It is shown that equations for electrical current in solid-state thermionic
and thermoelectric devices converge for devices with a width equal to the mean
free path of electrons, yielding a common expression for intensive electronic
efficiency in the two types of devices. This result is used to demonstrate that
the materials parameters for thermionic and thermoelectric devices are equal,
rather than differing by a multiplicative factor as previously thought

Humphrey, T. E.

Lewis, R. A.

O\u27Dwyer, Mark F.

Zhang, C.

English

arXiv

It is shown that the equations for electrical current in solid-state thermionic and thermoelectric devices converge for devices with a width equal to the mean free path of electrons, yielding a common expression for the intensive electronic efficiency in the two types of devices. This result is used to demonstrate that the material parameters for thermionic and thermoelectric refrigerators are equal, rather than differing by a multiplicative factor as previously thought

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University of Wollongong Research Online Faculty of Engineering - Papers (Archive) Faculty of Engineering and Information Sciences July 2005 Solid-state thermionics and thermoelectrics in the ballistic transport regime T. E. Humphrey University of Wollongong Mark F. O'Dwyer University of Wollongong, markod@uow.edu.au C. Zhang University of Wollongong, czhang@uow.edu.au R. A. Lewis University of Wollongong, roger@uow.edu.au Follow this and additional works at: https://ro.uow.edu.au/engpapers  Part of the Engineering Commons https://ro.uow.edu.au/engpapers/162 Recommended Citation Humphrey, T. E.; O'Dwyer, Mark F.; Zhang, C.; and Lewis, R. A.: Solid-state thermionics and thermoelectrics in the ballistic transport regime 2005. https://ro.uow.edu.au/engpapers/162 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.edu.au Solid-state thermionics and thermoelectrics in the ballistic transportregimeT. E. Humphrey,a M. F. O’Dwyer, C. Zhang, and R. A. LewisSchool of Engineering Physics, University of Wollongong, Wollongong, New South Wales (NSW) 2522,AustraliaReceived 3 February 2005; accepted 27 May 2005; published online 22 July 2005It is shown that the equations for electrical current in solid-state thermionic and thermoelectricdevices converge for devices with a width equal to the mean free path of electrons, yielding acommon expression for the intensive electronic efficiency in the two types of devices. This result isused to demonstrate that the material parameters for thermionic and thermoelectric refrigerators areequal, rather than differing by a multiplicative factor as previously thought. © 2005 AmericanInstitute of Physics. DOI: 10.1063/1.1977191Solid-state thermionic devices may be distinguishedfrom thermoelectric devices according to whether electrontransport is ballistic or diffusive.1 There is, however, little todistinguish the underlying thermodynamics of the two typesof devices, with both achieving reversibility under the sameconditions2,3 and both being governed by the same “materialparameter.”4–6 In a previous work, a proportionality factor ofF0 /F1/2, where Fn is a Fermi integral, was found betweenthe two material parameters.6Here we show that the equations for diffusive and bal-listic transport converge for devices with a width equal to themean free path of electrons. This results in a common equa-tion for the intensive electronic efficiency for both thermi-onic and thermoelectric devices, and a proportionality factorbetween their material parameters equal to unity. We explainthe discrepancy between this and a previous work by point-ing out an inconsistency in the energy dependency of therelaxation time in the two equations for conductivity used inRef. 6.Current density in both thermionic and thermoelectricdevices can be expressed asJ =  jkdk , 1where jkk is the net “energy-resolved” current of elec-trons flowing in the direction opposite to the temperaturegradient with momentum in the range k around k. In ther-mionic devices with a width less than the mean free path,most electrons travel ballistically from one reservoir to an-other. In this case the energy-resolved current density isgiven by with dependence upon k implicitjbkk = qDrxrf0k , 2where Drk is the density of states DOS in the reservoirs,k is the probability that electrons are transmitted betweenthe reservoirs, xrk is the velocity in the direction of trans-port given by the dispersion relation Ek in the reservoirs,and f0= f0k,C ,TC− f0k,H ,TH is the difference inthe Fermi occupation of states in the cold/hot reservoirs,wheref0k ,,T = 1 + exp	Ek − kT −1 3and where C/H is the electrochemical potential and TC/H thetemperature of electrons at the cold/hot ends of the device.Here we follow a previous work,6 and assume the trans-mission probability depends upon the total momentum of theelectrons rather than the momentum in the direction of trans-port only, allowing a direct comparison with thermoelectricsin which the energy of mobile electrons is also restricted inall three dimensions this assumption is made implicitly inRef. 