Novel Thermoelectric Materials: From Quasicrystals to DNA

Novel Thermoelectric Materials: From Quasicrystals to DNA Enrique Maciá Dept. Física de Materiales, Facultad CC. Físicas, Universidad Complutense de Madrid, E-28040, Madrid, Spain June 28, 2010

Abstract Some time ago it was proposed on sound theoretical basis that the best thermoelectric materials are likely to be found among materials exhibiting a sharp singularity in the density of states (DOS) close to the Fermi level, along with a substantial depletion of the DOS at the Fermi level. In this Chapter I will describe the thermoelectric properties of two different classes of materials exhibiting these required spectral features in their electronic structures. The rst class of materials comprises representatives of quasicrystalline alloys exhibiting semiconductor-like, rather than metallic electronic transport properties, along with extremely low thermal conductivity values. Accordingly, quasicrystals can be regarded as an unexpected instance of the so-called electron crystal-phonon glass representatives. Thus, quasicrystals occupy a very promising position in the quest for novel thermoelectric materials, naturally bridging the gap between semiconducting materials and metallic ones. The other class of compounds is of great interest to those researchers working in the eld of nanotechnology as it refers to the thermoelectric properties of molecular junctions and the possibility of designing Peltier cells at the nano-scale. In particular, I will focus on the electronic structure and transport properties of DNA based devices, with a special attention to the possible use of a thermoelectric signature for different codons of biological interest in order to explore new sequencing techniques based on physical processes instead of the usual chemical ones.

Contents 1. Introduction 1.1. Basic de nitions . . . . . . . 1.2. Transport coef cients . . . . 1.2.1. Bulk materials . . . 1.2.2. Molecular junctions 1.3. ZT optimizing strategies . .

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E-mail address: emaciaba@ s.ucm.es

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Novel Thermoelectric Materials: From Quasicrystal to DNA 2. Quasicrystals and related complex metallic alloys 2.1. Basic notions . . . . . . . . . . . . . . . . . . 2.2. Transport properties . . . . . . . . . . . . . . . 2.3. Electronic structure model . . . . . . . . . . . 2.4. Improving the thermoelectrical performance . .

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11 11 14 17 21

3. DNA based thermoelectric devices 3.1. Physical motivations . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Thermoelectric properties of polymer materials . . . . 3.1.2. Thermoelectricity in molecular junctions . . . . . . . 3.1.3. The DNA case . . . . . . . . . . . . . . . . . . . . . 3.2. Electronic structure and transport properties of DNA molecules 3.3. Thermopower of single-stranded oligonucleotides . . . . . . . 3.3.1. Analytical expressions . . . . . . . . . . . . . . . . . 3.3.2. Transport curves . . . . . . . . . . . . . . . . . . . . 3.4. Thermoelectric codon sequencing . . . . . . . . . . . . . . . 3.5. Duplex DNA model . . . . . . . . . . . . . . . . . . . . . . . 3.6. Thermopower of double-stranded DNA chains . . . . . . . . . 3.6.1. Analytical expressions . . . . . . . . . . . . . . . . . 3.6.2. Transport curves . . . . . . . . . . . . . . . . . . . .

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4. Conclusion

PACS Keywords: Thermoelectric materials. Key Words:quasicrystal, DNA AMS Subject Classi cation: .

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1. 1.1.

Introduction Basic de nitions

During the last few years we have witnessed a growing interest in searching for novel, high performance thermoelectric materials (TEMs) for energy conversion in small scale power generation and refrigeration devices. The ef ciency of thermoelectric devices depends on the hot side temperature, Th , the cold side temperature, Tc , and on the so-called thermoelectric gure of merit (FOM), usually denoted Z, according to the expression [1] p Tc 1 + ZTm 1 p ; (1) = 1 Th 1 + ZTm + 1 where Tm (Th + Tc )=2 is the average over the operational temperature range. Since Eq.(1) is a monotonously growing function of the dimensionless variable ZTm , the Carnot ef ciency C = 1 Tc =Th can only be obtained in the limit ZTm ! 1. Thus, new materials for thermoelectric applications are evaluated in terms of the dimensionless FOM which depends on the transport coef cients of the constituent materials in the form ZT =

P (T ) T; (T ) + l (T ) e

(2)

where T is the temperature, P (T ) S 2 , is the so-called thermoelectric power factor, (T ) is the electrical conductivity (conductance), S(T ) is the Seebeck coef cient, and e (T ) and l (T ) are the charge carrier and lattice contributions to the thermal conductivity (conductance), respectively. P is generally used to compare thermoelectric performances of materials that have similar thermal conductivities. In order to improve the thermoelectric performance, segmented devices where different materials are joined together are currently considered. In this case we must take into account the compatibility factor given by the expression [2] p 1 + ZT 1 s= ; (3) ST since materials with dissimilar s vales cannot be ef ciently combined in a single device. Therefore, both ZT and s must be simultaneously optimized. The thermoelectric compatibility of several materials of current technological interest has been recently reviewed in [3].

1.2. 1.2.1.

Transport coef cients Bulk materials

Within the linear response theory the electrical, j, and thermal, h, current densities through a given material are respectively related to the applied voltage and temperature gradients according to the expression j h

=

L11 L12 L21 L22

rV rT

;

(4)

Novel Thermoelectric Materials: From Quasicrystal to DNA

3

where the matrix elements are tensors that reduce to scalar quantities for high symmetry systems. In practice, we do not directly observe the transport matrix elements Lij , but certain transport coef cients related to them by the following experimental setups: Electrical conductivity (T ): The sample is kept at constant temperature (rT 0): The electrical current is measured and, taking into account Ohm's relation j = E= rV, from Eq.(4) one gets (5)

(T ) = L11

Seebeck coef cient S(T ): The sample is electrically insulated to prevent any electric current owing through it (j = 0). One subjects the sample to a thermal gradient and observes the presence of an electric eld which is given by E =S rT (Seebeck effect), hence from Eq.(4) one gets S(T ) =

