Large enhancement of the thermoelectric figure of merit in a ridged quantum well

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1 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well 1 Large enhanceent of the theroelectric figure of erit in a ridged quantu well Avto Tavkhelidze Tbilisi tate University, Chavchavadze ave. 13, Tbilisi 179, Georgia E-ail: avtotav@gail.co Abstract. Recently, new quantu features have been observed and studied in the area of ridged quantu wells (RQWs). Periodic ridges on the surface of the quantu well layer ipose additional boundary conditions on electron wave function and reduce the quantu state density. As a result, the cheical potential of RQW increases and becoes ridge height dependent. Here, we propose a syste coposed of RQW and an additional layer on the top of the ridges foring periodic series of p + n + junctions (or etal n + junctions). In such systes, charge depletion region develops inside the ridges and effective ridge height reduces, becoing a rather strong function of teperature T. Consequently, T-dependence of cheical potential agnifies and eebeck coefficient increases. We investigate in the syste of seiconductor RQW having abrupt p + n + junctions or etal n + junctions on the top of the ridges. Analysis ade on the basis of oltzann transport equations shows draatic increase in for both the cases. At the sae tie, other transport coefficients reain unaffected by the junctions. Calculations show one order of agnitude increase in theroelectric figure of erit ZT relative to the bulk aterial. PAC: 73.5.Lw, 74.5.Fy 1. Introduction Quantu wells are considered the ost reliable low-diensional systes for theroelectrics [1]. However, iproveents in theroelectric properties over bulk aterials are insufficient for ost applications. In this work, we present ridged quantu wells (RQWs) having advanced theroelectric properties. RQW layer has periodic ridges on the surface. Its operation is based on the effect of quantu state depression (QD). Periodic ridges ipose additional boundary conditions on the electronwave function. uppleentary boundary conditions forbid soe quantu states for free electron, and the quantu state density in the energy ρ(e ) reduces. According to Pauli s Exclusion Principle, electrons rejected fro the forbidden quantu states have to occupy the states with higher E. Thus, cheical potential μ increases. In seiconductors, QD reduces ρ(e ) in all energy bands including the conduction band (C). Electrons rejected fro the filled bands occupy the quantu states in the epty bands, and the electron concentration in the C increases []. This corresponds to donor doping (we will refer to it as QD doping). The QD transfers electrons to higher energy levels. If initially the seiconductor is intrinsic, then the QD doping will odify it to n-type. It is coparable with a conventional donor doping fro the point of increase in μ. However, there are no donor atos. QD doping does not introduce scattering centres and consequently allow high electron obility. There are distinctions and siilarities between the QD forbidden quantu state and a hole. tate is forbidden by the boundary conditions and cannot be occupied. However, it is not forbidden in an irreversible way. If the boundary conditions change (e.g., owing to charge depletion), then it can recobine with the electron (like the hole recobines with the electron). As the QD forbidden state is confined to the boundary conditions (acroscopic geoetry), it is not localized in the lattice and cannot ove like a hole.

2 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well Density of states in RQW (figure 1) reduces G ties ρ( E ) = ρ ( E) / G, where ρ ( E ) is the density of states in a conventional quantu well layer of thickness L (a = ) and G is the geoetry factor. In the first approxiation, for the case L, w >> a and within the range 5 < G < 1, the following siple expression can be used G L / a. (1) where a is the ridge height and L is the RQW layer thickness (figure 1). Density of QD Figure 1. Cross-section of a ridged quantu well. forbidden quantu states is ρ FOR ( E ) = ρ ( E) ρ ( E) / G = ρ ( E)(1 G ). () To deterine the nuber of rejected electrons n, () should be integrated over electron confineent energy range FOR ( E) = (1 G ) de ρ ( E) = (1 G ) n = de ρ n. (3) Here, n = de ρ ( E) is the nuber of quantu states (per unit volue) within electron confineent energy range (which depends on RQW and substrate band structures). RQW retains quantu properties at G ties ore widths with respect to the conventional quantu well. Previously, QD was studied experientally [3] and theoretically [4] in ridged etal fils. Theroelectric aterials are characterized in ters of diensionless figure of erit ZT [5]. Here, T is the teperature and Z is given by Z = σ /( κ e + κ l ), where is the eebeck coefficient, σ is electrical conductivity, κ e is electron gas theral conductivity, and κ l is lattice theral conductivity. The difficulty in increasing ZT is that aterials having high usually have lowσ. When σ is increased, it leads to an increase inκ e, following Wiedeann Franz law, and ZT does not iprove uch. Another approach is to eliinate the lattice theral conductivity by introducing vacuu nanogap between the hot and cold electrodes [6 8] and using electron tunnelling. Cooling in such designs was observed in [9] and theoretically studied in [1, 11]. However, vacuu nanogap devices appear extreely difficult to fabricate. In this work, we present a solution that allows large enhanceent of without changes in σ, κ e, andκ l. It is based on RQW having series of p + n + or etal n + junctions on the top of the ridges RQW (figure ). Depletion region width d(t ) depends rather strongly on teperature. The ridge effective height aeff ( T ) = a d( T ) and consequently the geoetry factor of RQW

