Effect of Quantum Confinement on Thermoelectric Properties of 2D and 1D Semiconductor Thin Films

Transcription

1 Effect of Quantum Confnement on Thermoelectrc Propertes of D and D Semconductor Thn Flms A. Bulusu and D. G. Walker Interdscplnary Program n Materal Scence Vanderblt Unversty Nashvlle, TN 3735 Wth devce dmensons shrnkng to nanoscales, quantum effects such as confnement and tunnelng become sgnfcant n electron transport. In addton, scatterng effects such as electron-phonon scatterng, electron-mpurty scatterng also affect carrer transport through small-scale devces. Currently most of the commonly used transport models are partcle based wth quantum correctons ncorporated to nclude quantum effects whle models based on the soluton to the Schrödnger wave equaton can be computatonally ntensve. In ths regard, the NEGF formalsm has been found to be very effcent n couplng quantum and scatterng effects. In ths paper the NEGF model s used to assess the effect of temperature on devce characterstcs of D and D thn flm superlattces whose applcatons nclude thermoelectrc coolng of electronc and optoelectronc systems. The effect of quantum confnement on the electrcal transport and ts mpact on the thermoelectrc fgure of mert s studed n the two cases. Results show a competng effect between the electrcal and thermal conductvty on the overall fgure of mert n the two dmensonally confned thn flms. INTRODUCTION The two fundamental aspects that dfferentate nanoscale devce modelng from bulk modelng are scatterng effects and quantum effects. Scatterng n nanostructures can sgnfcantly affect carrer as well as thermal transport n devces. Inelastc electron-phonon scatterng can cause electrons to lose or gan energy wth the lattce whle other scatterng effects such as carrer-carrer, carrer-defect, and carrer-boundary scatterng can lead to energy redstrbuton n devces where transport s near ballstc. Quantum effects manfest themselves n the form of electron tunnelng as well as carrer confnement. These effects usually domnate when the devce length scales are of the order of the de Brogle wavelength assocated wth the devce. Quantum confnement can lead to reduced densty of states avalable for the carrer whle electron tunnelng can have a detrmental mpact n the form of leakage currents n very small szed transstors []. However, quantum effects can be very useful as a hgh performance alternatve to very-largescale ntegraton (VLSI) [] n the form of resonant tunnelng dodes. In general carrer transport models can be classfed nto two types. The frst type s the classcal partcle models that are based on fndng the soluton to the Boltzmann transport equaton (BT. The second type nvolves a drect soluton to the Schrödnger wave equaton. Quantum effects n the BTE based models are ncluded by ncorporatng a quantum correcton to the transport model. Common methods of ncorporatng quantum effects are the densty gradent formalsm and the effectve potental method [3]. The densty gradent formalsm s derved from the equaton of moton for the one partcle Wgner functon where quantum correctons are ntroduced by expressng the mean potental energy as a power seres n h. The transport equaton for the Wgner dstrbuton functon can now be wrtten n the form of a modfed Boltzmann Transport Equaton. The effectve potental method nvolves a localzed wavepacket representaton of the carrers assocated wth an effectve potental obtaned by summng the nhomogeneous potental ntroduced n the Hamltonan, over all the carrers. The generaton of the effectve potental determnes the onset of quantzaton n the system. The quantum correcton methods have been found to gve an excellent match wth the Posson-Schrodnger solver for the case of carrer confnement and tunnelng. More recently, attempts have been made to combne quantum correctons wth the Monte Carlo technque [4] whch s a numercal soluton to the BTE. The results have been found to match well wth the Schrodnger soluton n the case of carrer confnement whle a reasonably close match was observed for the case of tunnelng. However, extenson of ths model to and 3 dmensons remans, computatonally tedous and