L12 L11

(6)

Peltier coef cient (T ): The sample is kept at constant temperature (rT 0) as an electrical current ows through the sample. We observe the presence of a thermal current associated with the voltage drop which is proportional to the electric current, i.e., h j (Peltier effect), so that from Eq.(4) one gets (T ) =

L21 : L11

(7)

According to the Onsager relations,[4, 5] the Peltier coef cient equals Seebeck coef cient multiplied by the temperature, i.e., = ST . Thus, Seebeck and Peltier effects are mutually conjugated phenomena. Thermal conductivity (T ): The sample is electrically insulated to prevent any electric current owing through it (j = 0). A thermal gradient is maintained and the ux of heat is measured. According to Fourier's law h = rT, so that from Eq.(4) one gets L12 L21 : (8) (T ) = L22 L11 Note that in the expression above thermal conductivity includes contributions from both charge carriers and atomic oscillations, i.e., (T ) = e (T )+ l (T ). By properly combining the nested relations given by Eqs.(5)-(8) we can express Eq.(4) in the form j h

=

ST

S + S2T

rV rT

:

(9)

Thus, measuring the transport coef cients (T ); (T ); and S(T ) we can completely determine the transport matrix describing the linear relations between currents and gradients. As we can see, in the limiting case S = 0 the transport matrix becomes diagonal and

4 j and h are completely decoupled from each other. Thus, the Seebeck coef cient, appearing in the non diagonal terms of the transport matrix, determines the coupled transport of electricity and heat through the considered sample. Since Eq.(2) indicates that large FOM values require large values of the Seebeck coef cient, in searching for promising TEMs one must then focus on materials exhibiting great couplings between the electrical and thermal currents. According to Wiedemann-Franz's law (WFL) in most materials thermal and electrical conductivities are mutually related, over certain temperature ranges, through the relationship (10) e (T ) = L0 T (T ); where L0 = (kB =e)2 0 is the Lorenz number, kB is the Boltzmann constant, e is the electron charge, and 0 is a constant whose value depends on the nature of the sample. Thus, for metallic systems 0 = 2 =3; and we get Sommerfeld's value L0 ' 2:44 10 8 V2 K 2 ; while for semiconductors described by Maxwell-Boltzmann statistics we have 0 ' 2:[4] The WFL expresses a transport symmetry arising from the fact that the motion of the carriers determines both the electrical and thermal currents at low temperatures. As the temperature of the sample is progressively increased, the validity of WFL will depend on the nature of the interaction between the charge carriers and the different scattering sources present in the solid. In general, the WFL applies as far as elastic processes dominate the transport coef cients, and usually holds for arbitrary band structures provided that the change in energy due to collisions is small compared with kB T .[6, 7] Accordingly, one expects some appreciable deviation from WFL when electron-phonon interactions, affecting in a dissimilar way to electrical and heat currents, start to play a signi cant role. On the other hand, at high enough temperatures the heat transfer is dominated by the charge carriers again, due to umklapp phonon scattering processes, and the WFL is expected to hold as well. In order to gain a deeper physical insight into the processes determining the temperature dependence of the transport coef cients we will rely on the Kubo-Greenwood formalism.[8, 9, 10] The central information quantities are the kinetic coef cients i+j

Lij (T ) = ( 1)

Z

+1

(E) (E 1

)i+j

2

@f @E

dE;

(11)

where (E) is the spectral conductivity function (de ned as the T ! 0 conductivity with the Fermi level at energy E), E is the charge carrier energy, (T ) is the chemical potential and f (E; T ) = [1 + exp((E ) )] 1 is the Fermi-Dirac distribution function, 1 where (kB T ) . In this formulation all the microscopic details of the system are included in the spectral conductivity function (E), which does not depend on T . Therefore, the temperature dependence of the transport coef cients appears in the charge carriers distribution, whereas all peculiarities of the scattering processes are incorporated in (E). This description is valid provided that the charge carriers are noninteracting and scattering is elastic. No assumption is made about the strength of disorder and the nature of the eigenstates. Consequently, the kinetic coef cients derived from Eq.(11) are valid for both extended and localized states. In fact, this formalism has been applied to study the thermoelectric properties of disordered systems undergoing an Anderson-driven metal-insulator transition,[11] as well as transport phenomena in quasicrystals and related approximants


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characterized by giant complex unit cells.[12, 13] On the other hand, it can be implemented within the Landauer-Büttiker formalism in order to account for charge transport through single-molecule nanodevices.[14] In those systems for which the electronic transport can be described within the Boltzmann approach one has (E) = e2 (E)n(E)v 2 (E), where (E) is the relaxation time, n(E) measures the charge carriers density, and v(E) is the group velocity of the carriers. In this Chapter we will consider relatively complex systems, like quasicrystals and molecular junctions, for which the applicability of the Boltzmann approach is not guaranteed, so that we will assume the more general relationship (E) = e2 N (E)D(E), where N (E) is the DOS and D(E) measures the diffusivity of the states. From the knowledge of the spectral conductivity function, after Eq.(11) one obtains the electrical conductivity (12)

(T ) = L11 (T ); the thermoelectric power, S(T ) =

1 L12 (T ) ; jejT (T )

(13)

and the electronic thermal conductivity, e (T )

=

1 L22 (T ) e2 T

T S 2 (T ) (T );

(14)

in a uni ed way. Then, by expressing Eqs.(12-14) in terms of the scaled variable x (E ) , the transport coef cients can be rewritten as (T ) =

S(T ) =

e (T )

=

T c2 J0

J0 ; 4

(15)

kB J1 ; jej J0

(16)

J0 J1 J1 J2

;

(17)

in terms of the reduced kinetic coef cients, Jn =

Z

1

xn (x) sech 2 (x=2)dx;

(18)

1

where c 2e=kB , and we have expressed the Fermi distribution derivative in terms of hyperbolic functions. Plugging Eqs.(15)-(17) into Eq.(2) one gets ZT = where '(T )

l (T )=T:

J0 J2

J12 ; J12 + c2 J0 '

(19)

6

Figure 1. (a) Tight-binding molecular junction model. The polymer is described in terms of a linear chain with an orbital per site (on-site energies " , " , and " ), where each lattice site represents a monomer (squares). The polymer chain is connected to leads modeled as semi-in nite one-dimensional chains of atoms (circles) with one orbital per site (on-site energy "M ). (b) Energy band structure of the molecular junction model sketched in (a). The bandwith of the contacts is WM = 4tM . The dashed horizontal line indicates the location of the contacts Fermi level. The segments below (over) the dashed line correspond to the HOMO (LUMO) orbitals of each monomer.

