cascaded in series ticularly promising are applications that utilize several [7]. Photodetectors based on the PTE in graphene are of graphene’s unique properties in a single system or de- thus highly promising for their use in many applications. vice. A perfect example of such devices are photode- However, there are a large number of design parameters tectors based on the photothermoelectric effect (PTE) in impacting the performance of these devices, including the graphene [2, 3], which combine a broadband and fast pho- light profile, device geometry, and material quality. toresponse with a high signal-to-noise ratio and minimal In this paper, we discuss the impact that these de- power consumption [4–7]. sign parameters have on the performance of PTE-based Figure 1 illustrates the PTE in graphene and how it graphene photodetectors, and we demonstrate how their may be used for photodetection. Optical absorption first performance may be optimized. We find that by vary- generates a population of excited electrons and holes, as ing the device geometry and material quality within an shown in panel (a). Because of its linear electronic band achievable range, the performance of these photodetec- structure [8], graphene is a broadband absorber with uni- tors can be tuned by more than one order of magnitude. form light absorption from the optical to the THz regime These results, and our study of how the performance [9]. Next, interparticle scattering quickly thermalizes scales with various geometrical and material parameters, these photoexcited carriers to an elevated temperature, should be highly valuable for the design of future pho- as shown in panel (b). Graphene’s small electronic heat todetectors based on the PTE in graphene. capacity means this electronic temperature can be much higher than the lattice temperature [10], and graphene’s fast electronic response means this thermalization pro- II. DEVICE OPERATION cess happens within a few tens of femtoseconds [11], en- abling photodetectors that can operate at very high fre- Figure 1(d) shows a schematic of a PTE-based pho- quencies [5]. Finally, electron-phonon scattering relaxes todetector. The graphene channel is oriented along the the electron temperature back to the lattice temperature, x-axis and is situated between source (S) and drain (D) as shown in panel (c). The weak interaction between contacts. Light is delivered to the graphene via an opti- electrons and acoustic phonons in graphene results in a cal waveguide oriented along the y-axis (out of the page). slow decay of the electron temperature [12]. Defects, dis- Absorption of light traversing the width of the graphene order, and optical phonon emission can speed up this channel then leads to a nonuniform electronic temper- process [13–16], but even so, the electronic temperature ature profile centered in the middle of the channel (at in graphene can remain elevated for several picoseconds, x = 0). An example of such a temperature profile is long enough to be efficiently detected [2, 10, 16–23]. shown in Fig. 1(e). This approach to optical detection has been used to This situation differs from typical thermoelectric de- design, for example, room-temperature THz detectors vices in that the hot “side” is the center of the channel with high bandwidth, fast response, and a high signal- √ while the cold sides are at the source/drain contacts. To to-noise ratio (noise-equivalent power < 40 pW/ Hz) generate a nonzero voltage from this temperature pro- [4, 6]; waveguide-integrated photodetectors operating at file, it is thus necessary to have a nonuniform Seebeck coefficient along the length of the channel. This is ac- complished via the split gates labeled “Vg1 ” and “Vg2 ”, which are used to form a pn junction that results in a See- ∗ aleandro.antidormi@icn2.cat beck coefficient that is antisymmetric across the length † aron.cummings@icn2.cat of the channel. An example of this nonuniform Seebeck 2 (a) (b) (c) noise-equivalent power (NEP), which specifies the input optical power at which the SNR is equal to 1 at a sam- pling bandwidth of 1 Hz. The PTE-based photodetectors operate at zero bias, such that the only source of noise is Johnson (thermal) noise, and the NEP is thus given by √ 4kB T R NEP = , (3) VPTE /Pin where Pin is the incident optical power, T is the ambient temperature, and kB T is the thermal energy. III. DEVICE OPTIMIZATION (d) 2Wg Vg1 Vg2 In this paper we focus on the optimization of three fig- Graphene Oxide ξ tox ures of merit: the photovoltage VPTE , the photocurrent IPTE , and the noise-equivalent power NEP, as defined z S Waveguide D in