3 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well 3 Figure. RQW with series of periodic junctions grown at the top of the ridges. becoes teperature-dependent, G = G( T ). All paraeters of RQW including μ becoe stronger functions of T than it will be in an RQW without the junctions. eebeck coefficient and theroelectric figure of erit increase. The objective of this work is to calculate ZT of RQW with series of p + n + and etal n + junctions and copare it with Z T of reference RQW (RRQW), in which G G( T ), and bulk aterial. Analysis was ade using the oltzann transport equations. First, we calculate μ for the syste of RQW and junctions and express it as μ = μ + μ J. (4) Here, μ is the cheical potential gradient in an RRQW and μ J is introduced by the junctions. ext, we insert μ in oltzann transport equations and calculate for the syste of RQW and junctions, expressing it as = + J, where is eebeck coefficient of RRWQ and J is introduced by the junctions. Finally, the reduced figure of erit ZT / Z T is calculated and μ dependences are presented for such traditional theroelectric aterials as i and Ge. Analysis was ade within the parabolic bands approxiation and the abrupt junction s approxiation. ince only heavy QD doping was considered, we neglected the hole contribution in this transport.. Charge and heat transport in the RWQ with junctions Cross-section of the syste of RQW and periodic junction is shown in figure. We assue that there is a teperature gradient T in the Y-diension. Consequently, depletion depth depends on the Y- coordinate, and geoetry factor gradient G appears in the Y-direction. Presence of G and T odifies the electron distribution function and causes electron otion fro the hot side to the cold side. This otion is copensated by theroelectric voltage. Let us write oltzann transport equations [1] for the syste of RQW and periodic series of junctions J 11 1 ( ε + e) T and J Q 1 = L ( + μ / e) L T = L μ / L ε. (5) Q Here, J is the electric current density, J is the heat current density, L are coefficients, ε is the i j electric field, and e is the electron charge. Within the parabolic bands approxiation, L are the functions of integrals of type [13]. Ω ( α ) ( f E) = + α ( E ) de ρ( E) τ( E) v y ( E μ) (6) where f is the electron distribution function, τ(e) is the electron lifetie, Ω ( α ) E i j v y is the electron velocity in the Y-direction, and α =, 1,. Let us find how QD affects ( ). The QD does not change dispersion relation and consequently v y. It reduces density of states (3D density) G ties, i.e. ρ(e ) = ρ ( E )/G and increases the transport lifetie G ties [], i.e., τ( E ) = G τ ( E ) (the latter fol-

4 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well 4 lows fro Feri s golden rule). Here, ρ ( E) and ( E) are the density of states and carrier lifetie, respectively, in the case a =. Consequently, product ρ( E ) τ( E) v y in the RQW is the sae as in the conventional quantu well of the sae width (3D case) and in the bulk aterial. This product does not depend on G and consequently it becoes independent of depletion depth. The QD changes the distribution function f, since it increases μ. The QD influences integrals Ω ( α ) ( E ) by changing μ alone. Therefore, integrals are the sae as in the bulk aterial having the corresponding cheical potential τ i j i j L L =. (7) where i j L are the coefficients of bulk aterial having the value of μ obtained by the conventional doping (for instance). Further, we have to find μ for the syste of RQW and junctions and insert it in (5) together with (7). Cheical potential of degenerated seiconductor ( < μ * < where μ * = μ / k T ) can be written as [14] (note that frequently Feri energy is used instead of cheical potential in seiconductor literature) [ ln( n / ) + ( )] 3/ n μ = k. (8) T C / where n is the electron concentration and C is the effective conduction band density of states. In the case of heavy QD doping ( n >> ni, where n i is intrinsic concentration), we can use (3) for n. Density of states reduces G ties in RQW, i.e. = C C / G, where C is the C effective density of states for bulk seiconductor. Inserting this and (3) in (8), we get [ n ( G ) ] + 3/ [ n ( G ] μ = k. (9) T ln con C con ) Figure 3 shows the reduced cheical potential dependence on geoetry factor in RQW, plotted C C.75.5 i μ / κ Β Τ Ge G Figure 3. Cheical potential dependence on geoetry factor in RQW for i at and Ge at n = c 3 (blue line). n = c 3 (red line)