1.2.2.

Molecular junctions

As an alternative to bulk materials the study of the thermoelectric properties of single molecules may underpin novel thermal devices such as molecular-scale Peltier coolers and provide new insight into mechanisms for molecular-scale transport. From the study of thermoelectric voltage over a molecule attached to two metallic leads one can gain valuable information regarding the location of the Fermi energy relative to the molecular levels. In particular, from the sign of the thermopower it is possible to deduce the conduction mechanism, with a positive sign indicating p-type conduction (the Fermi level is closer to the highest occupied molecular orbital (HOMO) level), whereas a negative sign indicates n-type conduction (the Fermi level is closer to the lowest unoccupied molecular orbital (LUMO) level). In fact, the extreme sensitivity of thermopower to ner details in the electronic structure suggests that one could optimize the device's thermoelectric performance by properly engineering its electronic structure. For instance, by shifting the Fermi level position in order to optimize the thermoelectric performance of a given molecular arrangement. In this way, the thermoelectric potential of some conducting polymers, like polythiophene and polyaminosquarine, has been recently reviewed on the basis of their electronic band structures [15]. Also, the thermoelectric properties of nanocontacts made of single-wall carbon nanotubes have been numerically studied, concluding that doped semiconducting nanotubes may exhibit very high gures of thermoelectric merit [16]. These results naturally rise the question regarding the possible use of suitable organic molecules to design novel thermoelectric devices.


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To start with, as a rst approximation we shall consider the charge carrier dynamics can be decupled from vibrational atomic motions in the molecule and that the coupling between the contacts and the molecule is weak enough, so that the lead-molecule-lead junction can be described in terms of three non-interacting subsystems according to the tight-binding Hamiltonian (Figure 1) [17] (20)

H = HM + HC + HL ; where HM =

N X

"n cyn cn

n=1

N X1

tn;n+1 (cyn cn+1 + cyn+1 cn );

(21)

n=1

describes the polymer chain, where N is the number of monomers, cy0 c1 + cy1 c0 + cyN +1 cN + cyN cN +1 ;

HC =

describes the molecule-metal coupling, where the leads and the end monomers, and HL =

1 X

"M cyl cl

tM (cyl cl+1

+

cyl+1 cl )

+

(22)

measures the coupling strength between +1 X

tM (cyl cl+1 + cyl+1 cl ); (23)

"M cyl cl

l=N +1

l=0

describes the contacts at both sides of the polymer chain, so that sites comprised between [ 1; 0] [ [N + 1; +1] belong to the leads, where cyj (cj ) is the creation (annihilation) operator for a charge at the jth site in the chain, "n are the on-site energies of the monomers, tn;n+1 is the hopping term between them, "M is the leads on-site energy and tM (> ) is their hopping term, so that the leads dispersion relation is given by E(k) = "M +2tM cos k. Within the transfer matrix framework, and considering nearest-neighbors interactions only, the Schrödinger equation corresponding to the Hamiltonian (20) can be expressed in the form N +1

= TN +1 TN ::::T1 T0

N

where

n

0

;

(24)

1

is the wavefunction amplitude for the energy E at site n and 0 1 tn;n 1 E "n Tn (E) = @ tn;n+1 tn;n+1 A ; 1 0

(25)

is the local transfer matrix. The lead-molecule-lead zero bias transmission coef cient, TN (E), describing the fraction of charge carriers transmitted through a chain of length N in the absence of any applied voltage, can then be obtained from the knowledge of the leads dispersion relation, Q E(k); and the matrix elements of the molecular junction global 0 transfer matrix M(E) n=N +1 Tn (E); by means of the relationship [18] TN (E) =

[M12

M21 + (M11

4 sin2 k : M22 ) cos k]2 + (M11 + M22 )2 sin2 k

(26)

8 From the knowledge of the transmission coef cient, the conductance through the molecular junction is determined using the Landauer formula [19] (27)

GN (EF ) = G0 TN (EF );

1 is the conductance quantum and E is the Fermi where G0 2e2 =h ' 1=12906 F energy. Several experimental measurements of the Seebeck coef cient in molecular junctions have been recently conducted at zero bias.[20, 21, 22] In such cases, the system can be described by only a single Fermi level and the Seebeck coef cient can be expressed in terms of TN (E) by means of the expression [23, 24]

SN (T ) =

jejL0

@ ln TN (E) @E

T:

(28)

EF

If an external electrical bias is applied through the system one must consider two distinct Fermi levels in the left and right electrodes, EFL and EFR , respectively. In that case the Seebeck coef cient can be obtained from the expression [25] TL SN (T; V ) =

jejL0

(E;V ) @TN (E;V ) + TR @TN@E @E L R EF EF ; TN (EFL ; V ) + TN (EFR ; V )

(29)

where TL and TR are the temperatures in the left and right contacts, respectively.

1.3.