Eqs. (1)-(3). As seen from these equations, optimiza- tion of the photodetectors involves tuning the interplay x Lch between the electron temperature profile Tel (x), the See- y beck profile S(x), and the channel resistance R. Our starting point will be the two-dimensional (2D) S(x) (e) heat equation, solved within the graphene sheet: Tel(x) ~ · [κ(x, y)∇Tel (x, y)] ∇ x ‒Lch /2 ‒Wg +Wg +Lch /2 −γ(x, y)Cel (x, y)Tel (x, y) +P (x, y) = 0, (4) where the x-axis (y-axis) is parallel (perpendicular) to FIG. 1. Schematic of the photothermoelectric effect in the graphene channel, κ(x, y) is the electronic thermal graphene. (a) Optical absorption generates an excited popu- conductivity, γ(x, y) is the electron cooling rate, Cel (x, y) lation of electrons and holes, (b) interparticle scattering ther- is the electronic heat capacity, and P (x, y) is the optical malizes photoexcited carriers to an elevated temperature, and power density incident on the graphene sheet. We assume (c) electron-phonon scattering relaxes the electron tempera- that κ, γ, and Cel are uniform along the y-axis, and their ture back to the lattice temperature. Panel (d) shows a cross variation along the x-axis is determined by the carrier section of a graphene PTE detector, where split gates generate a pn junction in the graphene channel, resulting in a thermo- density induced by the split gates and the native doping electric voltage between the source and drain contacts. Panel of the graphene. (e) shows examples of the temperature profile Tel (x) and the We decompose the optical power density into compo- Seebeck coefficient S(x) present in these devices. nents along the x and y axes as 1 −y/Lα P (x, y) = Pin · G(x) · e , (5) Lα coefficient is shown in Fig. 1(e). Finally, this combination of a nonuniform electron tem- where Pin is the total optical power incident on the perature and Seebeck coefficient generates a thermoelec- graphene sheet, G(x) is the profile of the optical power tric voltage between the source and drain electrodes [2, 3], along the length of the channel, and Lα is the length over Z which light is absorbed by the graphene as it traverses dTel (x) the width of the channel, which is determined by the VPTE = S(x) dx, (1) dx imaginary part of the graphene index of refraction and the degree of coupling to the waveguide, and is typically where S(x) is the spatially dependent Seebeck coefficient on the order of 50-100 µm [5]. In the following sections, and Tel (x) is the electronic temperature relative to the we solve the heat equation in different situations to de- lattice temperature. This voltage may be measured di- termine the optimal device parameters. rectly in an open-circuit configuration, or measured as a photocurrent in a closed circuit, A. Light profile IPTE = VPTE /R, (2) where R is the channel resistance. In the design of photodetectors, it is known that con- The signal-to-noise ratio (SNR) of the photodetector fining the incoming light can lead to enhanced photore- may be quantified via the figure of merit known as the sponse, and this can be achieved in different ways. In 3 one recent waveguide-integrated photodetector, careful and the resulting photovoltage is design of the waveguide was used to confine a 1.55-µm optical mode to a full width of ∼80 nm [5]. In another VPTE = 2S · Tξ , approach, the metal split gates above the graphene chan- where (11) nel were used to confine the 1.55-µm waveguide mode to  ξ  a full width of ∼100 nm [7]. Tξ = Tc erfcx √ , Here we quantify the gain in detector performance that 2Lc arises from confining the incident light profile along the with erfcx(x) the scaled complementary error function. x-axis. To isolate this effect, we assume a perfect pn The scaling of the photovoltage with the light profile junction at x = 0, such that S(x) = +S (−S) for x < 0 width is shown in Fig. 2. Here it can be seen that reduc- (x > 0). The photovoltage in Eq. (1) then reduces to ing the size of the light profile below the cooling length can significantly enhance the performance of PTE-based VPTE = 2S · Tel (x = 0). (6) photodetectors. For the examples mentioned at the be- Here we have assumed that the channel is sufficiently long ginning of this section [5, 7], confining 1.55-µm light to a such that Tel = 0 at the source and drain contacts. To spot width of ∼100 nm is responsible for a 2× to 3× en- find the electron temperature at x = 0, we start with the hancement of the photoresponse. For photodetectors op- 2D heat equation of Eq. (4). Because here we are only erating in the mid-IR or beyond, this degree of light con- concerned with transport along the x-axis, we reduce