dffcult. Quantum transport models that nvolve the soluton to the Schrödnger wave equaton can be used to study current flow over small scales where the transport can be ether ballstc or can nvolve some type of scatterng. The man models used to model ballstc transport are Quantum Transmttng Boundary Model (QTBM) [5] and the Quantum Devce Analyss by Mode Evaluaton (QDAM [6]. QTBM nvolves formulatng the boundary condtons for a gven problem by calculatng the transmsson and reflecton coeffcents for a known boundary potental. These boundary condtons are then used n a numercal soluton based on the fnte element method to obtan the wave functon over the entre problem doman. Whle ths method s sutable for solvng the Schrödnger equaton for varous boundary potentals, ncluson of dsspaton due to scatterng becomes very dffcult to solve usng ths method. QDAME nvolves dscretely samplng a devce s densty of states usng standng wave boundary condtons. The standng Correspondng author; greg.walker@vanderblt.edu

2 waves are decomposed nto travelng waves and njected from the contacts from whch ther occupances are assgned. Shrnkng devce dmensons present an ncreasng need for a smple quantum transport model that can effectvely couple quantum and scatterng effects. The non-equlbrum Green s functon formalsm provdes a framework for natural couplng of quantum and scatterng effects. Open boundary condtons allow the source and dran contacts to be coupled to the devce through smple self-energy terms that helps elmnate workng wth huge matrces that are of the sze of the source and dran reservor systems and nstead work wth matrces that are the sze of the devce Hamltonan. In addton, the NEGF formalsm allows for the rgorous ncorporaton of both elastc and nelastc scatterng effects usng the concept of Buttker probes where scatterng s treated as another contact, allowng t to be coupled to the devce usng self-energy terms. We present a bref synopss of the formalsm n the next secton whle a more thorough and detaled development can be found n [7] and [8]. A sgnfcant amount of research has gone nto enhancng the thermoelectrc fgure of mert ZT S σ/κ where S s the Seebeck coeffcent, σ s the electrcal conductvty and κ s the thermal conductvty. In ths regard, quantum well heterostructures have ganed attenton as thermoelectrc materals due to ther reduced thermal conductvtes [9,, ]. Phonon nterference effects caused by phonon-nterface scatterng gve rse to bandgaps at the nterface of the thn flms n a superlattce, affectng phonon transport through these structures. In addton, electron-phonon scatterng n the superlattce affects electron transport through the devce, whch eventually affects the thermoelectrc fgure of mert. Chen [, 3] studed the phonon transport n SGe superlattces usng the BTE for phonons. Phonon nterface scatterng was ncluded through a combnaton of dffuse and specular scatterng. Whle the emphass of present research s on heat transfer through phonon transport n superlattces, the effect of quantum confnement on the electrcal conductvty s also mportant n the performance of thermolectrcs. Usng the structure n [, 3] as a model system we use the NEGF formalsm to study the effect of temperature and quantum confnement on the electrcal propertes of D thn flm thermoelectrc layer. The effect of carrer confnement on the electrcal conductvty and the thermoelectrc fgure of mert are studed by reducng one of the macroscopc dmensons of the D flm leadng to confnement of electrons n two dmensons. The D and D flms are modeled to be ballstc n nature. Future work s focused towards studyng the devce characterstcs under the nfluence of temperature and electron-phonon scatterng n S and SGe nanowre superlattces whle ncorporatng the effects of phonon nterface scatterng. THE NEGF FORMALISM In general an solated devce and ts energy levels are descrbed usng a Hamltonan matrx where H s the Hamltonan; U s the Hartree potental and α the energy egen state of the electron. ( H U ) ψ ( r) ε ψ ( r). () α α α The electron densty matrx n real space s gven by [ ρ ( E, r, r' ] de f ( E ε µ ) δ ([ EI H ]). () k where f s the Ferm functon. For a two