ZT optimizing strategies

The mathematical structure of Eq.(2) suggests two complementary approaches in order to optimize ZT , namely, power factor enhancement or/and thermal conductivity reduction. To begin with, one should ascertain for materials having an as low as possible lattice contribution to the thermal conductivity, l .[26] In fact, promising materials rendering high ZT values, have been obtained on the basis of the so-called electron crystal-phonon glass approach,[27] by exploiting thermal conductivity reduction techniques such as alloy scattering in amorphous materials the design of nanostructured semiconductor superlattices and low dimensional systems to reduce the phonon mean free path, the recourse to cluster-based aggregates, as clathrates or lled skutterudites containing rattling heavy atoms which signi cantly degrade the phonon propagation. As we will see in Section 2 quasicrystalline alloys also exhibit very low thermal conductivities due to the existence of umklapp phonon-phonon scattering processes occurring at all scales in reciprocal space. Polymers inherently possess a low thermal conductivity which gives them a signi cant advantage over conventional TEMs, as we will discuss in Section 3. More drastic scenarios in order to de nitively suppress the thermal conductivity lattice contribution have recently


9

considered the transport properties of molecules (Lorentz gas) through completely rigid structures.[28] Nevertheless, it seems unlikely that one can arbitrarily increase the ZT value by only reducing the thermal conductivity, since the phonon mean free path cannot be reduced below the interatomic spacing.[29] Thus, materials suitable for a power factor enhancement should be sought in order to further increase the FOM. To this end, however, the coupling of S(T ); (T ) and e (T ) transport coef cients among them makes obtaining large ZT values a dif cult task. In fact, electrical conductivity generally decreases as the Seebeck coef cient increases (and vice versa, see Eq.(31) below), so that optimization of the FOM appears as a very appealing challenge from the viewpoint of materials engineering.[30] Traditionally, semiconductors have been considered as the most appropriate materials for thermoelectric applications. This criterion is based on the following facts. On the one hand, insulators exhibit too low power factors ( ! 0 in Eq.(2)). On the other hand, making use of the Wiedemann-Franz law given by Eq.(10) into Eq.(2), one obtains ZT =

S2 L0 +

l

(30)

:

T

Then, taking into account that the phonon contribution to the thermal conductivity is almost negligible as compared to the electronic one in metals (i.e., l e ), and assuming S = 10 VK 1 as a representative value for these materials, typical values of about ZT ' 10 3 are obtained for metals and alloys at room temperature. Accordingly, in order to optimize the FOM value we must look for a material located between metals and insulators. This explains the traditional interest in narrow gap semiconductors in the quest for good TEMs, the BiTe-SeTe alloys family (ZT ' 0:9) being the most widely used TEM at room temperature.[1, 31] According to experimental data accumulated over the past ve decades it seems that a practical barrier to increasing the ZT value exists for the considered materials at about ZT ' 1, a gure which provides a practical reference value. However, no de nitive theoretical justi cation for the possible existence of such a limiting ZT value (if any) has been given to date. Some indication supporting a possible FOM limiting value is obtained if one assumes that the Seebeck coef cient can be expressed as [32] S(T ) =

kB [A(T ) e

(31)

ln (T )],

where A(T ) includes scattering effects, the carrier mobility, and its effective mass. In this case the power factor can be expressed as an explicit function of the electrical conductivity transport coef cient. Making use of Eq.(31) the extreme condition dP=d 0 gives 2 Pmax = (2kB =e) , with = exp(A 2): In this way, one obtains a universal value for the ratio of the maximum electrical power factor at optimum electrical conductivity Pmax

=

2kB e

2

' 2:97

10

8

V2 K

2

,

(32)

which depends on fundamental constants only. Quite interestingly, the value Pmax

' 2:44

10

8

V2 K

2

= L0 ,

(33)

10 was reported from a linear regression analysis of suitable transport data for several established thermoelectric materials,[33] in reasonable agreement with the theoretical estimation. On the other hand, plugging the Pmax value given by Eq.(32) into Eq.(2), and assuming that the WFL holds in the form e = L0 T , one obtains (ZT )max =

2kB e

2

T L0 T + l (T )e2

A

:

(34)

In the high temperature regime, the lattice contribution to the thermal conductivity progressively decreases due to the onset of the phonon umklapp processes, so that the optimum 2

2kB FOM value approaches the limiting value (ZT )max L0 1 = 12= 2 ' 1: 216 in e this case. A deeper insight into the thermoelectric properties of different materials can be obtained by inspecting their electronic structure. In this way one realizes that good TEMs should exhibit two characteristic and complementary features near the Fermi level:

a relatively at band, leading to the presence of a sharp, narrow feature in the DOS. According to the Mott's expression S=

2 k2 B

3jej

@ ln N (E) @E

T;

(35)

EF

such a feature gives rise to high Seebeck coef cient values; a broad, dispersive band leading to high electrical conductivity. In fact, charge carriers mobility is proportional to the second-rank effective-mass tensor mij , which in turn, is determined by the local curvature of the dispersion relation E(k) according to the expression 1 1 @2E = : (36) mij ~ @ki @kj Since both spectral features usually exclude each other this explains why electrical conductivity generally decreases as the Seebeck coef cient increases (and vice versa). The appealing question regarding what electronic structure will provide the largest possible FOM was addressed some time ago, concluding that (i) the best TEM is likely to be found among materials exhibiting a sharp singularity (Dirac delta function) in the DOS close to the Fermi level, and (ii), in that case, the effect of the DOS background contribution onto the ZT value may be quite dramatic, the FOM value being inversely proportional (in a marked non-linear way) to the DOS value near the singularity.[34] One class of materials exhibiting the main features of the proposed electronic structure is based on alloys between transition metals and main group elements in which there exists a hybridization between the partially lled d-bands of the transition-metal and the (s,p)-bands of the remaining atoms. This hybridization results in the formation of a series of relatively narrow, spiky features as well as a deep well (pseudogap) close to the Fermi level, as we will discuss in more detail in the next Section.


