Enhanced Thermionic-Dominated Photoresponse in Graphene

Walter Burke Institute for Theoretical Physics and Institute of Quantum ... Significant field emission enhancement in ultrathin nano-thorn covered NiO...
0 downloads 0 Views 605KB Size
Download PDF
Letter pubs.acs.org/NanoLett

Enhanced Thermionic-Dominated Photoresponse in Graphene Schottky Junctions Joaquin F. Rodriguez-Nieva,*,† Mildred S. Dresselhaus,†,‡ and Justin C. W. Song*,§ †

Department of Physics and ‡Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States § Walter Burke Institute for Theoretical Physics and Institute of Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, United States ABSTRACT: Vertical heterostructures of van der Waals materials enable new pathways to tune charge and energy transport characteristics in nanoscale systems. We propose that graphene Schottky junctions can host a special kind of photoresponse that is characterized by strongly coupled heat and charge flows that run vertically out of the graphene plane. This regime can be accessed when vertical energy transport mediated by thermionic emission of hot carriers overwhelms electron−lattice cooling as well as lateral diffusive energy transport. As such, the power pumped into the system is efficiently extracted across the entire graphene active area via thermionic emission of hot carriers into a semiconductor material. Experimental signatures of this regime include a large and tunable internal responsivity 9 with a nonmonotonic temperature dependence. In particular, 9 peaks at electronic temperatures on the order of the Schottky barrier potential ϕ and has a large upper limit 9 ≤ e/ϕ (e/ϕ = 10 A/W when ϕ = 100 meV). Our proposal opens up new approaches for engineering the photoresponse in optically active graphene heterostructures. KEYWORDS: Graphene, hot carriers, Schottky junction, photocurrent

V

dominates over more conventional electronic cooling channels, for example, electron−lattice cooling. Indeed, we find that 1⊥q can be significant in graphene (see Figure 1c) when kBTg ≈ ϕ/ 2, dominating over acoustic and optical phonon cooling12,13 in pristine graphene Schottky junctions for not too high barrier heights. 1⊥q also overwhelms in-plane (lateral) diffusive energy

ertical heterostructures comprising layers of van der Waals (vdW) materials have recently emerged as a platform for designer electronic interfaces. 1 Of special interest are heterostructures that feature tunable interlayer transport characteristics, as exemplified by g/X Schottky junctions;2−10 here “g” denotes graphene, and X is a semiconductor material, such as Si, MoS2, or WSe2. These junctions are characterized by Schottky barriers ϕ that span 2 orders of magnitude ϕ ≈ 0.01− 1 eV and exhibit in situ control through applied bias or by using gate potentials.7−11 The wide range of ϕ values achievable across the g/X interface, combined with the unique graphene photoresponse mediated by long-lived hot carriers (elevated electronic temperatures, Tg, different from those of the lattice, T0),12−17 make graphene Schottky junctions a prime target for accessing novel vertical energy transport regimes.18 Here we show that specially designed graphene Schottky junctions can host an enhanced thermionic-dominated photoresponse driven by strongly coupled charge and energy currents. Such photoresponse proceeds, as illustrated in Figure 1a, via the thermionic emission of graphene hot carriers with energy larger than the Schottky barrier. At steady state, an equal number of cold carriers are injected at the Fermi surface through an ohmic contact, giving a net flow of heat 1⊥q out of the graphene electronic system balancing the energy pumped into the system. Strikingly, thermionic emission yields strong heat transport running vertically out of the hot electron system, which © 2016 American Chemical Society

transport. We find that the values of 1⊥q can be competitive with disorder-assisted cooling14−16 in more dirty devices. Graphene is essential to our proposal due to a unique combination of electronic characteristics. First, fast intraband Auger-type scattering19−21 allows the absorbed photon energy flux, 1 qin , to be efficiently captured as heat by ambient carriers in graphene; this process results in a thermalized hot carrier distribution.20,21 Second, graphene is characterized by slow electron−lattice cooling mechanisms12−17 that enable Tg > T0 to drive a strong thermionic current. This is due to the large optical phonon energy in graphene12,13 as well as the weak electron-acoustic phonon coupling (for a detailed comparison between cooling rates, see eq 8 below). Third, the bias and gate-tunable work function allows an experimentally accessible way to optimize device operation, predicted here to occur at Received: May 13, 2016 Revised: September 3, 2016 Published: September 6, 2016 6036

