Black Body for Thermophotovoltaic

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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 47, NO. 1, JANUARY 2000 241

Very Large Radiative Transfer over Small Distancesfrom a Black Body for Thermophotovoltaic

ApplicationsJanet L. Pan, Henry K. H. Choy, and Clifton G. Fonstad, Jr.

Abstract The maximum amount of radiated heat intensitywhich can be transferred from a black body of refractive index

to an object of refractive index located a shortdistance away is shown to be 2

s m a l l e r

times the free space Planckdistribution, where s m a l l e r is the smaller of and ,and where and are assumed greater than unity. Theimplication is that the radiative power spectral density within athermophotovoltaic cell could be designed to be much greaterthan the free space Planck distribution. The maximum radiativeintensity transferred occurs when the index of the black bodymatches that of the object at wavelengths where the Planckdistribution is sizeable. A simple expression is found for thetransferred radiative intensity as a function of the refractiveindices of, and the distance separating, the black body and theobject. This expression is interpreted in terms of the specific blackbody modes which are evanescent in the space between the blackbody and the object and which make the largest contribution tothe transmission of radiation. The black body, the object, and thegap region are all modeled as lossless dielectrics.

Index Terms Layered media, multilayers, photovoltaic cells,photovoltaic power systems, stratified media.

I. INTRODUCTION

W HEN AN object is placed a short distance away froma black body, the amount of heat transferred from theblack body has been observed to be much greater [ 1], [2] thanthe free space Planck distribution. An increase of the heattransferred of many orders of magnitude beyond the free spacePlanck distribution has been calculated numerically [ 3], [4].The heat transferred has been calculated [ 5] to increase as theinverse of the square of the distance separating the black bodyand the object, though other calculations [ 3], [4] show a morecomplicated dependence on this distance. This effect has beentermed [ 4], [6], [7] the spacing effect or microscale radiativetransfer . Recently, this effect has been proposed [ 6][9] asa means of extracting more power from a black body in thethermophotovoltaic generation of electricity.

Manuscript received November 23, 1998; revised June 26, 1999. The reviewof this paper was arranged by Editor P. N. Panayotatos.

J. L. Pan was with the Department of Electrical Engineering and ComputerScience, Massachusetts Institute of Technology, Cambridge, MA 02139 USA.She is nowwith the Department of Electrical Engineering, Yale University, NewHaven, CT 06520-8284 USA (e-mail: [email protected]).

H. K. H.Choy andC. G. Fonstad,Jr.are with theDepartment of ElectricalEn-gineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA.

Publisher Item Identifier S 0018-9383(00)00170-2.

Much of this transferred radiative intensity is believed tocome from black body modes which are evanescent in thespace between the black body and the object. The amount of radiated heat which is transmitted from the black body to theobject is usually calculated [ 3][5], [7] numerically from theFresnel [ 10] equations. However, an absolute maximum onthe amount of heat which can be transferred from the black

body also has not been predicted. The dependence of the heattransferred on the refractive indices of the black body andobject also has not been studied in the literature. This paperaddresses these issues.

The problem to be considered is shown in Fig. 1. Radiationfrom within the black body (labeled or medium 1) isincident on the surface of the black body at an angle of mea-sured from the normal direction . (One in-plane direction islabeled in the figure.) A gap (labeled GAP or medium 2)of width separates the black body from the object (labeled ormedium 3). The refractiveindicesof boththeblack body and the object are assumed to be greater than unity, and therefractive index in the gap region is taken to be unity. All three

regions, the black body, the gap, and the object, are assumed tobe lossless dielectrics. It is assumed that both theblack body andthe object are infinitely large in the transverse ( ) direction. Itis also assumed that both the black body and the object are heldat fixed temperatures, with the black body temperature muchhigher than that of the object.