11

Figure 2. Electron diffraction pattern corresponding to an AlCuFe icosahedral quasicrystal. A 5/10 fold symmetry axis around the origin can be clearly appreciated. If you connect all the vertices of a regular pentagon by diagonals you obtain the so-called Pythagorean pentagram shown on the upper right corner. At their intersecting points the diagonals form a smaller pentagon at the center, and the diagonals of this pentagon form a new pentagram enclosing a yet smaller pentagon. This progression can be continued ad in nitum, creating smaller and smaller pentagons and pentagrams in an endless succession exhibiting a selfsimilar nesting characteristic of the fractal structures (Courtesy of J. Reyes-Gasga).

2. 2.1.

Quasicrystals and related complex metallic alloys Basic notions

Quasicrystals (QCs) are metallic alloys representative of a novel condensed matter phase which can be regarded as a natural extension of the notion of a crystal to structures with quasiperiodic, rather than periodic, long-range order. Thus, QCs show an essentially discrete diffraction pattern (typical of well ordered systems), although exhibiting unusual symmetry arrangements of the diffraction spots, related to icosahedral, octogonal, decagonal and dodecagonal symmetries, which also display self-similar features (Figure 2).[12, 35] Shortly after the discovery of thermodynamically stable QCs of high structural quality in the AlCu(Fe,Ru,Os), AlPd(Mn,Re), ZnMg(RE), and Cd(Yb,Ca) icosahedral systems,[36] it was progressively realized that these materials occupy an odd position among the well-ordered condensed matter phases. In fact, since QCs consist of metallic elements one would expect they should behave as metals. Nonetheless, it is now well established that transport properties of stable QCs are quite unusual by the standard of common metallic alloys, as most of their transport properties resemble a more semiconductor-like than metallic character.[12, 35] For the sake of comparison in Table 1 we list a number of characteristic physical properties of both metals and QCs. By inspecting this Table one realizes that quasicrystalline alloys signi cantly depart from metallic behavior, resembling either ionic or semiconducting materials. Thus, high-quality QCs provide an intriguing ex-

12 ample of solids made of typical metallic atoms which do not exhibit most of the physical properties usually signaling the presence of metallic bonding. PROPERTY MECHANICAL TRIBOLOGICAL ELECTRICAL

MAGNETIC THERMAL OPTICAL

METALS ductility, malleability relatively soft easy corrosion high conductivity resistivity increases with T small thermopower paramagnetic high conductivity large speci c heat Drude peak

QUASICRYSTALS brittle (I) very hard (I) corrosion resistant low conductivity (S) resistivity decreases with T (S) large thermopower (S) diamagnetic very low conductivity (I) small speci c heat no Drude peak, IR absorption (S)

Table 1. Comparison between the physical properties of quasicrystalline alloys versus typical metallic materials. I (S) stands for ionic (semiconducting) materials typical properties. In every alloy system the true QC is accompanied by compositionally related classical crystals, having huge unit cell sizes, often forming micro-twinned networks with aperiodic symmetries. These crystals not only have very similar compositions, but also structures closely resembling that of the true QC, from which they can nevertheless be distinguished. For these reasons such crystals are called approximants. Approximant phases should not be confused with giant-unit-cell intermetallics, sometimes also termed complex metallic alloys, exhibiting complex structures that contain some hundred up to several thousand atoms in the unit cell. Examples are the Mg32 (Al,Zn)49 Bergman compound (Figure 3), with 162 atoms in the unit cell, orthorhombic 0 -Al74 Pd22 Mn4 (Figure 4) with 258 atoms in the unit cell,[37, 38] -Al4 Mn (586 atoms in the unit cell),[39] cubic -Al3 Mg2 (1168 atoms in the unit cell),[40] and the heavy-fermion compound YbCu4:5 , comprising as many as 7448 atoms in the supercell.[41] These giant unit cells contrast with elementary metals and simple intermetallics whose unit cells in general comprise from single up to a few tens atoms only. The giant unit cells with lattice parameters of several nanometers provide translational periodicity of the crystalline lattice on the scale of many interatomic distances, whereas on the atomic scale, the atoms are arranged in clusters with polyhedral order, where icosahedrallycoordinated environments play a prominent role. The structures of complex metallic alloys thus show duality; on the scale of several nanometers, these alloys are periodic crystals, whereas on the atomic scale, they resemble cluster aggregates. The high structural complexity of complex metallic alloys together with the two competing physical length scales—one de ned by the unit-cell parameters and the other by the cluster substructure—may have a signi cant impact on the physical properties of these materials, such as the electronic structure and lattice dynamics. On this basis, complex metallic materials are expected to exhibit novel transport properties of interest in the eld of thermoelectrics, like a combination of a relatively large thermopower with low thermal conductivity, and electrical and thermal resistances tunable by varying the composition.


13

Figure 3. (a) Successive atomic shells of the six-cell Bergman cluster. (b) Body-centered packing of the Bergman clusters, sharing a hexagonal face of the fourth shell produce the structure of the (Al,Zn)49 Mn32 Bergman phase. (Courtesy of Janez Dolinšek).

Figure 4. A view of the 0 -AlPdMn skeleton structure along the [010] direction. Mn atoms form a planar attened-hexagon lattice and are located in the centres of pseudo-Mackay icosahedra. The two interpenetrating polyhedra that form the outer shell of the pseudoMackay cluster (a 12-atom Pd icosahedron, white atoms) and a 30-atom Al icosidodecahedron (black atoms) are shown. (Courtesy of Janez Dolinšek).

14

Figure 5. Temperature dependence of the Seebeck coef cient for different quasicrystalline samples. (Courtesy of Roberto Escudero).

2.2.