DOI: 10.1021/acs.nanolett.6b01965 Nano Lett. 2016, 16, 6036−6041

Letter

Nano Letters

on the order of e/ϕ = 10 A/W for ϕ = 100 meV. For a discussion of net values of 9 in eq 1, see Figure 2c. Because the incident photon energy ℏω (e.g., in the visible) can be many multiples of ϕ, we anticipate that g/X Schottky photodetectors can provide significant gains in the internal responsivity compared to those in conventional (photovoltaicbased) photodetectors, which are limited by 9 PV ≤ e/ℏω .22 In particular, the ultrafast electron energy relaxation times in graphene yield multiple hot carriers per absorbed photon20,21 in stark contrast to photovoltaic-based schemes that yield a single electron−hole pair per absorbed photon. Naturally, the external responsivity of the device is also affected by the absorption coefficient of the photoactive material. Whereas the absorption coefficient of 2.3% per layer in graphene23 is small compared to, for instance, typical values of 10−50% in Si,24 this small value can be increased using optical waveguides3 and plasmon enhanced absoption.25 These external enhancement mechanisms will not be discussed here. In addition, g/X photodetectors also enable a boosted photoresponse compared to previous photothermoelectricbased schemes.17 This is because the vertical structure allows to circumvent lateral electronic heat diffusion, which drastically reduces the operating electronic temperatures and efficiencies in photothermoelectric-based schemes. Another important feature of the g/X photoresponse is the possibility of using the temperature dependence of 9 as a diagnostic of thermionic-dominated photoresponse. Including losses to the lattice via disorder-assisted cooling, we find that 9 is nonmonotonic, peaking at an optimal operating hot carrier temperature kBTg ≈ ϕ/2 (Figure 2c). Because Tg can be controlled by the incident light power and ϕ via gate voltage, nonmonotonic 9 as a function of Tg provides an easily accessible experimental signature of the strongly coupled charge and energy thermionic transport that is engineered across the g/X interface. Indeed, nonmonotonic temperature behavior does not occur in photovoltaic-based devices, where responsivity is mainly independent of pump power or photon intensity. We begin by modeling vertical transport across the g/X device, as depicted in Figure 1a. To describe thermionic transport over the barrier, we adopt a quasielastic but momentum nonconserving approximation.26 This approximation is valid because at high T a number of momentum scattering mechanisms at the Schottky junction are possible, such as scattering by defects, intrinsic phonons, and substrate phonons. Furthermore, the typical energy exchange in these processes is small on the Schottky barrier scale. As a result, we generically write the electron and heat current across the g/X Schottky junction as

Figure 1. (a) Schematic of thermionic-dominated heat transport in a graphene Schottky junction. Hot carriers with energies ε close to the Schottky barrier height ϕ are thermionically emitted into a semiconductor material in the out-of-plane direction ẑ, while cold carriers are injected through an Ohmic contact at the Fermi level, μ, generating a net vertical heat current 1⊥q . (b) Normalized ideal responsivity, α = 9 0ϕ/e , is shown as a function of normalized graphene electronic temperature, kBTg/ϕ, with e as the electron charge (see eq 1). Curves are obtained for normalized ambient temperatures kBT0/ϕ = 0, 0.25, 0.5, 0.75, 1.0 indicated with different colors for increasing T0; characteristic 9 0 ≈ e/ϕ ∼ 10A/W can be large (for ϕ = 100 meV, see text). (c) Thermionic cooling channel, 1⊥q , compared with acoustic phonon cooling (clean case), 1 ac q , disorder-assisted op cooling, 1 dis q , and optical phonon cooling 1 q (see text and eq 8), shown for ambient temperature T0 = 300 K (solid lines) and T0 = 0 op (dotted lines). Note that 1⊥q overwhelms 1 ac q and 1 q (clean case)

and is competitive with 1 dis q (dirty case). Parameter values used: μ = ϕ = 100 meV, kFl = 50, and G0 = 10 mS/μm2, see eqs 4 and 8.

