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Overcoming the black body limit in plasmonic and graphene near-field thermophotovoltaic systems

Ognjen Ilic, Marinko Jablan, John D. Joannopoulos, Ivan Celanovic, and Marin Soljačić

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Abstract

Near-field thermophotovoltaic (TPV) systems with carefully tailored emitter-PV properties show large promise for a new temperature range (600 – 1200K) solid state energy conversion, where conventional thermoelectric (TE) devices cannot operate due to high temperatures and far-field TPV schemes suffer from low efficiency and power density. We present a detailed theoretical study of several different implementations of thermal emitters using plasmonic materials and graphene. We find that optimal improvements over the black body limit are achieved for low bandgap semiconductors and properly matched plasmonic frequencies. For a pure plasmonic emitter, theoretically predicted generated power density of 14Wcm2 and efficiency of 36% can be achieved at 600K (hot-side), for 0.17eV bandgap (InSb). Developing insightful approximations, we argue that large plasmonic losses can, contrary to intuition, be helpful in enhancing the overall near-field transfer. We discuss and quantify the properties of an optimal near-field photovoltaic (PV) diode. In addition, we study plasmons in graphene and show that doping can be used to tune the plasmonic dispersion relation to match the PV cell bangap. In case of graphene, theoretically predicted generated power density of 6(120)Wcm2 and efficiency of 35(40)% can be achieved at 600(1200)K, for 0.17eV bandgap. With the ability to operate in intermediate temperature range, as well as high efficiency and power density, near-field TPV systems have the potential to complement conventional TE and TPV solid state heat-to-electricity conversion devices.

© 2012 Optical Society of America

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Figures (8)
Fig. 1
Fig. 1 Schematic illustration of a near-field TPV system. Plasmonic emitter, characterized by the plasma frequency ωp and damping γ, operates at temperature T1. Next to it, a distance D away, is the photovoltaic cell at temperature T2, characterized by the bandgap energy ωg, photon absorption coefficient α0, and the refractive index n. The parallel and the perpendicular wave vectors are q and kz, respectively.

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Fig. 2
Fig. 2 Contour plot of Π(ω,q) from Eq. (1) for (a) γ = 5 × 10−4eV, (b) γ = 5 × 10−3eV. Dashed line shows the dispersion relation from Eq. (17), and the solid cyan line shows the dispersion relation of a surface plasmon in air. In (c) and (d), solid (dashed) line corresponds to the spectral transfer function Π(ω,q) calculated using Eq. (1) (Eq. (5)), for four different values of q. On a separate scale, β is plotted as a function of ω0 (magenta), where the x-axis for ω0 and ω is shared. Plasma frequency, PV gap frequency and PV absorption coefficient are ωp = 0.6eV, ωg = 0.36eV and α0 = 1.3 × 104cm−1, respectively. Separation is D = 10nm (1/D ≈ 20eV/h̄c).

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Fig. 3
Fig. 3 Contour plot of logarithm of transfer ratio versus ωp, ωg, for the plasmonic emitter-PV system in Fig. 1. Plasmon damping is γ = 5 × 10−3eV, and temperature is T1 = 600K, with other parameters being the same as in Fig. 2.

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Fig. 4
Fig. 4 Plot of χ′ (blue) and χ″ (red) for a semiconductor as a function of frequency, with ωg = 0.36eV and n = 3.51. Three lines correspond to the absorption coefficient α calculated using square-root dependence, Eq. (3), with α0 = 1.3 × 104cm−1, experimental values for InAs from Ref. [19], and using step-like dependence, Eq. (8), with same α0. We see that both PV cell approximations, χ′ ≈ 0.85, and χ″/χ′ ≪ 1, are satisfied in this frequency range.

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Fig. 5
Fig. 5 Radiation transfer ratio, as a function of emitter temperature T1 for a pure plasmonic emitter-PV cell near-field TPV system. The ratio is plotted for several values of ωp and γ, with other parameters being the same as in Fig. 2, namely ωg = 0.36eV, D = 10nm.

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Fig. 6
Fig. 6 Silver-PV near-field TPV system. (a) Radiation transfer ratio as a function of PV gap frequency ωg and separation D, for T1 = 1235K. (b) Contour plot of integrand in Eq. (1) and Eq. (2), as a function of ω, q in regions below and above the vacuum light line, respectively. This plot corresponds to Htot for ωg = 0.36eV, D = 10nm point from plot (a). The y-axis is shared between two plots. Two dashed lines correspond to light lines for n = 1 (vacuum) and n = 3.51 (PV cell).