This paper develops a physical model which can be used tounderstand the increase of the heat transferred from the black body beyond the amount expected from the free space Planck distribution. This paper ascertains the maximum radiationwhich can be transmitted from the black body to the object. Thelarge amount of radiative intensity which can be transferredfrom a black body will be found to come from the largenumber of optical modes existing within a black body havinga refractive index greater than unity. The number of opticalmodes existing within a black body having a refractive indexgreater than unity is found to be much larger than the numberof propagating optical modes which are radiated into freespace. A simple expression is found for the maximum radiativeintensity transferred as a function of the refractive indices of,and the distance separating, the black body and the object. Thisexpression is interpreted in terms of the specific black bodyevanescent modes which make the largest contribution to thetransmission of radiation.

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PAN et al. : VERY LARGE RADIATIVE TRANSFER 243

III. POWER TRANSMISSION THROUGH EVANESCENT MODESLEAVING THE BLACK BODY

Of all the optical modes within the black body, the numberwhich are evanescent in medium 2 of Fig. 1 correspond to thosemodes which are incident on the black body surface at an anglegreater than that for total internal reflection: .Of these evanescent modes, the number which tunnel a distance

into the object is

(3.1)

where is the coefficient for the transmission of power inmedium 1 of Fig. 1 to medium 3, where is given in Ap-pendix A, and where is the angle that the surface normalmakes with the wave vector of the incident mode within theblack body. Appendix A shows that a good approximation forthe transmission coefficient is

valid forvalid for

(3.2)

where is

(3.3)

where is

(3.4)

where is the imaginary part of the wave vector associated withthe evanescent wave in the gap region

(3.5)

where the gap is assumed to be free space with the associatedwave vector

(3.6)

where is the wavelength of the radiation in free space, andis the GoosHnchen phase shift given in (A.6) of Appendix

A.As discussed in Appendix B, the maximum power transmis-

sion occurs when the refractive index of the black body is thesame as (matched to) that of the object. Under these condi-tions, the number of modes which tunnel from the black bodyto the object is found from (3.1), (B.5), (C.4), and (C8) for asmall distance to be

valid for (3.7)

and for a large distance to be

valid for (3.8)

where is the incomplete gamma function [ 17], andis the largest possible value of

(3.9)

[The assumption of matched conditions was invoked in (3.7)and (3.8) inwritingtheupperlimitof the integralin (3.1) as .This means that under matched conditions, all the modes inmedium 1 can become propagating modes in medium 3, car-rying power in the direction in Fig. 1.]

Equations (3.7) and (3.8) are expected to be numerically ac-curate whenever and , respectively. For ex-ample, (3.8) is expected to be numerically accurate whenever

. Numerically, is within4% of whenever .

Using the fact that the total transmitted power

(3.10)

comes from both modes which propagate and which tunnelthrough the gap region, the total transmitted power spectraldensity per unit solid angle is, in units of W/cm -ster-Hz

valid for (3.11)

valid for (3.12)

where we have used (2.6), (3.7), and (3.8). Since we are as-sumingthat theblack body is fixed at a temperature much higherthan that of the object, this transmitted power density is also thenet transmitted power density from the black body to the object,as the object is assumed to radiate a much smaller power densityback to the black body.

Equations (3.11) and (3.12) are discussed more fully in Ap-pendixes B and C, and Fig. 3 therein.

IV. DISCUSSION

Fig. 2 shows the power spectral density[in units of the free space Planck distribution, (2.1)] transmittedfrom a black body to an object a distance (in units of , thewavelength in free space) away. The curves in the figure wereobtained by plotting (3.11) for , and (3.12) for

. The calculations assume that the black body and the objecthave the same refractive index. The three curves shown are forrefractive indices of 2, 3, and 4 (labeled, respectively, from thelowest to the uppermost curve). These values of refractive indexare characteristic of semiconductors in the infrared. [At infraredwavelengths, GaAs and Ge have refractive indices [ 14] of 3.3and 4.0, respectively, and PbSnTe [ 15] has a refractive indexgreater than six (depending on the Sn mole fraction).]
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244 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 47, NO. 1, JANUARY 2000

Fig. 2. The power spectral density I ( ! ) = 2 [in units of the freespace Planck distribution, (2.1)] transmitted from a black body to an object adistance L (in units of , the wavelength in free space) away. The curvesin the figure were obtained by plotting (3.11) for L < 1 , and (3.12) for

L > 1 . The calculations assume that the black body and the object havethe same refractive index. The three curves shown are for refractive indices of 2, 3, and 4 (labeled, respectively, from the lowest to the uppermost curve). Atinfrared wavelengths, GaAs and Ge have refractive indices [ 14] of 3.3 and 4.0,respectively, and PbSnTe [ 15] has a refractive index greater than 6 (dependingon the Sn mole fraction).