kBTg ≈ ϕ/2, for a range of technologically achievable temperatures and barrier materials. Indeed, whereas other Schottky junctions (e.g., Au/Si, Ag/Si) may also display vertical energy currents, their large Schottky barriers (ϕ ≈ 1 eV) and fast electron−lattice cooling render the thermionic-dominated regime impractical for these systems. An important optoelectronic figure of merit is the conversion between incoming photon energy flux, 1 qin , and detected photocurrent, 1⊥e , encoded in the (internal) responsivity 9 = 1⊥e /1 qin . Importantly, our model yields a large upper limit for 9 . As we explain below, energy ϕ is transported per carrier extracted across the g/X junction (Figure 1a) yielding a limiting internal responsivity 9 0 , occurring in the thermionicdominated regime (i.e., 1⊥q = 1 qin ) given by 9 ≤ 9 0,

⎛e⎞ 9 0 = ⎜ ⎟α(Tg̃ , T0̃ ) ⎝ϕ⎠

⎡ 1⊥ ⎤ ⎢ e⎥= ⎢ 1⊥ ⎥ ⎣ q⎦

⎡ ⎛

⎞

⎣ ⎝

B g⎠

⎞⎤

⎛

∫−∞ dε⎡⎣⎢ ε ⎤⎦⎥G(ε)⎢⎢f ⎜⎜ k εT ⎟⎟ − f ⎜⎝ k εT ⎟⎠⎥⎥, ∞

e

B 0

⎦

2

G (ε ) = (1)

2πe Dg (ε)Dc(ε)|T(ε)|2 ℏ

(2)

Here G(ε) is a parameter with units of electrical conductance which characterizes the Schottky interface (see discussion below), Dg (Dc) is the density of states of graphene (the conduction band of the semiconductor), f(x) = 1/(ex + 1) is the Fermi distribution function, T(ε) is the energy-dependent tunneling transition matrix element between graphene and

Here e is the electron charge, T̃ g,0 = kBTg,0/ϕ are the dimensionless graphene and ambient temperatures (temperature of the lattice and semiconductor), respectively, and α is a dimensionless function (see text below) plotted in Figure 1b. The function α can take values close to unity, allowing 9 0 to be 6037

DOI: 10.1021/acs.nanolett.6b01965 Nano Lett. 2016, 16, 6036−6041

Letter

Nano Letters

function of Tg and T0; naturally the Tg values displayed can be attained via a suitably chosen 1 qin . The depletion width, for example, in g/Si interfaces,6 can be many times larger than the electron wavelength. As a result, only electrons with energies above the effective barrier ϕ formed at the g/X interface contribute to the current; in this way, the photocurrent is thermally activated. Here, we adopt a phenomenological approach to capture the essential physics independent of the microscopic details of the device. To this end, we approximate G(ε) = G0Θ (ε−ϕ) in eq 2 with Θ the step-function in order to aggregate the microscopics of the junction into a single variable that can be easily measured in experiments. This approach does not describe field emission, which is considered negligible because we are limiting our discussion to the zero bias behavior, that is, the closed circuit photocurrent. We emphasize that this approximation does not affect the qualitative behavior of 1⊥e , 1⊥q or α for the range of temperatures of interest, Tg ≲ϕ; further, this approximation represents a conservative estimate of the particle current, because G(ε) is typically a monotonically increasing function due to the larger density of states available for scattering at larger ε in graphene. As a side remark, we note that G0 is not the zero-bias junction conductance; the latter quantity is suppressed by a factor e−ϕ/T with respect to G0, as discussed in the paragraph following eq 10. Using a steplike transmission, heat and charge currents can then be expressed in terms of nondimensional integrals by defining x = ε/ϕ in eq 2, yielding

Figure 2. (a) Schematic of a hot carrier photodetector formed by a graphene-X Schottky junction, where X is a semiconductor material with bandgap ΔX. Here we consider graphene as the photoactive material for absorption, that is, photon energy ℏω < ΔX. (b) Ratio ζ = 1 qloss /1⊥q modeled via eq 10 as a function of Tg with T0 = 0 (solid line) and T0= 300 K (empty circles). Note that the latter has a smaller range, Tg ≥ T0. (c) The responsivity 9 (solid lines) for the g/X junction exhibits a nonmonotonic electronic temperature dependence peaking at Tg ≈ ϕ/2kB, shown for ζ modeled in eq 10 (panel b) with ζ* = 5, 1, 0.2 (blue, green, red, respectively); dimensionless 9 is shown on the left vertical axis. The dashed line indicates an ideal case 9 = 9 0 . Here we used values G0 = 2, 10, 50 mS/μm2, μ = ϕ = 100 meV and kFl = 50 yielding sizable 9 (right vertical axis).