5 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well 5 according to (9), for i and Ge aterials for different n. The reason for choosing n values will be discussed in ection 4. Further, we rewrite (9) as [( G ) /( G ] 3/ ) + ( G G ) ncon C μ = μ (1) + kt ln where G is constant. Introduction of μ μ( G ) defines reference aterial as n + -type seiconductor with electron concentration of ( D = n G ), or RRQW having constant geoetry factor 3/ ( G / T ) = and G = G. ext, we calculate the gradient of (1) taking into account that C T. The result is μ = + θ T. (11) μ ln ξ G G G G 1 1 G where θ ( ) k + T + ξ G T (1) 3/ and ξ ncon / C. Inserting (11) and (7) in oltzann equations, we find charge and heat currents in the syste of RQW + J 1 11 ( ε + μ / e) ( L L e) T 11 θ J = L / and (13a) J Q 1 ( ε + μ / e) ( L L e) T 1 θ = L /. (13b) Further, eebeck coefficient, electrical conductivity, and electron gas heat conductivity can be found fro (13a) and (13b) in a conventional way = L L θ / e L = ( θ / e), (14) κ e = L L ( θ / e) L L θ / e L L = κe, (15) 11 σ σ = L =. (16) Electrical and theral conductivity in the syste of RQW and junctions reain unaffected (with respect to RRQW or bulk aterial having the sae μ value) by a series of junctions, and change according to (14). To calculate θ and then, we have to find G / T first (1). 3. Geoetry factor teperature dependence in RQW with p + n + and etal n + junctions Depletion region reduces the effective height of the ridge fro a to a eff ( T ) = a d( T ). (17) Here, d(t ) is the depletion region depth. Differentiating (17), inserting (1), and taking into account that G = G( T ), we find ( G T ) = ( dg / dt ) = ( G / a )[ d d( T ) / dt ]. (18) / eff

6 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well 6 Let us find G / T in RQW with series of p + n + junctions first (figure 4). Within the abrupt junction approxiation, d(t ) has the following for [15]: d ε ( T ) = e D ( A A + D kt ϕ bi ) e 1/. (19) Here, ε is the dielectric perissibility of the aterial, ϕ = ϕ + ϕ is the built-in potential, A bi bin bip D is the QD doping concentration. k is the oltzann constant, is the acceptor concentration in p-type layer, and Figure 4. Energy diagra of p + n + junction at the top of the ridge. Ridge (left) is QD-n doped and additional layer (right) is acceptor doped. The built-in potential is (figure 4) ( E + μ γ ) e ϕ bi = ϕ bin+ ϕ bip= g +, () where Eg = EC EV and γ is the cheical potential of p + -type layer. For the case shown in figure 4, both μ and γ are negative as μ is easured fro the RQW conduction band botto and γ is easured fro the p + -type layer valence band top. Paraeter γ is deterined using the forula siilar to (8) for p + -type seiconductor. Inserting () in (19) and introducing ν n con / A, we write d ( T ) = e ε n con ( E g (1 G + μ + γ k T ) )[1 + ν (1 G )] 1/. (1) Differentiating (1) and inserting d μ / dt found fro (9) and d γ / dt found fro forula analogical to (8), and further inserting result in (18) gives [( μ / T ) + ( γ / T ) 5 k ( 3/ ) k ξ ( G ) ( 3 ) δ ] p n = E ( G T ) k. () 3/ where δ = ( A / V ), V is the valence band effective density of states in p + -type layer, and is the characteristic energy of the syste of RQW and p + n + junctions and equals to E E β G = Eg + μ + γ k T ) + + G ( G ) G ν G k + ν ( G ) ( 1 G T + ξ. (3) G