to finement could enhance the performance by 5× to 10×. a one-dimensional (1D) heat equation by averaging along the y-axis, giving  d2 Tel (x) Tel (x) αPin 2 − 2 + G(x) = 0, dx Lc κWch  where (7) VPTE  α = 1 − exp(−Wch /Lα ), p Lc = κ/γCel ,  with α the fraction of light absorbed, Wch the channel       width, and Lc the electronic cooling length. The cooling /Lc length depends on the dominant energy relaxation mech- anism in graphene [13–16], but is typically measured to FIG. 2. Photovoltage as a function of the width of the in- be on the order a few hundred nanometers up to 1 µm cident light profile ξ, from Eq. (11). The light profile width [18, 20, 21]. Here, Lc is independent of x, owing to the is normalized by the electron cooling length Lc , and VPTE is equal and opposite doping of the graphene channel by normalized by its value at zero width. The inset shows the shape of the Gaussian optical profile. the split gates. To explore the upper limit of light confinement, and also for mathematical convenience, we start with an in- finitely narrow light profile centered at x = x0 , G(x) = B. Channel length δ(x − x0 ). The resulting electron temperature profile is Tel,δ (x; x0 ) = Tc exp (−|x − x0 |/Lc ) , In the previous section we assumed that the distance where (8) between the source and drain contacts Lch was much longer than the cooling length Lc , such that the integral αPin Lc Tc = · . in Eq. (1) was taken from −∞ to +∞. Here we examine 2κ Wch the impact of a finite channel length on photodetector To account for a finite width of the optical mode in the performance. Assuming an ideal pn junction as before, waveguide, we now assume a Gaussian light profile, the photovoltage in Eq. (1) becomes 1 VPTE = 2S · [Tel (0) − Tel (Lch /2)] , (12) exp −x2 /2ξ 2 ,  G(x) = √ (9) 2πξ where we have taken the light profile to be symmetric where ξ defines the width of the light profile, as shown around x = 0 such that Tel (−Lch /2) = Tel (Lch /2). Fol- in the inset of Fig. 2. The electron temperature is then lowing the approach of Eq. (10) at x = Lch /2 gives the convolution of this Gaussian light profile with the   temperature arising from the delta-function light profile, VPTE = 2S · Tξ · 1 − e−Lch /2Lc . (13) Z The resistance of the graphene channel is R = Tel (x) = Tel,δ (x; x0 )G(x0 )dx0 , (10) (Lch /Wch )/σ, where σ is the conductivity, and from Eq. 4 (2) the photocurrent is thus given by In these situations, the split gates lose electrostatic control over the region between them, resulting in a uni- 1 − e−Lch /2Lc   Wch form carrier density determined by the native doping of IPTE = 2S · Tξ · σ · · . (14) Lc Lch /Lc the graphene layer. Uniform doping means a uniform Seebeck coefficient, and thus via Eq. (1) the region be- Similarly, from Eq. (3) the noise-equivalent power is tween the split gates will not contribute to the photore- r p ! sponse when the temperature profile is symmetric around Pin kB T Lc Lch /Lc NEP = · · . (15) x = 0. This may limit the overall performance of the 2S · Tξ σ Wch 1 − e−Lch /2Lc photodetector and counteract the gains made by confin- ing the light. Each of these quantities is plotted in Fig. 3 as a function In this section we evaluate the photodetector perfor- of the channel length, relative to the cooling length. The mance when the oxide thickness is smaller than the split- photocurrent and photovoltage are normalized to their gate separation, and compare it to the case of an ideal maximum values, while the NEP is normalized to its min- pn junction. To model this situation, we approximate imum value. the Seebeck coefficient, thermal conductivity, and cool- Figure 3 makes it clear that the optimal channel length ing length as piecewise functions of the form depends crucially on the sensing mode of the photode-  tector, with the photovoltage maximized at long Lch , the  {S, κ, Lc } , x ≤ −Wg photocurrent maximized at short Lch , and the NEP mini- {S(x), κ(x), Lc (x)} = {S0 , κ0 , L0 } , |x| < Wg (16) mized for an intermediate Lch ≈ 2.5Lc . This latter value  {−S, κ, L } , x ≥ W c g arises from the solution of the transcendental equation 1 + x = ex/2 , where x = Lch /Lc . This optimal value where Wg is half the spacing between the split gates. of the NEP arises from competition between the thermal The values of S, κ, and Lc are determined by the volt- noise and the photoresponse, as seen in Eqs. (3) and (15). age applied to the split gates, while the values of S0 , κ0 , and L0 are determined by the native doping of the  graphene layer, which depends on the substrate, encap- sulation, and device fabrication steps. The photovoltage from Eq. (1) then becomes )LJXUHRIPHULW 1(3 VPTE = S · [Tel (Wg ) + Tel (−Wg )]  VPTE +S0 · [Tel (Wg ) − Tel (−Wg )] . (17) IPTE Here we have assumed for simplicity that the channel length is long, as considering the impact of shorter chan-  nel length does not alter our main conclusions. In Eq.       (17) we see that the photovoltage depends in general on Lch /Lc both S and S0 . However, the explicit dependence on S0 disappears when the light spot is centered at x = 0. FIG. 3. Photodetector figures of merit as a function of the To find the temperature at x = ±Wg , we first assume channel length Lch , from Eqs. (13)-(15). The channel length an infinitely narrow light source incident at x = x0 . We is normalized by the electron cooling length Lc , the photo- then solve the heat equation in each region of the channel voltage and photocurrent are normalized by their maximum and match the temperature and the heat flux at x = x0 values, and the NEP is normalized by its minimum value. and at x = ±Wg . Finally, we convolve the temperature with a Gaussian light profile centered at x = 0, giving   W exp Lcg C. Split-gate separation VPTE = 2S · Tc ·     W W cosh L0g + r sinh L0g In the previous sections we assumed a perfect pn junc-     2  Wg ξ 1 − tion at the point of light absorption, which is reasonable erf(x+  · exp − · exp · 0 ) + erf(x0 ) when the oxide thickness is larger than the spacing be- Lc 2L20 2    2   tween the split gates. However, to achieve good electro- Wg ξ + static control over the graphene layer, thin oxide layers + cosh · exp · erfc(xc ) , L0 2L2c are ideal, and they are often thinner than the split-gate where (18) spacing. For example, in Ref. 5 the split gates were em- √ ± bedded in the silicon waveguide with a spacing of 80 nm, x0,c = (Wg /ξ ± ξ/L0,c ) / 2, while the oxide, composed of either Al2 O3 or hBN, was r = (Lc /L0 ) · (κ0 /κ) . ∼10 nm thick. Meanwhile, Ref. 7 utilized a traditional split-gate design, as shown in Fig. 1(d), with a split-gate Given the complexity of this expression, there is not much separation of 100 nm and an oxide thickness of 40 nm. insight to be gained through visual inspection. Thus, in 5 Fig. 4 we plot the photovoltage as a function of the spot 2.5 × 1011 cm−2 . In panel (b) we consider moderate dop- size ξ and the split-gate separation Wg . ing by setting EF0 = EF = 100 meV, such that L0 = Lc and κ0 = κ. Finally, in panel (c) we consider high na- D  0.1 tive doping by setting EF0 = 300 meV, corresponding to /RZQDWLYH a carrier density n0 ≈ 1013 cm−2 . We also investigate 0.2 0.3 GRSLQJ the role of Pauli blocking in this high-doping case, which is accomplished by eliminating the first term within the  curly brackets in Eq. (18). 0.4 0.5 0.6 0.7 0.8 As expected, for all cases the photoresponse is maxi- mized for the limiting case when ξ = Wg = 0. However,  for finite values of ξ or Wg , the photoresponse exhibits E  nonmonotonic behavior. This is indicated by the dashed 0.2 0RGHUDWH GRSLQJ line in each panel, which shows the value of ξ that max- imizes VPTE for each fixed value of Wg . In contrast to Wg P Fig. 2, the photoresponse is not maximized at ξ = 0, but  rather at ξ ≈ Wg . Intuitively this makes sense: because 0.2 the region between the split gates does not contribute 0.3 to the photoresponse, reducing the light spot size much 0.4 0.5 0.6 below Wg can have a deleterious effect. In the other di- 0.7 0.8  rection, making the light spot too large also reduces the F  photoresponse, following Fig. 2. +LJKGRSLQJ 3DXOLEORFNLQJ With respect to the optimal value of Wg , Figs. 4(b)  and (c) indicate that a smaller Wg is always better, as the region between the split gates does not contribute to 0.1 the photoresponse. However, this trend does not hold 0.2 for the low-doping case in panel (a). Here, the dotted 0.4 0.3  line shows the value of Wg that maximizes VPTE for each      fixed value of ξ. The fact that the optimal value of Wg  P is finite is a consequence of the competing roles played by L0 and κ0 . Faster electron cooling between the split FIG. 4. Photovoltage VPTE as a function of the light spot size gates, corresponding to smaller L0 , reduces the electron ξ and split-gate separation Wg , from Eq. (18). Panel (a) is for temperature and thus the photoresponse. However, lower low native doping of the graphene layer, (b) is for moderate thermal conductivity slows the spread of heat away from doping, and (c) is for high doping, when Pauli blocking has the light source, which raises the electron temperature. set in. The dashed lines show the value of ξ at which VPTE is This can be seen mathematically via the definition of Tc maximized for