semiconductor electronic states, and energies ε are referenced from the Fermi energy μ (see Figure 1a). The function T(ε) contains all the microscopic information about the relevant mechanisms that couple graphene with the material X, such as phonons or hot-spots formed by defects. Two important assumptions are present in eq 2. First, we neglected hole transport between graphene and the valence band of X assuming that the barrier height for hole transport is much larger than the corresponding one for electron transport. Second, we assume that the Fermi level and the temperature in graphene and X are spatially fixed. In a more realistic scenario, the pumping power may cause the temperature and Fermi level to spatially vary in the out-of plane direction. In this case, both quantities need to be determined self-consistently by appropriate balance equations. However, we do not anticipate these will introduce new qualitative features. When light heats graphene electrons so that Tg > T0, eq 2 describes the short-circuit charge current (photocurrent) and the energy current flow, shown schematically in Figure 1c. At steady state, Tg is determined by energy balance of the incident absorbed power in graphene, 1 qin , and the energy being dissipated by the graphene electronic system, 1 out q , that

1⊥e =

∫1

∞

dxΔf (x),

1⊥q =

G0ϕ2 e2

∫1

∞

dxxΔf (x) (4)

where Δf(x) = f(x/T̃ g) − f(x/T̃ 0). It is straightforward to show that the integrals on the right-hand side of eq 4 are related to complete Fermi integrals Fk(ξ) =

∫0

∞

dx

xk e x−ξ + 1

(5)

⎛n ⎞ k+1 ∞ n via ∫ dxx nf (x /T̃ ) = ∑k = 0 ⎜ ⎟T̃ Fk( −1/T̃ ). In the low1 ⎝k ⎠ temperature regime, T̃ ≪ 1, the value of Fk behaves as Fk(−1/ ̃ T̃ ) ≈ Γ(k + 1) e−1/T with Γ the gamma function. In the hightemperature regime, T̃ ≫ 1, Fk takes values F0(−1/T̃ ) ≈ ln(2) and F1(−1/T̃ ) ≈ π2/12. A key feature of thermionic-dominated [1 qloss = 0] transport is the strong charge-energy current coupling manifested in 9 0 . Using eq 4 above, we obtain eq 1 with α given by ∞

includes both the thermionic channel, 1⊥q , and other dissipative

α(Tg̃ , T0̃ ) =

channels, 1 qloss (e.g., electron−lattice cooling and diffusive heat transport discussed below). Explicitly, we have 1 qin = 1 qout = 1⊥q (Tg , T0) + 1 qloss(Tg , T0)

G0ϕ e

∫1 dxΔf (x) ∞

∫1 dxxΔf (x)

(6)

The general behavior of α(T̃ g, T̃ 0) can be most easily understood by first setting T0 = 0. In this case, α in eq 1 adopts the simple form

(3)

where we have fixed T0 to the temperature of the ambient environment, that is, there is no backflow of hot electrons into graphene. The latter assumption results from the large heat capacity and fast electron−lattice cooling in highly doped semiconductors such as Si.27 In what follows, we shall analyze the energy/charge characteristics of g/X Schottky junctions as a

−1 ⎡ ⎛ −1 ⎞ ⎤ ⎜ ⎟ F ⎢ 1⎝ T̃ ⎠ ⎥ g ⎥ α0(Tg̃ ) ≡ α(Tg̃ , T0̃ = 0) = ⎢1 + Tg̃ ⎛ ⎢ −1 ⎞ ⎥ F0⎜ T̃ ⎟ ⎥ ⎢⎣ ⎝ g ⎠⎦

6038

(7)