7 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well 7 where β = a / d. eff In the case of etal n +, chottky junction will for at the top of the ridges (figure 5). Figure 5. Energy diagra of etal n + junction at the top of the ridge. Ridge (left) is QD-n doped. Within abrupt junction approxiation, the etal n + junction depletion layer depth is [15] d ε ( T ) = e 1 D k T ϕ bi e 1/. (4) Here, ϕ bi = Φ χ + μ / e where Φ is the etal work function and eχ is the seiconductor electron affinity easured fro the botto of conduction band. Repeating the above-described steps for etal n + junction, we found [( / T ) ( 5/ ) k ( 3/ ) k ξ ( G ) ] ( G T ) = E μ. (5) Here, β G G E = ( eφ eχ + μ k T ) + k T + ξ. (6) G ( G ) G Investigation of () and (5) shows that both ( G / T ) p n and ( G / T ) n strongly depend on G, and this can be used to increase eebeck coefficient considerably. However, ( G / T ) p n diverges and changes the sign for the value of G for which E = (the sae happens in the case of etal n + junction). Care should be taken to keep G far enough fro the divergence point and avoid the change of G / T sign ( will also change its sign). At the sae tie, G should be chosen so that ( ) p n ( G / T ) p n is positive and high enough to obtain large enhanceent in value. Additionally, γ (acceptor doping of p-type layer) can be varied to attain the desired ( G / T ) p n value. In the case of etal n + junction, Φ can be varied instead ofγ. Equations () and (5) are obtained on the basis of (1) describing dependence of G on the diensions. However, (1) is valid only for a range of geoetry factors 5 < G < 1 and for a<<w. Let us try to find G for the arbitrary geoetry. This requires solving of the tie-independent chrödinger equation in the ridged geoetry [4]. Unfortunately, there is no analytical solution in the ridged well (solution contains infinite sus). However, there are fairly accurate nuerical ethods. Matheatically, there is no difference between QD and the electroagnetic ode of depression, and Helholtz equation and the sae boundary conditions are used in both the cases. Helholtz spectru calculation can be found in the literature related to Casiir effect. Casiir energy exhibits strong dependence on

8 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well 8 the photon spectru and consequently on the geoetry of the vacuu gap [16]. A nuber of geoetries, including double-side ridged geoetry [17] and double-side corrugated geoetry [18], were investigated. ew, optical approach for arbitrary geoetry was also developed [19]. Unfortunately, none of the above-described ethods allows siple analytical solution like (1). Consequently, we have to accept the liit 5 < G < 1 for further analysis. However, the above-listed nuerical ethods allow one to go well beyond those liits. 4. eebeck coefficient of RQW with p + n + and etal n + junctions It is reasonable to calculate eebeck coefficient of RQW with junctions relative to RRQW, in which geoetry factor is constant ( G / T ) =. This will allow coparison of diensionless figure of erit ZT with Z T, using siilarity of electric (16) and heat conductivities (15). In (1) (1) G is arbitrary. To siplify the coparison, we choose G so that RWQ with junctions and reference RRQW have the sae μ value. Equation of cheical potentials leads to G = G (9). Inserting this in (1) and further inserting the obtained result in (14), we obtain for eebeck coefficient of RQW with junctions = k T 1 + G, ξ. (7) e G T, Here, p n and n are the eebeck coefficients of RQW with p + n + and etal n + junctions, respectively. 3D eebeck coefficient of reference RRQW do not differ fro eebeck coefficient of the bulk aterial, and within the parabolic bands approxiation it is equal to [] k r + 5/ Fr + 3/ ( μ*) = μ *. (8) e r + 3/ Fr + 1/ ( μ*) Here, r refers to scattering paraeter and it is assued that electron lifetie τ( E) E, μ * = μ / k T is the reduced cheical potential, and F(μ *) are Feri integrals. In the case of no ipurities (QD doping) and low energies, acoustic phonons are responsible for electron scattering and r =. It should be noted here that RQW has G ties ore width [] with respect to the conventional quantu well. As 3D ρ(e) in wide quantu wells tends to ρ(e) of the bulk aterial of the sae width, using (8) for RRQW is a good approxiation. However, in the case of thin layers oscillatory behaviour of transport coefficients should be considered [1]. To find the ratio Z / Z, we use relations (15) and (16) and obvious relation between lattice theral conductivitiesκ l = κ l, all together leading to Z ( ) p n, / Z, =. (9) Figure 6 shows the dependence ( μ ) according to (8). The in the sae figure is deterined by first, calculating p n ( G) by inserting () in (7), and then (9) was used to convert X-axis so that was obtained fro p n ( G). The ratio Z p n / Z in the sae figure is calculated according to (9). We present μ dependences as they allow the understanding of possible μ ranges, within which real devices can operate without changing sign of ( G / T ) p n () and (3) and consequently sign. r