Fig. 3.I = 2

(solid line) andI = 2

(dashedline) [(B.6) and (C.9), respectively, in units of the free spacePlanckdistribution,(2.1)] which can be transmitted from a black body to an object a distanceL . (in units of ) away under matched conditions for a black bodyindex of 3.3. The calculations were done by putting a lower bound and anupper bound on the power transmission and tunneling coefficients. The actualtransmitted power spectral density, I ( ! ) , must lie below the solidline ( I ) and above the dashed line ( I ) in thefigure. I ( ! ) asymptotically approaches I (solidline) as L ! 0 , and asymptotically approaches I (dashed line)as L ! 1 .

Fig. 2 shows t hat f or , the p ower transmittedfrom the black body is a strong function of the refractive index,and is proportional to the square of the refractive index. Thefigure also shows that for , the power trans-

mitted from the black body has only a weak dependence on therefractive index, and is inversely proportional to the square of the refractive index. For , the power trans-mitted from the black body in excess of the free space Planck distribution, , is inversely proportional to the fourth powerof thedistance . All these observations canbe understood froma simple physical picture which we describe in this section.

For small distances ( ) separating the black body from the object, (3.11) becomes

(4.1)

Equation (4.1) can be understood by observing that for

(4.2)

where is defined in (3.5). The two terms in (4.1) come from

the two terms in (4.2). These terms say that under matchedconditions, all the black body modes in (2.2) are trans-mitted into the object with the transmission coefficient givenin (4.2). This gives a total number of transmitted modes of ap-proximately , whichis approximately the expression given in (4.1). The strongdependence of on the refractive index shown inFig. 2 for thus comes from the number ( )of black body modes, and the quadratic dependence of the

tunneling coefficient, (4.2), on the refractive index.The quadratic dependence on the spacing of for is also a consequence of (4.2).

For large distances ( ) separating the black body from the object, (3.11) becomes

(4.3)

which is roughly inversely proportional to both the square of therefractive index and the fourth power of the distance . In (4.3),

corresponds to the average transmission coefficient for thosemodes within the black body which are incident on the black body surface within the escape cone .

The second term inside the braces in (4.3) can be understood

as evaluated for those evanescent modesleaving the black body which have a significant tunneling co-efficient, (3.4), at large . The latter correspond to those modeswith

(4.4)

in (3.4). These evanescent modes have incident wave vector di-rections which fill the solid angle of those greater thanby the amount

(4.5)
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The functions and have a further physicalsignificance in that they are, respectively, the small and largelimits of the full expression (A.9), for . In other words, itis easy to show that

valid forvalid for .

(A.13)

APPENDIX BMAXIMUM POWER WHICH CAN BE TRANSFERRED THROUGH

EVANESCENT MODES

It is easy to show that the maximum power transmission frommedium 1 in Fig. 1 to medium 3 occurs when the refractive in-dices of media 1 and 3 are the same. Using (A.11) and assumingthat the refractive indices of media 1 and 3 are the same, themaximum number of modes which can tunnel from medium 1into medium 3 is

(B.1)

[The assumption of the same refractive index in both medium 1and medium 3 was invoked in (B.1) in writing the upper limitof the integral as . This means that under matched condi-tions,allthe modes in medium1 canbecomepropagatingmodesin medium 3, carrying power in the direction in Fig. 1.] Thisintegral can be done by invoking the definitions for

(B.2)

and the definitions for and , (3.6) and (A.7), and the defini-tion for

(B.3)

Thus, we have the identities

(B.4)

which can be used in (B.1) to obtain

(B.5)

Using (3.10), the total maximum possible transmittedpower spectral density per unit solid angle is, in units of W/cm -ster-Hz