DOI: 10.1021/acs.nanolett.6b01965 Nano Lett. 2016, 16, 6036−6041

Letter

Nano Letters Importantly, α0 is a decreasing function of temperature T̃ g, as shown in the black dashed curve of Figure 1b. In particular, for T̃ g ≪ 1 α0 takes values α0 ∼ (1+ T̃ g)−1 of order unity, and for T̃ g≫ 1 α0 decreases with inverse temperature as α0 ≈ 12 log(2)/(π2T̃ g) (see Figure 1b). This latter fact means that, although 9 0 is expressed in units of e/ϕ in eq 1, 9 0 cannot grow indefinitely by making ϕ smaller; 9 0 reaches a saturating value 9 0 ≈ 12 log(2)e/(π 2kBTg) for kBTg ≫ ϕ, as shown in Figure 1c. For finite values of T0, the qualitative behavior of α does not depart significantly from that of α0. As shown in Figure 1b, where α(T̃ g, T̃ 0) is obtained by numerical integration of eq 6 for different values of T̃ 0, the function α largely follows the α0 curve and only shifts slightly from α0 with increasing T̃ 0. Further, the range of operating hot electron temperatures is now smaller,T̃ g ≥ T̃ 0, as illustrated in Figure 1b by curves that now start at T̃ g = T̃ 0. Although α is finite at T̃ g = T̃ 0, there is no net current at equal temperatures (as indicated by the empty circles at the beginning of the curves in Figure 1b); a nonvanishing α at T̃ g = T̃ 0 arises from the differential ratio that characterizes the responsivity 9 . Considering losses, eq 4 yields 1⊥q that can be sizable (see blue curves in Figure 1c). In plotting Figure 1c, we numerically integrated eq 4 and used ϕ = 100 meV and G0 = 10 mS/μm2 (see below for G0 estimates). Further, we find that 1⊥q compares favorably with intrinsic electron−lattice cooling in graphene: (i) single-acoustic phonon cooling for pristine graphene 1 ac q (green curves), (ii) optical phonon cooling

1 qloss ≈ 1 qdis = γdis(Tg3 − T03) as an illustrative example, see eq 8. Adopting the same procedure as described above, we find

⎛ T̃ 3 − T̃ 3 ⎞ g 0 ⎟ ζ(Tg̃ , T0̃ ) = ζ ⎜ ∞ , *⎜ ∫ dxxΔf ⎟ ⎝ 1 ⎠

3 3 1 dis q = γdis(Tg − T0 ),

⎡ ⎛ ℏω ⎞ ⎛ ℏω ⎞⎤ ⎢N ⎜ op ⎟ − N ⎜ op ⎟⎥ = 1 op γ q ⎟ op⎢ ⎜ ⎝ kBT0 ⎠⎥⎦ ⎣ ⎝ kBTg ⎠

(8)

is competitive with 1 dis q , as shown in Figure 1c. The hot carrier thermionic cooling channel, 1⊥q , and the strong charge-energy current coupling it produces (eq 1) can manifest itself in large and nonmonotonic responsivities in g/X photodetectors (Figure 2a). Accounting for energy balance in eq 3, we find a net responsivity given by 90 , 1+ζ

ζ=

1 qloss 1⊥q

G0

(10)

find that 1⊥q scales as T2g and rises less steeply than the T3g power law of supercollision cooling. Hence, there is a “sweet spot” for observing a competitive thermionic channel 1⊥q . The optimal value occurs for temperatures kBTg/ϕ ≈ 0.5 (see Figure 2b) with minimum ζmin≈ 1.85ζ*. This can be estimated from eq 10 in the limit T̃ g ≪ 1 and T0 = 0, where the above-mentioned optimal values are obtained from minimization of the equation ̃ ζ/ζ* ≈ T̃ 2ge1/Tg (Figure 2b). The responsivity in eq 9 mirrors ζ to display a nonmonotonic dependence on Tg, peaking at a temperature kBTg ≈ ϕ/2, as shown in Figure 2c. Peak responsivities in the range 9 ≈ 1−10 A/W are obtained within our model. Indeed, for large G0 = 50 mS/μm2 (corresponding to ζ* = 0.2), 9 starts to approach the ideal case, 9 = 9 0 (dashed black line). The nonmonotonic dependence of 9 as a function of Tg provides a clear fingerprint of the competition between thermionic energy transport and conventional electron−phonon cooling. Because the Schottky barrier heights can be tuned by the applied gate voltage, the peak temperature kBTg ≈ ϕ/2 is gate tunable. Further, the scaling of ϕ and the device conductance G0 also provide experimental knobs with which to adjust the responsivity of the device. The optimal responsivity occurring at kBTg ≈ ϕ/2 is an important characteristic for the design of graphene photodetectors. Indeed, given that Tg < 2000 K in realistic situations,