dmensonal flm the Ferm functon s gven by consderng the electrons to have free flow along the x and y drectons and experence confnement along the z drecton. E mck BT f N D ln[ exp where N. (3) kbt πh The Ferm functon for a D flm s calculated n a smlar manner by consderng the electron to have free flow only n the x drecton. f D s gven by N mck BT f D f ( x) where N. (4) 4 π h h k x The value of x s gven by x. m k T δ(ei-h) s the local densty of states. Usng the standard expanson for the delta functon we get ( ) δ EI H E I H E I H. (5) π From the above equaton, the delta functon can also be wrtten as δ ( EI H ) G( G ( ; π G ( E I H. (6) G( s the retarded Green s functon whle G ( s ts conjugate complex transpose. The Green s functon can be nterpreted as the mpulse response of the Schrödnger equaton that wll gve us the n th component of the wave functon f the system s gven an mpulse exctaton at the m th component. The densty of states n real space s gven by the spectral functon, whch can be nterpreted as the avalable densty of states that are flled up accordng to the Ferm functon so as to obtan the electron densty. The dagonal elements of the spectral functon represent the local electron densty of states, A( r, r', πδ ( EI H ) G( G (. (7) Thus the electron densty matrx for an solated devce can also be wrtten n the form [ ] de ρ f ( E ε µ )[ A( E )] k. (8) π k c B

3 Now consder the case of a smple nanotransstor consstng of source and dran contacts. Let µ and µ be the chemcal potentals of the source and the dran. The dstrbuton of electrons n the source and dran s sad to follow the Ferm dstrbuton where f and f are the Ferm functons of the source and dran. The dfference n the Ferm levels of the source and dran causes electrons to flow from the source to the dran through the channel. The average number of electrons N at steady state wll le between f and f.e. f < N < f. When no scatterng s ncorporated the channel can be consdered to be ballstc n nature and s expected to have zero resstance to current flow. However, expermental measurements [4] have shown that the maxmum measured conductance of a oneenergy level channel approaches a lmtng value G q / ħ 5.6(KΩ) -. The reason for ths lmt to conductance arses from the fact that current n the contacts s carred by nfnte transverse modes whle the number of avalable modes n the channel s lmted. Ths means that the densty of states n the contacts s spread over a large energy range whle the channel densty of states les specfcally between µ and µ. Upon couplng the contact and the channel, some of the densty of states from the contact spread nto the channel whle the channel loses some of ts densty of states to the contact. As a result, the couplng causes the densty of states n the channel to spread out over a wder range of energy levels resultng n a reducton n the number of states lyng between µ and µ. Consequently, the overall effect of the couplng s to broaden the range of energy levels of the channel leadng to a reducton n the number of states n the range µ to µ that are avalable for electron flow through the channel. In the NEGF formalsm the couplng of the devce to the source and dran contacts s descrbed usng self-energy matrces Σ and Σ. The self-energy term can be vewed as a modfcaton to the Hamltonan to ncorporate the boundary condtons. Accordngly, equaton and 4 can be rewrtten as ( H U Σ Σ ) ψ ( r) ε ψ ( r) (9) α α α G( E I H Σ Σ. () The rate at whch an electron s extracted from the reservor s dependent on, an nfntesmal term added to the energy of the devce egenstate that wll ensure that the system behaves rreversbly as long as t exceeds the spacng between the energy levels n the reservors. Approprate values of wll ensure that an electron always travels from the reservor nto the channel and not vce-versa. The broadenng of the energy levels ntroduced by connectng the devce to the source and dran contacts s ncorporated through the Gamma functons Γ and Γ gven by Γ ( Σ Σ ) Γ ( Σ Σ ) () The self-energy terms affect the Hamltonan n two ways. The real part of the self-energy term shfts the devce egenstates or energy level whle the magnary