(B.6)Equation (B.6) was derived by placing an upper limit of

on the full expression, (A.9), for . Thus, theactual transmitted power spectral density issmaller than that given in (B.6)

(B.7)

Moreover, (A.13) says that is a good approximationfor for . Thus, (B.6) is expected to be a goodapproximation for the actual transmitted power spectral density

for

for

(B.8)The solid line in Fig. 3 is a plot of the maximum possibletransmitted power spectral density, in(B.6), in units of the free space Planck distribution, (2.1), as afunction of the distance (in units of ) separating the black body from the object for a matched refractive index of 3.3.From the above discussion, it is clear that the actual transmittedpower spectral density, , would lie below thesolid line in Fig. 3, and would approach the solid line as .The curves in Fig. 2 were obtained by plotting (B.6) for

.

APPENDIX CMINIMUM POWER WHICH CAN BE TRANSFERRED THROUGH

EVANESCENT MODES UNDER MATCHED CONDITIONS

Using (A.12) and assuming that the refractive indices of media 1 and 3 are the same, the minimum number of modeswhich can tunnel from medium 1 into medium 3 is

(C.1)Since the GoosHnchen phase shift, (A.6), for the TE mode isdifferent from that for the TM mode, the number of tunnelingmodes having the two different polarizations are considered sep-arately

(C.2)


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248 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 47, NO. 1, JANUARY 2000

Using the definition of the GoosHnchen phase shift, (A.6),we have for the TE mode

(C.3)

which can be used in (C.2) to obtain

(C.4)

where is the incomplete gamma function [ 17]

(C.5)

For the TM mode, we have, for

(C.6)

which can be used in (C.2) to obtain

(C.7)

It is difficult to obtain a simple, closed form result for the righthand side of (C.7). Physically, we expect that at these incident

angles, which are much greater than that for total internal reflec-tion and much greater than the Brewster angle, the transmissionof TM modes should not differ too much from that of TE modes.In fact, it is easy to show that

(C.8)

Equation (C.8) shows that ,and we will assume for therest of this paper.

Equation (3.10) can be used to calculate the minimum powerspectral density which can be transmitted per unit solid angle,in units of W/cm -ster-Hz

(C.9)

Equation (C.9) was derived by placing a lower limit of on the full expression, (A.9), for . Thus, the

actual transmitted power spectral density isgreater than that given in (C.9)

(C.10)

Moreover, (A.13) says that is a good approximationfor for . Thus, (C.9) is expected to be a goodapproximation for the actual transmitted power spectral density

for

for(C.11)

The dashed line in Fig. 3 is a plot of the minimum possibletransmitted power spectral density, , in(C.9) in units of the free space Planck distribution, (2.1), as afunction of the distance (in units of ) separating the black body from the object for a matched refractive index of 3.3.From the above discussion, it is clear that the actual transmittedpower spectral density, , would lie above the
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dashed line in Fig. 3, and would asymptotically approach thedashed line as . The curves in Fig. 2 were obtained byplotting (C.9) for .

ACKNOWLEDGMENT

The authors would like to thank Prof. S. Prasad for helpful

discussions.

REFERENCES[1] G. A. Domoto, R. F. Boehm, and C. L. Tien, Experimental investiga-

tion of radiative transfer between metallic surfaces at cryogenic temper-atures, ASME J. Heat Transf. , ser. C, vol. 92, no. 3, pp. 412417, Aug.1970.

[2] C. M. Hargreaves, Anomalous radiative transfer betweenclosely-spaced bodies, Phys. Lett. A , vol. 30, no. 9, pp. 491492, Dec.1969.

[3] R. F. Boehm andC. L. Tien,Smallspacing analysis of radiative transferbetween parallel metallic surfaces, ASME J. Heat Transf. , ser. C, vol.92, no. 3, pp. 405411, Aug 1970.

[4] E. G. Cravalho, C. L. Tien, and R. P. Caren, Effect of small spacingon radiative transfer between two dielectrics, J. Heat Transfer, Trans. ASME , ser. Series C, vol. 89, pp. 351358, Nov. 1967.