Here the prefactors are γac = ℏD2μ4kB/8πρ(ℏvF)6, γdis = 2D2μ2k3B/ρc2ℏ (ℏvF)4kFl and γop = ℏ2ω3op/24πρa4v2F with D as the deformation potential, ρ is the graphene mass density, kFl is the dimensionless disorder parameter, a is the lattice constant, ωop is the optical phonon frequency, and N(x) is the Bose distribution. For the cooling mechanisms, we used μ = 100 meV, D = 20 eV, ρ = 7.6 × 10−7 kg/m2, kFl = 50, a = 1.4 Å, and op ℏωop = 0.2 eV. Indeed, 1⊥q overwhelms both 1 ac q and 1 q and

9=

e 2γdisϕ

where the characteristic ζ is set by e2γdisϕ/G0. As expected, increasing the prefactor γdis for 1 dis q increases the losses to phonon scattering embodied in ζ. Alternatively, increasing the conductance across the g/X interface enhances the thermionic channel. In calculating 9 in eq 9, we use the same parameter values as in Figure 1, μ = ϕ = 100 meV and kFl = 50. The value of G0 can be estimated from conductance measured in the dark state, GD, obtained in actual g/X devices at equilibrium Tg = T0 (for example, g/Si Schottky junctions in refs 7 and 8). Indeed, under an infinitesimally small potential bias ΔVb we can approximate Δf in eq 6 as Δf(x) = [ex/(ex + 1)2](eΔVb/T0) due to the small chemical potential difference eΔVb between graphene and X. Integrating over x in eq 6, we obtain GD = G0/ (1 + eϕ/kBT0). In a typical scenario kBT0 ≪ ϕ, the conductance in the dark state is exponentially suppressed with increasing temperature as GD ≈ G0 exp(−ϕ/kBT0), which is in agreement with the qualitative behavior observed in refs 2−4 and 7−10. To give an estimate of the range of conductances achievable in g/X devices, GD in these experiments report GD ∼ 0.1 − several × μS/μm2 for ϕSi ∼ 0.3 eV with T0 at room temperature. This gives G0 in the 1−100 mS/μm2 ballpark (for 9 in Figure 2c, we used G0 = 2, 10, and 50 mS/μm2, which correspond to ζ* = 5, 1, and 0.2, see eq 10). As shown in Figure 2b, ζ exhibits a clear nonmonotonic dependence on Tg characterized by the following two regimes: (i) small T̃ g ≪ 1, 1⊥q is exponentially suppressed by the ̃ transport barrier ϕ, thus 1 dis q dominates; (ii) large Tg ≫ 1, we

dis 1 op q (magenta curves), and (iii) disorder-assisted cooling 1 q (red curves), where we consider the degenerate limit (μ ≫ kBTg) for all cases:12−14

1 ac q = γac(Tg − T0),

ζ = *

(9)

where ζ quantifies the losses. To estimate ζ for actual devices, we consider the disorder-assisted cooling power in graphene,14 6039

DOI: 10.1021/acs.nanolett.6b01965 Nano Lett. 2016, 16, 6036−6041

Letter

Nano Letters Present Address

Schottky barriers in the 100 meV ballpark allow operation of the g/X photodetector near optimal responsivities (i.e., near minimum ζ). These values of ϕ can be achieved, for instance, in graphene/WS2 devices.10 Although g/X photodetectors allow in situ control of ϕ by electrostatic doping, it is important to note that several parameters of the model vary implicitly with ϕ. On the one hand, changes in ϕ also induce changes in graphene doping, thus modifying the electronic cooling power. Further, when ϕ becomes smaller than the incoming photon energies, photoemission of primary carriers over the barrier competes with thermalization by electron−electron interactions. In this case, a smaller amount of the incident power is captured in the hotcarrier distribution. Naturally, there are other mechanisms for losses that affect the responsivity. For instance, lateral (in-plane) heat currents, 1 q = −∇·(κ ∇Tg), can transport heat toward the contacts in small devices. To estimate this effect, we use the Wiedemann− Franz relation, κ∥(Tg) = (π2/3e2)( k2BTgσ), where σ is the inplane electrical conductivity of graphene. For the relevant regime of moderate to high temperatures, Tg ≳ ϕ, we can approximate 1⊥q ≈ G0kB2Tg2/e 2 = γ ⊥(Tg)Tg [cf. eq 2]. As a