9 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well p-n / (k /e) 6 4 p-n Ge i 6 4 Z p-n T / Z T Z /Z p-n Ge i μ / k T Figure 6. solid lines, ( ) dotted line, and Z ) / ( μ Z μ) dashed lines (belong to right Y- μ p n ( axis). Dependences and Z μ) / Z are calculated for i and Ge aterials. p n ( Traditional theroelectric aterials i and Ge are chosen as exaples. Dependences are plotted for the following paraeters: Material i, n = c -3, a = n, γ = 3. 3, β =. 1, T = 3 K; Material Ge, n = c 3, a = 35 n, γ =, β =. 5, T = 3 K. For both i and Ge, we choose n so that, for value G=1, μ was close to the optiu value μ OPT / k T = r + 1/ [19]. Figures 5 and 3 allow finding of ( G) and G) / Z ( G) dependences as well. For this, p n Z p n ( μ should be deterined fro the desired point in figure 5 and next G can be found using figure 3. ext, L can be found by first deterining d fro n and G values using (1) and next, deterining aeff fro (17) and inserting it in (1) instead of a. For the above paraeter values and G = 1, we got L = 18 n for i and L = 75 n for Ge. It should be noted here that RQW exhibits quantu properties at G ties ore widths with respect to the conventional quantu well []. Figure 6 shows large enhanceent of figure of erit in p + n + junction RQW both for i and Ge aterials. The dependences are quite identical for aterials having rather different band gaps (1.1 ev for i and.66 ev for Ge []). This shows that despite E g entering expressions for depletion depth (1) and ( G / T ) p n (), alost siilar and Z p n / Z dependences can be obtained by atching such paraeters as β andγ. Figure 7 shows dependences in the case of etal n + junctions. The was deterined by first calculating n ( G) by inserting (5) in (7) and then using (9) for X-axis to convert n ( G) to. oth and Z n / Z dependences are siilar for i and Ge. Two curves are not distinguishable on the given scale.

10 A. Tavkhelidze. Large enhanceent of theroelectric figure of erit in a ridged quantu well n / (k /e) 6 4 -n i Ge 6 4 Z -n T / Z T Z / Z -n i Ge μ / k T Figure 7. solid lines, ( ) dotted line, and Z ) / ( μ Z μ) dashed lines. μ n ( Curves are plotted for the following paraeters: Material i, n = c 3, a = 15 n, β =. 55, ( ) = Φ χ k T / e, T = 3, K; Material Ge, n = c 3, a = 8 n, β =.4, ( Φ χ) = 15 k T / e, T = 3 K. RQW width L deterined in the above-described way are L = 6 n for i and L = 85 n for Ge. As the analysis shows, high theroelectric figure of erit can be obtained in both p + n + junction and etal n + junction cases. Metal n + junction RQW sees to be siple in fabrication when copared with p + n + junction RQW, since etal fil deposition is ore straightforward than epitaxial growth of the seiconductor layer. However, p + n + junction allows ore precize regulation of ( G / T ), as it can be done by atching the acceptor concentration (γ value), which is less coplex than finding etal with required work function ( Φ value in the case of etal-n + junction). 5. Conclusions Theroelectric transport coefficients were investigated in the syste of ridged quantu wells and periodic series of p + n + and etal n + junctions at the top of the ridges. Analysis was ade on the basis of oltzann transport equations. It was shown that the eebeck coefficient increases considerably. At the sae tie, electrical and theral conductivities reain unaffected by the series of junctions. This allows large enhanceent of theroelectric figure of erit. Dependence of eebeck coefficient on geoetry factor G and junction paraeters was investigated and the analytical expression was obtained. eebeck coefficient changes sign for soe value of G. Dependences of and ZT on cheical potential were presented for both p + n + and etal n + junction RQW (separately for i and Ge aterials). Calculations show one order of agnitude increase in theroelectric figure of erit with respect to the bulk aterial. Acknowledgent The author thanks the Physics Departent of ew York University, where part of this work was done, for their hospitality.

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