each value of Wg , and vice versa for the dotted in Eq. (8). Therefore, under the right conditions it is line in panel (a). The thin solid lines are contours of VPTE , acceptable to have a finite Wg and lower cooling length if relative to the maximal value at (ξ = 0, Wg = 0). these are counteracted by a small thermal conductivity. In Fig. 4 we consider three levels of native doping of the graphene layer between the split gates: low doping Finally, let us briefly discuss the magnitude of the pho- in panel (a), moderate doping in (b), and high doping toresponse in Fig. 4. In each panel, VPTE is normalized to in (c), where Pauli blocking has set in. In the region its maximal value at (ξ = 0, Wg = 0), with the thin solid under the split gates, we assume the Fermi level has been lines showing the contours of VPTE . Clearly, it is impor- tuned to EF = ±100 meV, corresponding to a carrier tant to have Wg , ξ  Lc for an optimal photoresponse. density of n ≈ 1012 cm−2 . We also let Lc = 1 µm, typical The photodetectors in Refs. 5 and 7 have Wg ≈ ξ ≈ 40- of experimental results [18, 20, 21]. Next, by assuming 50 nm, putting them at ∼95% of the optimal photore- the electrical conductivity scales quadratically with EF sponse in Figs. 4(a) and (b), but at only ∼25% in panel (i.e., linearly with the carrier density) and the graphene (c). Pauli blocking between the split gates should thus density of states scales linearly with p EF , we find that obviously be avoided – this may not be an issue for pho- κ0 /κ = (EF0 /EF )2 and L0 /Lc = EF0 /EF [13], where todetectors operating at 1.55 µm, as native p-doping of EF0 is the Fermi level in the region between the split graphene is typically in the range of 100-300 meV, but gates. for the mid-IR and beyond this is a crucial design issue, In Fig. 4(a), we consider the case of low native dop- even for cases where the light is strongly confined and ing by setting EF0 = 50 meV, corresponding to n0 ≈ the split-gate separation is small. 6 D. Channel width (dashed) lines for the case of relaxing (reflecting) edges. Here we have assumed that Lc = 1 µm and Lα = 50 Thus far we have examined photodetector performance µm. We see that the behavior of the graphene edges has considering transport along the length of the channel. In little impact on the photocurrent or the NEP; the former this section we examine in more detail the role of the is maximized for wide channels, as an increasing Wch channel width Wch , and determine its optimal value. To corresponds to a linear decrease in the channel resistance. do so, we solve the 1D heat equation as before, but now Meanwhile, the NEP is minimized for a channel width for the temperature profile along the y-axis, perpendicu- of Wch ≈ 65 µm, depending on the boundary conditions. lar to the graphene channel. Averaging the temperature For relaxing graphene edges, assuming Wch , Lα  Lc , along the x-axis (over ±Lc ), the heat equation becomes the optimal value of Wch is given by the solution of (1+2ζ)(1+2x) = ex , where ζ = Lc /Lα and x = Wch /Lα . d2 Tel (y) Tel (y) βPin 1 −y/Lα For the values in Fig. 5, this gives Wch ≈ 1.34Lα ≈ − + e = 0, 67 µm. For reflecting edges, the optimal channel width is dy 2 L2c 2κLc Lα Wch ≈ 1.26Lα ≈ 63 µm, from the solution of ex = 1+2x. where (19) Similar to the scaling of the NEP with channel length, L Z c  Lc  this optimal value of the NEP arises from the nonmono- β= G(x)dx = erf √ tonic competition between the thermal noise and the pho- 2ξ toresponse. −Lc It is only for the photovoltage that the graphene edges when G(x) is a Gaussian light profile. play an appreciable role. For perfectly reflecting edges, The edges of the graphene channel play an impor- VPTE decreases monotonically with increasing Wch , a tant role in the resulting temperature profile. Recent consequence of loss of PTE signal due to heat spreading experimental work has demonstrated strong energy re- along the y-axis. Meanwhile, for perfectly relaxing edges, laxation at graphene edges, indicative of a resonant elec- VPTE is maximized for an intermediate value of Wch , tron cooling mechanism [22]. However, the edges in long enough to suppress the impact of heat relaxation at that study were highly disordered, while perfectly clean the edges but short enough to minimize loss of tempera- and straight edges may theoretically not exhibit any en- ture due to heat spreading. Assuming Wch , Lα  Lc , ergy relaxation. To study these limiting cases, we solve the optimal value of Wch is given by the solution of Eq. (19) for two sets of boundary conditions: Tel (y = (1 + 2ζ)(1 + x) = ex . For the values in Fig. 5, this gives 0) = Tel (y = Wch ) = 0 for perfectly relaxing edges, and Wch ≈ 0.31Lα ≈ 15 µm. dTel /dy|y=0 = dTel /dy|y=Wch = 0 for perfectly reflecting (i.e., nonrelaxing) edges.  