(J.C.W.S.) Institute of High Performance Computing Singapore, and Division of Physics and Applied Physics, Nanyang Technological University. Funding

We are grateful for useful discussions with M. Baldo, M. Kats, and L. Levitov. We also thank V. Fatemi, A. Frenzel, and K. Tielrooij for a critical reading of the text. J.F.R.N. and M.S.D. acknowledge financial support from the National Science Foundation Grant DMR-1507806. J.C.W.S. acknowledges support from a Burke Fellowship at Caltech. Notes

The authors declare no competing financial interest.

■

(1) Geim, A. K.; Grigorieva, I. V. Nature 2013, 499, 419−425. (2) Miao, X.; Tongay, S.; Petterson, M. K.; Berke, K.; Rinzler, A. G.; Appleton, B. R.; Hebard, A. F. Nano Lett. 2012, 12, 2745−2750. (3) Wang, X.; Cheng, Z.; Xu, K.; Tsang, H. K.; Xu, J.-B. Nat. Photonics 2013, 7, 888−891. (4) Britnell, L.; Gorbachev, R. V.; Jalil, R.; Belle, B. D.; Schedin, F.; Katsnelson, M. I.; Eaves, L.; Morozov, S. V.; Mayorov, A. S.; Peres, N. M. R.; Neto, A. H. C.; Leist, J.; Geim, A. K.; Ponomarenko, L. A.; Novoselov, K. S. Nano Lett. 2012, 12, 1707−1710. (5) Yu, W. J.; Li, Z.; Zhou, H.; Chen, Y.; Wang, Y.; Huang, Y.; Duan, X. Nat. Mater. 2012, 12, 246−252. (6) Li, X.; Zhu, H.; Wang, K.; Cao, A.; Wei, J.; Li, C.; Jia, Y.; Li, Z.; Wu, D. Adv. Mater. 2010, 22, 2743−2748. (7) Chen, C.-C.; Aykol, M.; Chang, C.-C.; Levi, A. F. J.; Cronin, S. B. Nano Lett. 2011, 11, 1863−1867. (8) Yang, H.; Heo, J.; Park, S.; Song, H. J.; Seo, D. H.; Byun, K.-E.; Kim, P.; Yoo, I.; Chung, H.-J.; Kim, K. Science 2012, 336, 1140−1143. (9) Britnell, L.; Gorbachev, R. V.; Jalil, R.; Belle, B. D.; Schedin, F.; Mishchenko, A.; Georgiou, T.; Katsnelson, M. I.; Eaves, L.; Morozov, S. V.; Peres, N. M. R.; Leist, J.; Geim, A. K.; Novoselov, K. S.; Ponomarenko, L. A. Science 2012, 335, 947−950. (10) Georgiou, T.; Jalil, R.; Belle, B. D.; Britnell, L.; Gorbachev, R. V.; Morozov, S. V.; Kim, Y.-J.; Gholinia, A.; Haigh, S. J.; Makarovsky, O.; Eaves, L.; Ponomarenko, L. A.; Geim, A. K.; Novoselov, K. S.; Mishchenko, A. Nat. Nanotechnol. 2012, 8, 100−103. (11) Di Bartolomeo, A. Phys. Rep. 2016, 606, 1−58. (12) Bistritzer, R.; MacDonald, A. H. Phys. Rev. Lett. 2009, 102, 206410. (13) Tse, W.-K.; Das Sarma, S. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 79, 235406. (14) Song, J. C. W.; Reizer, M. Y.; Levitov, L. S. Phys. Rev. Lett. 2012, 109, 106602. (15) Graham, M. W.; Shi, S.-F.; Ralph, D. C.; Park, J.; McEuen, P. L. Nat. Phys. 2012, 9, 103−108. (16) Betz, A. C.; Jhang, S. H.; Pallecchi, E.; Ferreira, R.; Feve, G.; Berroir, J.-M.; Placais, B. Nat. Phys. 2012, 9, 109−112. (17) Gabor, N. M.; Song, J. C. W.; Ma, Q.; Nair, N. L.; Taychatanapat, T.; Watanabe, K.; Taniguchi, T.; Levitov, L. S.; Jarillo-Herrero, P. Science 2011, 334, 648−652. (18) Ziman, J. Principles of the Theory of Solids; Cambridge University Press: Cambridge, England, 1972. (19) Koppens, F. H. L.; Mueller, T.; Avouris, Ph.; Ferrari, A. C.; Vitiello, M. S.; Polini, M. Nat. Nanotechnol. 2014, 9, 780−793. (20) Tielrooij, K. J.; Song, J. C. W.; Jensen, S. A.; Centeno, A.; Pesquera, A.; Zurutuza Elorza, A.; Bonn, M.; Levitov, L. S.; Koppens, F. H. L. Nat. Phys. 2013, 9, 248−252. (21) Song, J. C. W.; Tielrooij, K. J.; Koppens, F. H. L.; Levitov, L. S. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 155429. (22) Sze, S. M. The physics of semiconductor devices; Wiley: New York, 2007.