6, between Eqs. 1.1 and 1.3. The theoretical differ-ences between thermionic devices in which the transmissionprobability is a function of k and kx are explored in detail inother papers.7–9In thermoelectric devices the energy-resolved diffusiveelectron current density may be obtained from the Boltz-mann transport equation under the relaxation-time approxi-mation, and can be written as10jdkk = qDlxl 2df0dxk , 4where Dlk is the local DOS, xl is the velocity of electronsin the direction of transport, determined from the local dis-persion relation Ek, and k=0Ekr is the relaxationtime in the direction of transport.In solid-state power generators and refrigerators with awidth close to the electronic mean free path, it is expectedthat Eqs. 2 and 4 should yield the same results. To showthis, we take the energy dependence of the relaxation time tobe r=−1/2, which corresponds to scattering that is domi-nated by acoustic phonons, and results in a mean free path inthe direction of transport, 	xkk, which is indepen-dent of energy.6 We also note that df0x /dxf0x− f0x+x /x when x is small, so that, for a piece of thermo-electric material L=	 in length, df0x /dxf0 /	 and Eq.4 becomesjdkk = qDlxl f0k . 5It can be seen that Eqs. 2 and 5 have the same form,where the product kDrkxrk in Eq. 2 plays the sameaElectronic mail: tammy.humphrey@unsw.edu.auJOURNAL OF APPLIED PHYSICS 98, 026108 20050021-8979/2005/982/026108/3/$22.50 © 2005 American Institute of Physics98, 026108-1Downloaded 22 Aug 2006 to 130.130.37.6. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsprole in determining the energy spectrum of the electronswhich carry current as the product Drkxrk does in Eq.5. This simple result provides an additional underpinningfor the observation of Ulrich et al.6 that thermionic and ther-moelectric devices refrigerate or generate power via thesame underlying physical mechanism. To show that there isno sharp transition in the behavior of a solid-state powergenerator or refrigerator as its width changes from L	 toL	, one can use the fact that the probability that an elec-tron can travel a distance L without suffering a collision is11P = exp− L/	 6to obtain an equation for the energy-resolved current densityuseful in solid-state power generators and refrigerators oflength L	 asjkk = qf0	Lxl Dl1 − P + xrDrPk , 7which can easily be generalized to the case where 	→	k.Equality between Eqs. 2 and 5 results in a commonexpression for the electronic efficiency12 in thermionic de-vices and intensive efficiency13,14 across a small section ofthermoelectric material ofPGb = VJ/Q̇H 8for a solid-state power generator andRb = Q̇C/VJ 9for a solid-state refrigerator, where the heat flux density inthe cold/hot reservoir of a thermionic device, or at the cold/hot ends of the small section of a thermoelectric material, isgiven byQ̇C/H =   Ek − C/Hjkqdk , 10where the −/+ refers to the cold/hot case, and it is assumedthat the temperature gradient, electric field =V /L, andcurrent have no components in the y and z dimensions. It isimportant to point out that Eq. 10 considers only heat car-ried by electrons. If phonon-mediated heat leaks are alsoconsidered, then two extra terms should be added to Eq. 10,the first being ±Tl /L, where l is the thermal conductivityof the lattice, while the second accounts for the “Joule” heatreleased by electrons which diffuse between the hot and coldreservoirs, part of which is then carried by phonons to eachend of the device. Shakouri et al.15 have shown that in thelimit L	 then exactly half of this heat VJ is deposited inthe hot and cold reservoirs while if L	, this heat is entirelydeposited in the reservoir receiving the net current of elec-trons.Finally, it can be shown that the above results yield thesame expression for the material parameter in thermionic andthermoelectric refrigerators. Assuming a dispersion relationof E=2k2 /2m* and three-dimensional reservoirs so that D=1/ 23. Eq. 2 may be used to obtain the same thermi-onic materials parameter as found in Ref. 6, expressed interms of the mean free path 	 asTI =	l4m*kh3kTC2. 