Transport properties

As we discussed in Section 1.3, it may seem surprising to propose metallic alloys as a suitable thermoelectric material. However, such a proposal makes sense due to the peculiar transport properties of QCs and their related phases.[12] In particular, there are three qualitative reasons supporting the potential of QCs as TEMs. 1. Their electrical conductivity steadily increases as the temperature increases up to the melting point, leading to a parallel FOM increase. 2. QCs bearing transition metals in the systems i-AlCu(Fe,Ru,Os) and i-AlPd(Mn,Re) exhibit signi cantly large thermoelectric power values (30-120 VK 1 ) as compared to those of typical metallic systems (1-10 VK 1 ) at room temperatures, and the temperature dependence of the Seebeck coef cient usually deviates from the linear behavior (characteristic of charge diffusion in ordinary metallic alloys described by Eq.(35)), exhibiting pronounced curvatures at temperatures above 50 100 K (Figure 5). In addition small variations in the chemical composition give rise to sign reversals in the S(T ) value. 3. Third, the thermal conductivity of QCs is about two orders of magnitude lower than that of common metals, within the range 1 5 Wm 1 K 1 at room temperature and it is mainly determined by the lattice phonons (rather than the charge carriers) over a wide temperature range. This low thermal conductivity of QCs is particularly remarkable in the light of Slack's phonon-glass/electron-crystal proposal for promising thermoelectric materials,[27] and it has considerably spurred the interest on the potential application of QCs as thermoelectric materials from an experimental viewpoint.


15

Therefore, the electronic transport properties of quasicrystalline alloys exhibit unusual composition and temperature dependences, resembling more semiconductor-like than metallic character. When taken together in Eq.(2), these unusual electronic and thermal transport properties clearly favour a FOM enhancement. Thus, QCs occupy a very promising position in the quest for novel TEMs, naturally bridging the gap between semiconducting materials and metallic ones.[42, 43] In fact, one of the main advantages of QCs is that one can ef ciently exploit the high sensitivity of their transport coef cients to stoichiometric changes in order to properly enhance their power factors, thereby optimizing the numerator in Eq.(2), without sacri cing their characteristic low thermal conductivity. This property is illustrated in Table 2, where we list the power factor and FOM values for several i-AlPdRe representatives. As we can see, P and ZT values differing by more than two orders of magnitude can be attained in a single QC system by slightly changing the sample's composition by a few atomic percent (hence preserving the quasiperiodic crystalline structure). We also note that both positive and negative values of the thermopower can be obtained in this way, which allows for both the n- and p-type legs in a typical thermoelectric cell to be fabricated from the same material.[44] Sample ( Al68:5 Pd22:9 Re8:6 Al69:4 Pd21:2 Re9:4 Al67:7 Pd23:2 Re9:1 Al67:8 Pd22:2 Re10:0

1 cm 1 )

110 95 90 180

S ( V K 1) 10 7 +55 +95

P ( Wm 1 K 1:1 0:5 27:2 162:5

ZT 2)

1K 1)

(Wm 1:16 1:2 0:86 0:76

0:0003 0:0001 0:01 0:06

Table 2. Room temperature values of the transport coef cients and FOM for several AlPdRe icosahedral QCs reported in the literature.[45] (*) Estimated. In Table 3 we list the transport coef cients and thermoelectric response data for those representatives of the different QC families yielding the best FOM values at room temperature. By inspecting this table two main conclusions can be drawn: 1) typically metallic, very small ZT values are obtained for those QCs exhibiting either e=a ' 1:75 or e=a ' 2:00, 2) isostructural Al71 Pd20 Mn9 and Al71 Pd20 Re9 icosahedral samples, with e=a ' 1:80 exhibit the largest ZT values. Furthermore, signi cantly enhanced FOM values are obtained at higher temperatures for closely related QCs exhibiting similar e=a values (Table 4). On the other hand, FOM values similar to those of i-AlPd(Re,Mn) samples have been reported for the cubic approximant phases 1/1-Al71:6 Re17:4 Si11 (e=a = 1:951, ZT = 0:10) and 1/1-Al75:6 Mn17:4 Si7 (e=a = 1:911, ZT = 0:07) at room temperatures.[56] These alloys have not only very similar compositions in the phase diagram, but also structures closely resembling that of related QCs, and exhibit most of their characteristic transport properties anomalies as well. Finally, in Table 5 we list the gure of merit and compatibility factors of different QCs at room temperature as derived from data reported in the literature making use of Eq.(3). We observe that the largest s values are comparable to those observed in usual TEMs, like Bi2 Te3 or SiGe (s ' 1 V 1 ) and PbTe (s ' 1:2 V 1 ).[3] The most promising QCs belong to the icosahedral AlPdMn family, which exhibits a s factor larger than those corresponding to other icosahedral phases, approaching the gure reported for

16 Sample

Ref.

e=a (

Zn57 Mg34 Er9 Al65 Cu20 Ru15 Ag42:5 In42:5 Yb15 Al62:5 Cu24:5 Fe13 Cd84 Yb16 Al64 Cu20 Ru15 Si1 Al71 Pd20 Re9 Al71 Pd20 Mn9

[48] [49] [50] [49] [51] [49] [52] [52]

2:090 1:751 2:000 1:774 2:000 1:761 1:801 1:801

1 cm 1 )

6170 250 5140 310 7000 390 450 714

S ( V K 1) +7 +27 +12 +44 +16 +50 +80 +90

P ( Wm 1 K 30 19 74 60 1 79 98 2 88 5 78

ZT 2)

1K 1)

(Wm 4:5 1:8 4:8 1:8 4:7 1:8 1:3 1:5

Table 3. Room temperature values of the transport coef cients and FOM for QCs belonging to different families as reported in the literature. The samples are listed according to the value of the so-called average electron per atom ratio, e=a, which is obtained from their stoichiometric composition by assuming the valence values Cu = +1, Ag = +1, Mg = +2, Cd = +2, Zn = +2, Yb = +2, Al = +3, In = +3, Ga = +3, Er = +3, Si = +4, Pd = 0, Fe = 2:66, Ru = 2:66, Mn = 3:66, and Re = 3:66. In so doing, one generally assumes that the Hume-Rothery mechanism plays a substantial role in QCs stabilization, and the transition atoms take electrons from the conduction band, hence adopting a negative effective valence.[46, 47] ( ) After Ref. [53]. ( ) Estimated upper limit.