result, we find a cooling length ξ = κ(Tg)/γ ⊥(Tg) coming from the thermionic channel that is independent of Tg. Using a uniform in-plane σ ∼ 1 mS,28 we find ξ∥ ≈ 0.6 μm, so that vertical energy extraction dominates over in-plane thermal conduction for sufficiently large devices with size L > ξ∥. We note that interactions with the substrate can result in cooling via surface optical phonons. These losses will vary for different substrate (X) choices and are only significant when X is a polar material.29 Importantly, we do not expect them to be relevant in nonpolar materials, for example, X = silicon. Lastly, it is interesting to note that g/X photodetectors can also operate at low photon energies, ℏω ≤ 2 μ. In this regime, conventional Drude absorption from ambient carriers directly captures incident radiation. This contrasts with conventional semiconductor photodetectors, that do not absorb light below the semiconductor bandgap. A tantalizing possibility is to use g/X Schottky junctions within the mid-IR−THz bandwidth where presently available technologies offer lackluster performance.30,31 In summary, graphene Schottky junctions host tunable interfaces across which energy transport can be engineered, exemplified by thermionic-dominated transport regime wherein energy and charge currents are strongly coupled. Fingerprints of the thermionic-dominated regime include high responsivities on the order of 9 ∼ 1−10 A/W, and a nonmonotonic dependence of 9 on electron temperature (or pump power) in g/X photodetectors. The large degree of in situ tunability allows optimization of the g/X interface for different applications and irradiation conditions; vertical hot carrier transport opens up new vistas to efficiently harvest photon energies over a wide spectral range, utilizing the entire exposed graphene area as a photoactive region.

■

REFERENCES

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (J.F.R.N.). *Email: [email protected] (J.C.W.S.). 6040

DOI: 10.1021/acs.nanolett.6b01965 Nano Lett. 2016, 16, 6036−6041

Letter

Nano Letters (23) Nair, R. R.; Blake, P.; Grigorenko, A. N.; Novoselov, K. S.; Booth, T. J.; Stauber, T.; Peres, N. M. R.; Geim, A. K. Science 2008, 320, 1308−1308. (24) Rajkanan, K.; Singh, R.; Shewchun, J. Solid-State Electron. 1979, 22, 793−795. (25) Yan, H.; Low, T.; Zhu, W.; Wu, Y.; Freitag, M.; Li, X.; Guinea, F.; Avouris, P.; Xia, F. Nat. Photonics 2013, 7, 394−399. (26) Rodriguez-Nieva, J. F.; Dresselhaus, M. S.; Levitov, L. S. Nano Lett. 2015, 15, 1451−1456. (27) Goldman, J. R.; Prybyla, J. A. Phys. Rev. Lett. 1994, 72, 1364. (28) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature 2005, 438, 197−200. (29) Price, A. S.; Hornett, S. M.; Shytov, A. V.; Hendry, E.; Horsell, D. W. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 161411. (30) Chan, W. L.; Deibel, J.; Mittleman, D. M. Rep. Prog. Phys. 2007, 70, 1325−1379. (31) Rogalski, A. Infrared Detectors; CRC Press: Boca Raton, FL, 2007.

6041

DOI: 10.1021/acs.nanolett.6b01965 Nano Lett. 2016, 16, 6036−6041