part of Σ causes the densty of states to broaden whle gvng the egenstates a fnte lfetme. The effect of the self-energy terms s shown n equaton Γ( [ H Σ ( ] H H Σ( where [ Σ( Σ ( ] Σ H. () The lfetme of the egenstate can be related to the self-energy through the equaton γ Im Σ. (3) τ h h Followng the above argument, the Green s functon for a gven egenstate can also be wrtten as G(. (4) ( E ε ') γ / where ε ε σ. ε corresponds to the energy of the egenstate whle σ corresponds to the energy ncrement due to couplng of the self-energy functon. The correspondng densty of states due to the broadenng, take the form ( D( π ( G( G ( ) A. (5) The electron densty for the open system s gven by de [ ] [ n ρ G ( ]. (6) π G n ( represents the electron densty per unt energy and s gven by G n n GΣ G [ Σ ] Γ f Γ f. (7) n where [ ] [ ] The source, dran and scatterng self-energy terms are used to obtan the fnal Green s functon from whch the net channel current s calculated as a dfference n the nflow and outflow currents usng equatons 8, 9 and. The entre calculaton s carred out self-consstently wth Posson s equaton to account for the change n channel potental wth the change n electron densty. de Inflow f ( E µ ) Trace[ ΓA] (8) h π I RESULTS de n Trace[ ΓG ] Outflow (9) h π q Trace h n [ Γ A] f Trace[ Γ G ] () Thermoelectrc effects n slcon D and D thn flms Two knds of thn flms were studed n ths paper as shown n fgure. The frst was a D Slcon thn flm wth nfnte 3

4 dmenson n x and y whle the z drecton had a thckness of 3nm. The second flm s a D thn flm wth nfnte dmenson n the y drecton whle the x and z drectons have thckness of 6nm. The thermopower for slcon thn flm was studed for varous temperature ranges of the source and dran contacts whle varyng the dopng n slcon. The source temperature was mantaned constant at 3K whle the dran contact was mantaned a hgher temperature relatve to the source. The appled bas ranged from to.5v. t 3nm Dran Current Densty (Ampere/m ) 4.E9 3.E9 dtk n 5x 8 cm -3.E9 dt5k.e9.e E9 -.E9 Slcon D flm -3.E9 Dran Voltage (V) Fgure. Current-voltage characterstcs of a ballstc slcon thn flm wth dopng level of 5x 8 cm -3. Channel temperature varaton ranges from K to 5K. Slcon D flm 6 nm Z X Y Fgure 3 shows the averaged predcted and measured Seebeck coeffcent values for D and D slcon flms for varous dopng levels n all temperature ranges consdered. The predcted values of S dffer from the measured values approxmately by a factor of ndcatng good agreement [5]. In addton, the calculated Seebeck coeffcents for the D and D slcon flms are very close to each other provng the fact that the Seebeck coeffcent s a materal property and s ndependent of the effects of quantum confnement. Fgure. Schematc dagram of D and D Slcon thn flms wth D and D confnement. Fgure shows the current-voltage characterstcs for a ballstc slcon D thn flm doped to electron concentraton of 5x 8 cm -3. The source-dran temperature dfference ranges from K to a maxmum of 5K. Intally for low bas condtons, the number of hgh energy electrons generated n the dran contact s hgher than the number of electrons arrvng at the dran through the channel leadng to negatve current values. As the bas n ncreased gradually, more electrons from the source are drawn towards the dran due to the appled bas leadng to hgher nflow at the dran leadng to postve current values. Seeebeck Coeffcent (V/K) E7.E8.E9.E Slcon D flm Slcon D flm Measured Fgure 3. Predcted vs. measured [5] values of Seebeck coeffcent for slcon at 3K as a functon of dopng levels. The effect of carrer confnement two dmensons has contrastng nfluences on the thermoelectrc fgure of mert ZT due to thermal and electrcal conductvty. Confnng phonons n two dmensons leads to ncreased phonon boundary scatterng resultng n a decrease n thermal conductvty than the D value. However, the decrease n avalable electron densty of states wth D confnement leads to a drastc reducton n the electrcal conductvty as seen n fgure 4. 4