The photovoltage is given by )LJXUHRI0HULW VPTE = 2S · Tav , (20) 1(3 W  where Tav = W1ch 0 ch Tel (y)dy is the average tempera- IPTE R ture along the y-axis,    VPTE Lα Lα Lc Tav ≈ Tξ · −A − , Lc Lc Lα        (   where   (21) Wch P coth W 2Lα · tanh 2Lc ch Wch , relaxing edges A= FIG. 5. Photodetector figures of merit as a function of the Lc /Lα , reflecting edges channel width Wch , from Eqs. (20)-(23). The photovoltage √  √  and photocurrent are normalized by their maximum values and we have used erf Lc / 2ξ ≈ erfcx ξ/ 2Lc . Note and the NEP is normalized by its minimum value. Here we that for reflecting graphene edges, the photovoltage re- have assumed a cooling length Lc = 1 µm and an optical duces to what we obtained in Eq. (11). absorption length Lα = 50 µm. The solid (dashed) lines are Using Eqs. (2) and (3), the photocurrent and the NEP for the case of perfectly relaxing (reflecting) graphene edges. are then given by Wch IPTE = 2S · Tav · σ · , (22) L r ch IV. MATERIAL OPTIMIZATION Pin 4kB T Lch NEP = · . (23) 2S · Tav σ Wch So far we have demonstrated how to optimize the per- The photovoltage, photocurrent, and NEP are plotted as formance of PTE photodetectors in terms of the device a function of the channel width in Fig. 5, with the solid geometry. Now we turn our focus to the properties of 7 graphene itself, and how the material quality correlates and (11) the photoresponse is given by RV = S/κ·Lc /Lα . with device performance. The photoresponse is typically For relaxing edges, we use Eqs. (20) and (21). quantified in terms of the photovoltage or photocurrent normalized by the input optical power, In Fig. 6, we plot the maximal photoresponse as a function of graphene material quality, quantified by the RV = VPTE /Pin , (24) carrier mobility µ. For each case, we have written the RI = IPTE /Pin . (25) conductivity as σ(E) = σmin + µ · en(E), where n(E) = E 2 /(π~2 vF2 ) is the carrier density and vF is the Fermi velocity. From this expression for σ(E) we calculate S, A. Thermopower / thermal conductivity / κ, and Lc from Eq. (26), and find the maximal value of cooling length the photoresponse as a function of EF (and Wch for the case of relaxing edges). We assume typical experimental As shown in the previous sections, the material quan- values of σmin = 0.2 mS and Lα = 50 µm. tities that directly relate to photodetector performance are the Seebeck coefficient S, the electronic thermal con- In Fig. 6, the optimal photoresponse reaches values of ductivity κ, and the electronic cooling length Lc . Each of 100−200 V/W, even for relatively low-mobility graphene. these are related to the electrical conductivity, σ, through Meanwhile, to date the highest measured response of a the relations [13, 26] waveguide-integrated photodetector in voltage mode is RV = 12.2 V/W [7], indicating that there is a lot of 1 K1 room left for optimization of these devices. We discuss S=− , eT K0 this device and another one in more detail in Section V. K12   2 κ= K2 − , Figure 6 is also striking in that the optimal photovolt- hT K0 p age saturates with increasing graphene quality, reaching Lc = κ/γCel , a maximum at µ ≈ 5000 cm2 /Vs. The reason for this where (26) is that the photoresponse is inversely proportional to the Z   thermal conductivity κ, which scales linearly with mobil- j ∂f Kj = (E − EF ) σ e(E) − dE, ity. As discussed earlier, a higher thermal conductivity ∂E means faster diffusion of heat away from the pn junction, e is the electron charge, h is Planck’s constant, σ e = reducing the electron temperature and thus the photore- σ/(2e2 /h), and f is the Fermi-Dirac distribution. sponse. Meanwhile, as shown in the inset of Fig. 6, S and The electron cooling is driven by a combination of su- Lc scale sublinearly with µ, yielding diminishing returns. percollisions and direct electron-phonon relaxation, and In the case of Lc , at high mobility the electron cooling is characterized by the cooling rate γ. Cooling aris- becomes limited by optical phonon emission [16]. ing from supercollisions and acoustic phonon emission is given by Eq. (13) of Ref. 13, while cooling due to optical phonon emission is given by Eq. (20) of the Supporting  Information of Ref. 16. UHIOHFWLQJ The Seebeck coefficient is often approximated by the  RVopt 9: Mott formula, S = (πkB )2 T /(3σ) · dσ/dE, but we find that this expression severely overestimates the maximal   value of S for high-mobility graphene. Meanwhile, when  S î 9. UHOD[LQJ EF > 2kB T the thermal conductivity is fairly well ap- Lc P   proximated by the Wiedemann-Franz law, κ = LσT ,  where L = (πkB /e)2 /3 is the Lorenz number. This rela-    tion generally holds for the situations we consider below.       