11The thermoelectric material parameter is given by6 =Tlk2q2F0−1, 12where  is the electrical conductivity and Fn is a Fermi in-tegral given byFn =1n + 10 nexp −  + 1d , 13where the reduced Fermi energy is = −E /kTC and E isthe height of the barrier in thermionic devices so that thetransmission probability in Eq. 2 becomes =0 for EE,=1 for EE or the conduction-band edge in a piece ofthermoelectric material. Using the same dispersion relationand density of states as for thermionic devices, we can writean equation for conductivity in thermoelectric materials as qJdd/dx=4m*q2	h3 Edf0EdEdE 14=	4m*kh3q2TF0 , 15where we have used the facts that df0 /dx= d /dxdf0 /d for dT /dx=0, that df0 /d=−df0 /dE, and haveintegrated by parts to obtain the final line. It can be seen bysubstitution that this yields =TI if T=TC.In Ref. 6 the thermionic and thermoelectric materialsparameters were expressed in terms of the mobility of elec-trons. In order to do this for the thermoelectric material pa-rameter, two different expressions for conductivity wereequated: = − q2 x2Dldf0dE dE , 16where =0E−1/2, and = nq = q Dlf0dE 17=−q2m* DlEdf0dE . 18Given that x2E and q /m*, it can be seen that the twoequations do not give the same result for conductivity as therelaxation time is assumed to be proportional to E−1/2 in thefirst and independent of energy in the second, and this diffi-culty results in a factor of F0 /F1/2 identified in Ref. 6.In summary, it has been shown that solid-state thermi-onic and thermoelectric devices with a width close to themean free path of electrons share a common equation forelectrical current density and electronic efficiency, and thattheir material parameters for refrigeration are equal. This re-sult means that in the specific case where the productkDrkxrk in a thermionic device is equal to the productDlkxl k in a thermoelectric device for all values of k026108-2 Humphrey et al. J. Appl. Phys. 98, 026108 2005Downloaded 22 Aug 2006 to 130.130.37.6. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspwhich make a contribution to the current above the barrierfor thermionic device or above/below the conduction band inn-p-type thermoelectric materials, then the performance ofthese two types of devices is expected to be indistinguishablein the regime where the device length is of the order of amean free path.ACKNOWLEDGMENTSOne of the authors T.E.H. is supported by a grant fromthe NSW Sustainable Energy Research and DevelopmentFund SERDF and the Australian Research Council. Threeof the authors M.F.O’D., C.Z., and R.A.L. are supported bythe Australian Research Council. One of the authors T.E.H.acknowledges helpful discussions with A. Shakouri.1G. S. Nolas, J. Sharp, and H. J. Goldsmid, Thermoelectrics Basic Prin-ciples and New Materials Developments Springer, Berlin, 2001.2T. E. Humphrey, R. Newbury, R. P. Taylor, and H. Linke, Phys. Rev. Lett.89, 116801 2002.3T. E. Humphrey and H. Linke, Phys. Rev. Lett. 94, 096601 2005.4C. B. Vining and G. D. Mahan, J. Appl. Phys. 86, 6852 1999.5A. Shakouri and C. LaBounty, Proceedings of the 18th International Con-ference on Thermoelectrics, ICT99, Baltimore, MD, 1999 IEEE, Piscat-away, NJ, 1999, p. 35.6M. D. Ulrich, P. A. Barnes, and C. B. Vining, J. Appl. Phys. 90, 16252001.7D. Vashaee and A. Shakouri, Phys. Rev. Lett. 92, 106103 2004.8M. F. O’Dwyer, T. E. Humphrey, R. A. Lewis, and C. Zhang, submitted toPhys. Rev. B; cond-mat/0506388.9T. E. Humphrey, M. O’Dwyer, and H. Linke, J. Phys. D 38, 2051 2005.10C. Kittel, Introduction to Solid State Physics, 8th ed. Wiley, New York,2005.11N. W. Ashcroft and N. D. Mermin, Solid State Physics Saunders CollegePublishing, Orlando, Florida, 1976, p. 249.12G. N. Hatsopoulos and E. P. Gyftopoulos, Thermionic Energy Conversion:Processes and Devices MIT, Cambridge, Massachusetts, 1973, Vol. 1.13C. Vining, Symposium on Thermoelectric Materials-New Directions andApproaches, edited by T. M. Tritt, MRS Symposia Proceedings No. 278Materials Research Society, Pittsburgh, 1997, p. 3.14G. J. Snyder and T. S. Ursell, Phys. Rev. Lett. 91, 148301 2003.15A. Shakouri, E. Y. Lee, D. L. Smith, V. Narayanamurti, and J. E. Bowers,Microscale Thermophys. Eng. 2, 37 1998.026108-3 Humphrey et al. J. Appl. Phys. 98, 026108 2005Downloaded 22 Aug 2006 to 130.130.37.6. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

A new comparison between solid-state thermionics and thermoelectrics

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