[5] J. J. Loomis and H. J. Maris, Theory of heat transfer by evanescentelectromagnetic waves, Phys. Rev. B , vol. 50, pp. 1851718524, 1994.

[6] R. S. DiMatteo, Enhanced semiconductor carrier generation via mi-croscale radiativetransfer;MPCAnelectrical power finance instrumentpolicy; interrelated innovations in emerging energy technologies, S.M.thesis, Dept. Elect. Eng. Comput. Sci. and Technology and Policy Pro-gram, Mass. Inst. Technol.,, Cambridge, June 1996.

[7] M. D. Whale, A fluctuational electrodynamic analysis of microscaleradiative transfer and the design of microscale thermophotovoltaic de-vices, Ph.D. dissertation, Dept.Mech. Eng.,Mass. Inst.Technol., Cam-bridge, June 1997.

[8] , , in Proc. 25thPhotovoltaic Specialists Conf. . Washington, DC, May1317, 1996.

[9] , , in 2nd NREL Conf. Thermophotovoltaic Generation of Electricity ,vol. 358. Colorado Springs, CO, 1995.

[10] M. Born andE. Wolf, Principles of Optics , NewYork: Macmillan, 1964.[11] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media

(in translation from Russian), J. B. Sykes and J. S. Bell, Eds, New York:Pergamon, 1960.[12] E. Yablonovitch andG. Cody, Intensityenhancement in texturedoptical

sheets for solar cells, IEEE Trans. Electron Devices , vol. ED-29, pp.300305, Feb. 1982.

[13] E. Yablonovitch, Statistical ray optics, J. Opt. Soc. Amer. , vol. 72, p.899, 1982.

[14] M. Shur, Physics of Semiconductor Devices . Englewood Cliffs, NJ:Prentice-Hall, 1990.

[15] G. P. Agrawal andN. K.Dutta, Long-Wavelength Semiconductor Lasers ,New York: Van Nostrand Reinhold, 1986.

[16] H. A. Haus, Waves and Fields in Optoelectronics . Englewood Cliffs,NJ: Prentice-Hall, 1984.

[17] I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series, and Prod-ucts (in translationfrom Russian by Scripta Technica), 4thed, A. Jeffrey,Ed, New York: Academic, 1980.

JanetL. Pan receivedthe S.B.,S.M., and Ph.D. degrees fromthe MassachusettsInstitute of Technology. Cambridge.

Currently, she is an Assistant Professor in the Department of Electrical Engi-neering, Yale University, New Haven, CT.

Dr. Pan is a member of Sigma Xi, Eta Kappa Nu, and Tau Beta Pi.

Henry K. H. Choy was born in Hong Kong, China, in 1974. He received theB.Eng. degree in electrical engineering from McMaster University, Hamilton,Ont., Canada, in 1996, and the M.S. degree in electrical engineering from theMassachusetts Institute of Technology, Cambridge, in 1998, where he is cur-rently studying compound semiconductor optoelectronic devices with Prof. C.G. Fonstad, Jr.

Clifton G. Fonstad, Jr. received the B.S. degree in1965from the Universityof Wisconsin, Madison,andthe M.S. and Ph.D. degrees in 1966 and 1970, re-spectively, from the Massachusetts Institute of Tech-nology (MIT), Cambridge.

Since 1970, he has been a faculty member in theDepartment of Electrical Engineering and ComputerScience at MIT, where he is currently a FullProfessor. He teaches undergraduate and graduatecourses in semiconductor devices and technology;he also conducts an active graduate student research

program concerned with the application of MBE-grown IIIV heterostructuresin a variety of advanced electronic and optoelectronic devices, including p-i-nphotodiodes, quantum-well laser diodes, and microscale thermophotovoltaic

cells. Most recently, he has been concerned with developing novel technologiesfor realizing very large scale monolithic optoelectronic integrated circuits, in-cluding epitaxy-on-electronics, aligned pillar bonding, and silicon-on-galliumarsenide. With his students and collaborators, he has published more than 140articles in refereed technical journal. He is also the author of Microelectronic Devices and Circuits, an undergraduate text on semiconductor device physics,models, and applications.

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