Note that for all calculations below we consider devices  FP29V operating at room temperature, T = 300 K. FIG. 6. Optimal photovoltage response as a function of graphene mobility, for the cases of relaxing and nonrelax- B. Photovoltage ing edges. For nonrelaxing edges the optimal photovoltage is given by RVopt = max{S/κ · Lc /Lα }. For relaxing edges, the EF Here we consider the impact that material quality has on the photovoltage. We are interested in the maximal optimal photovoltage is RVopt = max {2S · Tav /Pin }, where EF ,Wch photovoltage that can be obtained for a given material, so Tav is defined in Eq. (21). The inset shows how the Seebeck we consider a perfect pn junction, a narrow light profile, coefficient and cooling length, at the point of optimal pho- and a long channel. We also compare relaxing and non- toresponse, scale with graphene mobility. relaxing graphene edges. For nonrelaxing edges the pho- tovoltage is maximized for Wch → 0, and from Eqs. (8) 8 C. Peltier cooling and the photocurrent with increasing graphene quality. This is shown in Fig. 7, where the solid lines indicate the optimal photocur- For the photocurrent the picture is more complicated. rent response, calculated by maximizing Eq. (28) over Up to now we have only considered the generation of elec- EF , for three different channel lengths, Lch = 0.3, 1, 3 trical signal via the Seebeck effect, but we have neglected µm. Similar to the photovoltage, the photocurrent re- other thermoelectric effects. Of particular relevance is sponse saturates with increasing graphene quality and the Peltier effect, which results in electronic cooling (or does not show significant improvement for graphene mo- heating) when a current is driven across a junction be- bilities above µ = 5000 − 10000 cm2 /Vs. To highlight tween dissimilar materials. This is captured by adding the impact of Peltier cooling, the dashed lines show the an extra term to the heat equation, ~j · ∇Π, where ~j is photoresponse when this effect is ignored. We attribute the electronic current density and Π = S · (Tel + T ) is the this strong Peltier effect to the unique electrical proper- Peltier coefficient [3]. ties of graphene, which permit both high mobility and a When the PTE photodetector is operated in an open- high Seebeck coefficient. circuit configuration, the detected signal is the photo- voltage, VPTE . In this configuration there is no current  flow and thus the Peltier effect plays no role. However, in a closed-circuit configuration, a photocurrent IPTE flows  P RIopt $: through the channel, leading to Peltier cooling at the pn junction. The Peltier term in the heat equation then be-   P comes VPTE /(RWch ) · T · dS/dx, where we assume low optical power such that Tel  T . Assuming a perfect Lch  P pn junction, and making use of Eqs. (2) and (6), the 1D heat equation of Eq. (7) can be rewritten as       2 d Tel (x) Tel (x) αPin  FP29V − + G(x) = 0, dx2 L2eff κWch FIG. 7. Optimal photocurrent response as a function of where (27) graphene mobility, calculated as the maximum of Eq. (28) 1 1 4S 2 σT over EF . The solid (dashed) lines show the photoresponse = + η δ(x), with (without) Peltier cooling. L2eff L2c κLch and η = 1 − exp(−Lch /2Lc ). Here we also assume re- flecting graphene edges. In this modified heat equation, Leff thus describes an V. COMPARISON WITH EXPERIMENTS effective cooling length that is comprised of the usual cooling length, Lc , and an additional term arising from Peltier cooling at the pn junction. Here we see that the Here we briefly compare the above models of photore- Peltier cooling strength grows as S 2 , arising from the sponse to a couple experimental examples of photodetec- Peltier term and the fact that IPTE ∝ S. Thus, en- tors, and discuss where room remains for performance hancing the Seebeck coefficient of graphene should yield optimization. We start with Ref. 7, which used care- diminishing returns and saturation of the photocurrent. fully designed split gates to confine the optical light pro- To quantify the role of Peltier cooling, we solve Eq. (27) file to a few tens of nanometers. With this device de- by making use of Eq. (18). In Eq. (18), we had assumed sign, the authors measured a photovoltage response of a finite region in the middle of the graphene channel (of RV = 12.2 V/W in one of their devices, which is the width 2Wg ) where the values of the thermal