5 .E5 Slcon D flm Slcon D flm Thermoelectrc effects n SGe D and D thn flms Electrcal Conductvty (mho/m).e4.e3.e.e.e.e7.e8.e9.e The thermoelectrc propertes of SGe D and D superlattce flms were studed n a smlar manner by mantanng the dran at hgher temperature than the source. For the D flm each of the semconductor thn flms was 5nm thck leadng to a combned thckness of 3nm. The source temperature was mantaned at 3K whle the dran temperature values were vared from 3K to 35K. Slcon and Germanum were both degenerately doped to values rangng from.5x 7 cm -3 to 8.5x 9 cm -3. The predcted value of the Seebeck coeffcent n the case of the D flm for the 8.5x 9 cm -3 dopng level was found to be 6µV/K, whch s a good match to the expermentally measured value of 3µV/K n [6]. Fgure 4. Comparson of electrcal conductvty values for a Slcon D flm and Slcon D flm. The change n ZT wth dopng for D and D slcon thn flm s shown n fgure 5. The competng effect of the decreasng thermal conductvty wth the decrease n electrcal conductvty for the two cases s very obvous especally at hgher dopng levels. At hgh dopng levels whle the number of electrons avalable for transport s hgh, the reduced densty of states n the D flm leads to tmes as much resstance to the flow of electrons compared to the D flm where more densty of states are avalable to transport the avalable electrons. Snce the Seebeck coeffcent does not change sgnfcantly for both cases, the ncrease n ZT by reduced thermal conductvty of the D flm can be offset by the reduced electrcal conductvty. It should be noted that n addton to the decrease n the flm thckness from nfnty to 6nm n the y-dmenson, the dfference n the thckness of the two flms from 3nm to 6nm n the z drecton may also lead to addtonal confnement effects n the z drecton causng the overall reducton n electrcal conductvty. Current work s focused towards modelng and comparng flms wth dentcal thckness n the z- drecton. ZT Values for D and D Slcon Thn Flms.E.E-.E-.E-3.E-4.E-5 E7 E8 E9 E D Slcon flm D Slcon flm Fgure 5. Change n ZT values wth D and D carrer confnement. k S (D) 3W/m-K [6], k s (D) 3 W/m- K [7]. The D superlattce flms were each modeled as havng a square cross-secton of 3nm x 3nm leadng to an over all superlattce cross-sectonal area of 6nm x 6nm. The superlattce flm was consdered to be nfntely long n the x-drecton. Whle the source was mantaned at a temperature of 3K, the temperature of the dran had to be ncreased to a mnmum value of 43K to notce any sgnfcant thermoelectrc effects. The reason for ths behavor was attrbuted to the fact that a 6nm thck devce behaves as a ballstc devce havng almost no resstance to electron flow. In order for the electrons to gan suffcent thermal energy so as to travel n a drecton opposte to the appled feld they have to be excted to very hgh energy levels. These energy levels were acheved only by ncreasng the dran temperature to at least 43K. Fgure 6 shows the Seebeck coeffcents of the D and D SGe flms. The dfference n the calculated values of S for both cases vared approxmately by 5µV/K provng that S s a materal property. Seebeck Coeffcent of SGe Thn Flms (muv/k) E7.E8.E9.E Fgure 6. Seebeck coeffcent for SGe D and D superlattce flms for varous dopng levels. D Flm D Flm As n the case of slcon flms the reduced densty of states due to confnement n D caused the electrcal conductvty for the SGe flms to decrease sgnfcantly when the y-dmenson of the D superlattce thn flm was reduced from nfnty to 6nm changng t nto a D superlattce flm as seen n fgure 7. 5

6 Electrcal Conductvty of SGe Thn Flms (mho/m).e6.e5.e4.e3.e.e.e D Flm D Flm.E7.E8.E9.E Fgure 7. Comparson of electrcal conductvty of SGe D and D superlattce thn flms for varous dopng levels. The effect of the change n electrcal conductvty and thermal conductvty wth ncreasng confnement s evdent from the change n ZT values. ZT of D SGe Thn Flm D Flm D Flm.E7.E8.E9.E 6.E-4 5.E-4 4.E-4 3.E-4.E-4.E-4.E Fgure 8. Thermoelectrc fgure of mert for the D and D SGe superlattce flms, k SGe (D).9W/m-K [6], k SGe (D).45 W/m-K. The value of the thermal conductvty n the case of S/SGe superlattce nanowres was found to reduce by approxmately a factor of due to alloy scatterng and ncreased boundary scatterng [8, 9]. Usng ths rato as a reference, the value of thermal conductvty