Sample Al68 Ga3 Pd20 Mn9 Al70:8 Pd20:9 Mn8:3 Al71 Pd20 (Re0:35 Fe0:65 )9 Al71 Pd20 Re9 Al71 Pd20 (Re0:45 Ru0:55 )9

Ref. [52] [44] [55] [52] [54]

e=a 1:820 1:820 1:859 1:850 1:850

T (K) 473 550 500 570 700

ZT 0:26 0:23 0:21 0:15 0:15

Table 4. High temperature gure of merit for QCs belonging to the AlPd(Mn,Re) icosahedral family as reported in the literature. T denotes the temperature maximizing the gure of merit and e=a the electron per atom ratio.

0:002 0:003 0:005 0:01 0:01 0:02 0:07 0:12

Novel Thermoelectric Materials: From Quasicrystal to DNA Sample Al62:5 Cu24:5 Fe13 Al64 Cu20 Ru15 Si1 Cd84 Yb16 Al67:8 Pd22:2 Re10:0 Al71 Pd20 Mn9

Ref. [49] [49] [51] [45] [52]

S( VK +44 +50 +16 +95 +90

1)

ZT 0:01 0:02 0:01 0:07 0:12

s (V 0:38 0:66 1:04 1:21 2:16

17

1)

Table 5. Room temperature thermopower, gure of merit, and compatibility factors for samples belonging to different quasicrystalline families. the so-called TAGS compounds at T = 550 K (s ' 2:7 V

2.3.

1 ).[3]

Electronic structure model

The experimental data reviewed in the previous Section clearly highlight the important role of band structure effects in the thermoelectric response of QCs and related alloys, suggesting that additional improvement may be attained by a judicious choice of both sample composition and processing conditions. In Figure 6 we show low temperature tunneling spectroscopy measurements corresponding to the quasicrystalline sample i-Al63 Cu25 Fe12 : These measurements reveal a broad pseudogap extending over an energy scale of about 0.6 eV (shown in the inset) along with some ne structure close to the Fermi level. The broad pseudogap stems from a Fermi surface pseudo-Brillouin zone interaction, while the dips may be respectively related to hybridization effects between d-Fe states and sp-states and d-orbital resonance effects. Recent tunnelling spectroscopy measurements performed in icosahedral QCs at low temperature (5:3 K) have provided additional experimental support for the existence of a large number of energetically localized features close to the Fermi level in the electronic structure of the 5-fold surface of an i-AlPdMn sample at certain local regions.[58] In order to make a meaningful comparison with experimental measurements one should take into account possible nite lifetime and temperature broadening effects. In so doing, it is observed that most ner details in the DOS are signi cantly smeared out and only the most conspicuous peaks remain in the vicinity of the Fermi level at room temperature.[59] These considerations convey one to reduce the number of main spectral features necessary to capture the most relevant physics of the transport processes. Following previous works we consider a realistic model for the electronic structure of transition metal bearing, Albased icosahedral QCs and related phases in terms of the spectral conductivity function given by, [13] 1

(E) =

1

(E

2 1)

+

2 1

+

(E

2

2 )2

+

2 2

:

(37)

This expression satisfactorily describes the electronic structure of these alloys in terms of a wide Lorentzian peak (related to the Fermi surface-Brillouin zone interaction) plus a narrow Lorentzian peak (related to sp-d hybridization effects).[46, 47] This model includes

0

Bias Voltage (mV)

100 0

-20

188 -300

190

192

194

196

198

200

202

204

206

-20 0

-100

Bia s V o lta g e (m V )

200

20

3 00

40

60

18

189 -60

190

191

192

193

194

-40

G (Arbitrary units)

G(V) (Arbitrary Units)

Figure 6. Differential conductance for the Al63 Cu25 Fe12 -Al tunnel junction at a temperature of T = 2 K for two different energy scales: 60 meV (main frame) and 300 meV (inset). Data le courtesy of Roberto Escudero. (Adapted from ref.[57]).

six parameters, determining the Lorentzian's heights ( = i ) and widths ( i ), their positions with respect to the Fermi level, i , and their relative weight in the overall structure, > 0. The parameter is a scale factor measured in ( cm eV) 1 units. The overall behavior of this curve agrees well with the experimental results obtained from tunneling and point contact spectroscopy measurements, where the presence of a dip feature of small width (20 60 meV), superimposed onto a broad (0:5 1 eV) asymmetric pseudogap has been reported.[60, 61, 62, 63] Suitable values for the electronic model parameters appearing in Eq.(37) can be obtained by properly combining ab-initio calculations with experimental transport data within a systematic phenomenological approach.[12, 64, 65, 66] Making use of Eq.(37) the kinetic coef cients given by Eq.(18) can be expressed in the polynomial form,[66]

J0 = A J00 + J02

2

+

1 X

J0;2n

2n

n=2

J1 = A J11

1

+

1 X

J1;2n

1

2n+1

n=2

J2 = A J20 + J22

2

+

1 X

n=3

where A

4

2

(

1

+

2)

1

J2;2(n

1)

!

!

;

;

2(1 n)

!