conductivity highest measured to date for waveguide-integrated pho- and cooling length (κ0 , L0 ) differed from those under the todetectors in voltage mode. Plugging their experimental split gates (κ, Lc ). To capture the situation represented parameters into the models presented above (σmin = 0.2 by Eq. (27), we let κ0 = κ and L0 = Leff . We write the mS, µ = 2000 cm2 /Vs, Wg = ξ = 50 nm, Lch = 4.8 Dirac delta function in the expression for Leff as δ(x) = µm with 3.2 µm covered by the split gates, Wch = 1.7 1/(2Wg ), and finally we take the limit Wg → 0. µm), we would predict RV ≈ 65 V/W for nonrelaxing This yields a modified expression for the photocurrent graphene edges and RV ≈ 35 V/W with relaxing edges. response, The former is more than 5× higher than the measured value, while the latter is approaching the same order of magnitude (less than 3×). This suggests that thermal   σ Lc σ Lc 2 RI = S · η 1 + 2T · η ·S , (28) relaxation at the graphene edges may be limiting the al- κ Lch κ Lch ready impressive device performance, due to the quite where we have taken the limits ξ → 0 and α → 1. narrow channel width (also note that our models do not The denominator captures the impact of Peltier cooling account for optical losses due to the metal split gates). and indicates diminishing returns in the photoresponse Indeed, our models indicate that by increasing Wch to 25 9 µm, the photoresponse can reach up to 56 V/W. peting roles of the Seebeck coefficient and the cooling Next we compare to Ref. 24, which achieved length on one side, and the thermal conductivity on the high-frequency (>67 GHz) operation in a waveguide- other. Meanwhile, the photocurrent saturates with in- integrated photodetector with high-mobility CVD creasing graphene quality because of Peltier cooling. This graphene, and measured a photoresponse of RV = 6 effect is often overlooked or ignored, but our analysis indi- V/W. Using their device parameters (σmin = 0.2 mS, cates it plays an important role in high-quality graphene, µ = 16000 cm2 /Vs, Wg = 500 nm, ξ = 300 nm , Lch = 10 as a consequence of graphene’s large mobility and See- µm with 8 µm covered by the split gates, Wch = 80 µm), beck coefficient. In general, minimal improvements in we predict RV = 32 V/W, which is significantly larger optimal photoresponse are expected for mobilities above than the measured value (more than 5×). The long chan- 5000 cm2 /Vs. nel width means that edge relaxation is not playing a Finally, comparisons of our models with experimen- role, so the discrepancy here may lie in the estimate of tal works suggest the important role played by resonant the cooling length Lc . As already commented by the au- electron cooling in graphene. This mechanism can occur thors of Ref. 24, a cooling length of Lc = 130 nm was at graphene edges and can strongly limit photodetec- needed to fit their models to their measurements, and we tor performance in narrow channel devices. Such edge find similar behavior with our models. This is quite a bit relaxation may be mitigated by proper choice of passi- lower than the cooling length of Lc ≈ 1 µm predicted by vation, which may either delocalize resonant states re- the theory of supercollisions plus phonon emission [3, 16]. sponsible for the fast cooling, or move them to ener- This discrepancy suggests that supercollisions and direct gies away from the regime of device operation. Reso- electron-phonon relaxation may not be enough to explain nant cooling may also be playing an important role in experiments, and one must consider the impact of reso- high-quality graphene, where electron cooling lengths are nant electron cooling [14, 15, 22]. Under this mechanism, much shorter than what is predicted by the other cooling a small number of resonant scatterers, arising from va- mechanisms. In our opinion, this issue deserves further cancy defects or covalent chemical adsorbates, may be detailed study. sufficient to suppress the electronic heat while having an insignificant impact on the carrier mobility. ACKNOWLEDGMENTS VI. SUMMARY AND CONCLUSIONS We are grateful for useful discussions with Klaas-Jan Tielrooij and Max Lemme. ICN2 is supported by the In summary, we provide a detailed analysis of the op- Severo Ochoa Centres of Excellence program, funded by timization of photodetectors based on the photothermo- the Spanish State Research Agency (AEI, Grant No. electric effect in graphene. First, we show how various SEV-2017-0706), and is funded by the CERCA program geometrical parameters – including light spot size, chan- of the Generalitat de Catalunya. 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