for SGe D thn flm was taken to be half of that of the D flm. Whle the Seebeck coeffcent remaned the same for both cases, the decrease n thermal conductvty due to ncreased confnement effects s more than compensated by the decrease n electrcal conductvty due to a sharp drop n avalable densty of states. As a result, the overall ZT value dropped by to 4 orders of magntude n the case of the D SGe superlattce flm compared to the D flm. ZT of D SGe Thn Flm CONCLUSIONS The non-equlbrum Green s functon formalsm was used to study the effects of quantum confnement on the thermoelectrc behavor of electrons n D and D slcon and SGe superlattce flms. The ncrease n confnement from D to D flm manfests tself n the form of reduced electron densty of states leadng to a sharp declne n the electrcal conductvty of the flms. Whle the ncreased confnement s beleved to decrease the thermal conductvty due to boundary scatterng, the smultaneous drop n electrcal conductvty causes the overall thermoelectrc fgure of mert to not change sgnfcantly n the case of slcon flms. In the case of the SGe superlattce flms, the decrease n electrcal conductvty s steeper causng the fgure of mert to reduce from to 4 orders of magntude for D confnement. ACKNOWLEDGMENTS We are grateful to Prof. Supryo Datta, Dept. of Electrcal and Computer Engneerng, Purdue Unversty, for provdng us wth the manuscrpt of hs book on whch much of ths work s based. Ths research was funded through the Vanderblt Dscover Grant REFERENCES [] Rahman, A., Ghosh, A and Lundstrom, M., 3, Assessment of Ge n-mosfets by Quantum Smulaton, Techncal Dgest of Internatonal Electronc Devces Meetng, pp [] Mazumder, P., Kulkarn, S., Bhattacharya, M., Sun, J. P. and Haddad, G.I., 998, Dgtal crcut applcatons of resonant tunnelng devces, Proceedngs of the IEEE, 86, 4, pp [3] Asenov, A., Watlng, J. R., Brown, A. R. and Ferry, D. K.,, The Use of Quantum Potentals for Confnement and Tunnelng n Semconductor Devces, Journal of Computatonal Electroncs,, pp [4] Tang, T. W. and Wu, B., 4, Quantum Correcton for the Monte Carlo Smulaton va the Effectve Conducton-band Edge Equaton, Semconductor Scence and Technology, 9, pp [5] Lent, C., S. and Krkner, D., J., 99, The Quantum Transmttng Boundary Method, Journal of Appled Physcs, 67,, pp [6] Laux, S., E., Kumar, A. and Fschett, M., V.,, Ballstc FET Modelng Usng QDAME: Quantum Devce Analyss by Modal Evaluaton, IEEE Transactons on Nanotechnology,, 4, pp [7] Datta, S.,, Nanoscale Devce Smulaton: The Green's Functon Formalsm, Superlattces and Mcrostructures, 8, pp [8] Datta, S., 5, Quantum Transport: Atom to Transstor, Cambrdge Unversty Press, New York. [9] Hcks, L.D. and Dresselhaus, M. S., 993, Effect of Quantum Well Structures on the Thermoelectrc Fgure of Mert, Physcal Revew B, 47,, pp [] Whtlow, L. W. and Hrano, T., 995, Superlattce Applcatons to Thermoelectrcty, Journal of Appled Physcs, 78, 9, pp

7 [] Hcks, L. D., Harman, T. C. and Dresselhaus, M. S., 993, Use of Quantum-well Superlattces to Obtan a Hgh Fgure of Mert from Non-conventonal Thermoelectrc Materals, Appled Physcs Letters, 63, 3, pp [] Chen, G., 998, Thermal Conductvty and Ballstc- Phonon Transport n the Cross-Plane Drecton of Superlattce, Physcal Revew B, 57, 3, pp [3] Chen, G., 999, Phonon Wave Heat Conducton n Thn Flms and Superlattces, Journal of Heat Transfer,, pp [4] Szafer, A. and Stone, A. D., 989, Theory of Quantum Conducton through a Constrcton, Physcal Revew Letters, 6, pp [5] Geballe, T. H. and Hull, G. W., 955, Seebeck Effect n Slcon, Physcal Revew, 98, 4, pp [6] Yang, B., Lu, J. L., Wang, K. L. and Chen, G.,, Smultaneous measurements of Seebeck coeffcent and thermal conductvty across superlattce, Appled Physcs Letters, 8,, pp [7] L, D., Wu, Y., Km, P., Sh, L., Yang, P. and Majumdar, A., 3, Thermal Conductvty of Indvdual Slcon Nanowres, Appled Physcs Letters, 83, 4, pp [8] L, D., Wu, Y., Fan, R., Yang, P. and Majumdar, A., 3, Thermal Conductvty of S/SGe Superlattce Nanowres, Appled Physcs Letters, 83, 5, pp [9] Dames, C. and Chen, G., Theoretcal phonon thermal conductvty of SGe superlattce nanowres, 4, Journal of Appled Physcs, 95,, pp