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Fig. 7
Fig. 7 Graphene-PV near-field TPV system. (a) Contour plot of integrand H(ω,q) in Eq. (1) as a function of ω, q for parameters T1 = 600K, D = 10nm, μ = 0.2eV and τ = 10−13s. PV cell parameters are ωg = 0.17eV, α0 = 0.7 × 104cm−1. Solid (magenta) line is the vacuum surface plasmon dispersion relation, Eq. (14), for the graphene sheet. (b) H(ω) evaluated for different parameters with T1, D same as in (a). For comparison, black ωp-line demonstrates H(ω) for a pure plasmonic emitter analyzed in section 2, with ωp = 0.3eV. (c) Contour plot of the heat transfer ratio vs. the two black bodies in the far field, as a function of T1 and D. In (c) and (d), τ = 10−13s and doping is μ = 0.25eV. (d) Electric power generated PPV as a function of T1 where the voltage across the PV diode terminals is Vo = 0.08V. The x-axis is shared between plots (a), (b) and plots (c), (d), respectively.

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Fig. 8
Fig. 8 Optimization of the flux ratio Hevan/Hbb for graphene-PV near-field TPV system, where ωg = 0.17eV, D = 10nm.

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Tables (2)

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Table 1 Ideal plasmonic emitter-PV cell near-field TPV system: radiated power exchange, Prad, generated electrical power PPV, efficiency, and the flux ratio (relative to the transferred power between two black bodies in the far field), tabulated for a set of parameters, up to two significant digits. We assume D = 10nm, γ = 0.01eV.

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Table 2 Comparison between a silicon-PV and a graphene-PV near field TPV system: radiated power exchange, Prad, generated electrical power PPV, efficiency, and the flux ratio (relative to the transferred power between two black bodies in the far field), tabulated for a set of parameters, up to two significant digits. We assume D = 10nm, τ = 10−13s.

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Equations (33)

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H evan s , p = 1 π 2 ∫ 0 ∞ d ω [ Θ ( ω , T 1 ) − Θ ( ω , T 2 ) ] ∫ ω / c ∞ d q q Im ( r 01 s , p ) Im ( r 02 s , p ) | 1 − r 01 s , p r 02 s , p e 2 i k z 0 D | 2 e − 2 | k z 0 | D
H prop s , p = 1 π 2 ∫ 0 ∞ d ω [ Θ ( ω , T 1 ) − Θ ( ω , T 2 ) ] ∫ 0 ω / c d q q ( 1 − | r 01 s , p | 2 ) ( 1 − | r 02 s , p | 2 ) 4 | 1 − r 01 s , p r 02 s , p e 2 i k z 0 D | 2
ε 2 ( ω ) = ( n + i α 2 k 0 ) 2 where α ( ω ) = { 0 , ω < ω g α 0 ω − ω g ω g , ω > ω g
H evan p = 1 π 2 ∫ 0 ∞ d ω [ Θ ( ω , T 1 ) − Θ ( ω , T 2 ) ] ∫ ω / c ∞ d q q Π ( ω , q )
Π ( ω , q ) = γ β ( ω , q ) 4 [ ω − ω 0 ( q ) ] 2 + [ γ + β ( ω , q ) ] 2 where β ( ω , q ) ≡ ω p 2 − 2 ω 0 ( q ) 2 2 ω 0 ( q ) χ ″ ( ω ) χ ′ ( ω )
Π ( ω , q ) = γ β ( ω , q ) 4 [ ω − ω 0 ( q ) ] 2 + [ γ + β ( ω , q ) ] 2 ≈ γ β ( ω , q ) 4 [ ω − ω 0 ( q ) ] 2 + β ( ω , q ) 2
A ( q ) = ∫ ω g ∞ d ω Θ ( ω , T 1 ) Π ( ω , q )
ε 2 ( ω ) = ( n + i α 2 k 0 ) 2 where α ( ω ) = { 0 , ω < ω g α 0 , ω > ω g
P rad = 1 π 2 ∫ 0 ∞ d q q [ ∫ 0 ∞ d ω Π ( ω , q ) h ¯ ω e h ¯ ω k T 1 − 1 − ∫ ω g ∞ d ω Π ( ω , q ) h ¯ ω e h ¯ ω − e V o k T 2 − 1 ]
P P V = 1 π 2 ∫ 0 ∞ d q q [ ∫ ω g ∞ d ω Π ( ω , q ) e V o e h ¯ ω k T 1 − 1 − ∫ ω g ∞ d ω Π ( ω , q ) e V o e h ¯ ω − e V o k T 2 − 1 ]
η T P V = P P V P rad
σ intra ( ω , T ) = i ω + i / τ e 2 2 k b T π h ¯ 2 ln   [ 2 cosh μ 2 k b T ]
σ inter ( ω , T ) = e 2 4 h ¯ [ G   ( h ¯ ω 2 ) + i 4 h ¯ ω π ∫ 0 ∞ G ( ε ) − G ( h ¯ ω / 2 ) ( h ¯ ω ) 2 − 4 ε 2 d ε ]
q = ε 0 2 i ω σ ( ω , T )
Π ( ω , q ) = Im   ( ε 1 − 1 ε 1 + 1 )   Im   ( ε 2 − 1 ε 2 + 1 ) | 1 − ( ε 1 − 1 ε 1 + 1 )   ( ε 2 − 1 ε 2 + 1 ) e − 2 q D | 2 e − 2 q D
ε 1 ( ω ) − 1 ε 1 ( ω ) + 1 = { ω p 2 ω p 2 − 2 ω 2 [ 1 + 2 i ω γ ω p 2 − 2 ω 2 ] if  γ ≪ ω p 2 − 2 ω 2 2 ω i ω p 2 2 ω γ [ 1 − i ( ω p 2 − 2 ω 2 ) 2 ω γ ] if  γ ≫ ω p 2 − 2 ω 2 2 ω
ω 0 ( q ) = ω p 2 1 − χ ′ e − 2 q D
ω 0 ( q ) = ω p 2 1 − e − q D
Π ( ω , q ) = ( γ / 2 ) 2 4 ( ω − ω 0 ( q ) ) 2 + γ 2
∫ d ω Π ( ω , q ) = γ 4   [ π + 2 tan − 1   ( 2 ω 0 γ ) ]
E ( r , ω ) = ∫ d 3 r ′ G E ( r , r ′ , ω ) j ( r ′ )
H ( r , ω ) = ∫ d 3 r ′ G H ( r , r ′ , ω ) j ( r ′ )
G E ( r , r ′ , ω ) = − ω μ 0 8 π 2 ∫ d 2 q 1 k z 1 ( s ^ T s s ^ + p ^ 2 + T p p ^ 1 + ) e i q ( r q − r q ′ ) + i k z 2 ( z − D ) − i k z 1 z ′
G H ( r , r ′ , ω ) = − n 2 ω 8 π 2 c ∫ d 2 q 1 k z 1 ( − p ^ 2 + T s s ^ + s ^ T p p ^ 1 + ) e i q ( r q − r q ′ ) + i k z 2 ( z − D ) − i k z 1 z ′
T = t 10 t 02 e i k z 0 D 1 − r 02 r 01 e 2 i k z 0 D
〈 j α * ( r , ω ) j β ( r ′ , ω ′ ) 〉 = Θ ( ω , T ) 2 π [ σ ( ω ) + σ * ( ω ) ] δ ( r − r ′ ) δ ( ω − ω ′ ) δ α β
S z = 2 Re   ( E x H y * − E y H x * )
E x H y * − E y H x * = n 2 * ω 2 μ 0 c ( 8 π 2 ) 2 ∫ d 3 r ′ ∫ d 2 q d 2 q ′ 1 | k z 1 | 2 [ g x α E g y α H * − g y α E g x α H * ] × Θ ( ω , T 1 ) 2 π [ σ + σ * ] δ ( z ′ ) e i ( q − q ′ ) ( r q − r q ′ ) e i k 21 ( z − D ) e − i k z 1 z ′ e − i k z 2 ′ * ( z − D )
g E = s ^ T s s ^ + p ^ 2 + T p p ^ 1 +
g H = − p ^ 2 + T s s ^ + s ^ T p p ^ 1 +
g x α E g y α H * − g y α E g x α H * = | T s | 2 k z 2 * n 2 * k 0 + | T p | 2 k z 2 n 2 k 0 | k z 1 | 2 k 0 2
r p G = 1 − ε G ε G , t p G = 1 ε G , where ε G = 1 + σ k z 0 2 ε 0 ω
2 ω Im ( r p G ) Im ( k z 0 ) = | t p G | 2 Re ( σ ) ε 0
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