;

(38)

=3, and the coef cients Jij only depend on the electronic

Novel Thermoelectric Materials: From Quasicrystal to DNA Sample Al71 Pd21 Re8 Al70 Pd22:5 Re7:5 Al70:5 Pd21 Re8:5 Al70 Pd21:4 Re8:6 Al70 Pd20 Re10

Ref. [70] [50] [68] [69] [69]

m( VK 1.14 0.64 1.48 1.87 -0.14

e=a 1.837 1.826 1.804 1.785 1.734

19

(eV) 1 -23.43 -13.05 -30.34 -38.37 +2.79

2)

1

Table 6. Phenomenological coef cient 1 values for several i-AlPdRe samples derived from thermopower curves reported in the literature. The low temperature slope, m; is related to 2 1 1 through the expression 1 ' 20:5m[ VK ] (eV) .[67] model parameters f i ; i ; g. Explicitly, we have J20

n0 , J00 q0

3

J ; 2 20

J11 =

2 (n0 q1 q02

n1 q 0 ) ;

where n0 "21 "22 , q0 n0 " ( 1 + 2 ) 1 , n1 = 1 "22 + 2 "21 , and q1 = ( 1 2 + 1 2 2 2 + 2 and " with "2i 1 2) ( 1 + 2 "2 . 1 "1 + 2) i i In the low temperature regime the kinetic coef cients reduce to the zeroth order terms in Eqs.(38) and l T 3 , so that the c2 J0 ' T 2 and J12 T 2 terms become negligible with respect to the J0 J2 constant term in Eq.(19) and we get ZT ' 4b 21 T 2 , where b e2 L0 ' 2:44 10 8 (eV)2 K 2 , and 1

4 1 1 "2

+ ""41 "42

4 2 2 "1

=

d ln (E) dE

1 2

;

(39)

EF

measures the slope of the DOS close to EF .[64] The value of 1 can be experimentally determined from the Seebeck coef cient slope at low temperatures from the expression S(T ) ' 2jej 1 b 1 T .[67] For the sake of illustration in Table 6 we list some representative values. As we can see the steeper slopes occur for samples with an e=a ratio close to 1:8. 2 T 2, J ! On the other hand, at high enough temperatures we have,[66] J0 ! AkB 1 2 2A(q1 T4 1 2 )kB T , J2 ! 21AbT =5, and l ! 0, so that the term J0 J2 dominates the denominator of Eq.(19), and we obtain ZT =

20 21b

+ 1+

1 1

2 2 2

2

1 : T2

(40)

The reliability of Eq.(40) can be estimated from the high-temperature ZT values derived from the electronic model parameters listed in Table 7 for suitable representatives of QCs and related metallic alloys exhibiting complex structures. In fact, these values compare well with the experimentally reported high-temperature ZT values corresponding to the samples i-Al71 Pd20 Mn9 (ZT ' 0:02 at T = 973 K),[52] and i-Al71 Pd20 Re9 (ZT ' 0:05 at T = 950 K).[55] Since ZT is a continuous function of T , the functional dependence of the FOM obtained in the low- and high-temperature limits guarantees that ZT must attain a maximum value

20 Sample i-Al63 Cu25 Fe12 1/1-Al73:6 Mn17:4 Si9 -Al73 Pd22:9 Mn4:1 -Al72:9 Pd22:9 Mn4:2

Ref. [57] [64] [65] [66]

1

1:07 0:21 0:72 0:83

(meV) 5 23 83 102

2

(meV) 16 29 72 50

1

(meV) 587 65 134 86

2

(meV) 55 22 85 81

ZT 0:001 0:015 0:046 0:049

Table 7. Thermoelectric gure of merit at T = 103 K for several complex metallic alloys calculated from their electronic structure model parameters making use of Eq.(40). The cubic approximant phase 1=1-AlMnSi has a unit-cell volume of 2 nm 3 , containing N = 138 atoms. The - and - AlPdMn phases have an orthorhombic unit-cell composed of McKay clusters as basic building blocks. Their unit-cell volume are V = 4:81 nm3 and V = 22:22 nm3 , containing N = 320 and N = 1500 atoms, respectively.[35] at some intermediate temperature. Making use of Eqs.(15)-(17) we can express Eq.(2) in the form 1 1 J0 J2 kl ZT = ; (41) 1 1 + ke J12 where the rst factor only depends on the charge carrier transport properties. Now, according to several measurements l T n (with n ' 1:5) in the 100 300 K interval for most QCs, whereas e T 3 above 200 K,[71] so that the ratio l = e T 3=2 progressively reduces as the temperature increases and Eq.(41) reduces to, J0 J2 J12

ZT '

1

1

;

(42)

so that electronic structure effects, described by the reduced kinetic coef cients Ji , play the major role in determining ZT in the intermediate temperature regime. Thus, by keeping terms up to 2 in Eq.(38), Eq.(42) takes the form ZT =

1 + (A2

A1 T 2 A1 )T 2 +

105 2 4 ; 676 A2 T

(43)

where A1 4b 21 and A2 = 26bJ02 =(5J20 ), and we have used the relation J22 = 2 7 J02 =5. Assuming A2 A1 , the largest ZT value will be obtained when A1 = A2 ,[72] hence minimizing the denominator in Eq.(43), which reduces to ZT =

4b 21 T 2 ; 420 2 4 4 1 + 169 b 1T

(44)

so that, for a given temperature, the FOM is an even function of 1 , vanishing when 1 = 0, and attaining two well de ned peaks at both sides of the minimum (Figure 7 inset). These features can be understood in terms of the location of the Fermi level in the sample's electronic structure as it is illustrated in Figure 8. In fact, since 1 depends on the slope of the DOS close to the Fermi level according to Eq.(39), when the Fermi level is located close to the pseudogap minimum we get 1 ' 0. If the Fermi level shifts to the left (right) from the pseudogap's minimum, we get progressively larger ZT values as the DOS slope at the

21

ZT

0

200

400

600

1 .6 0

1 .6 5

e /a

1 .7 0

0 .0 -3 0 - 2 0 -1 0

0 .4

0 .8

1 .2

ξ1

0

10

1 .7 5

20

30

1 .8 0

1 .8 5


σ S (µ W m K ) 2

-1

-2

Figure 7. (Main frame) Composition dependence of the room temperature thermopower factor of several i-AlPdRe samples.[73] (inset) Dependence of the FOM on the 1 parameter value as determined from the Eq.(44).

Fermi level steepens, eventually attaining ZT ' 1:27 around 1 ' 17 at room temperature. This a signi cantly large gure, comparable to those reported for current benchmark TEMs. Quite interestingly, the overall behavior of the ZT ( 1 ) curve shown in the inset of Figure 7 closely correlates the composition dependence of the room temperature thermopower factor reported for several i-AlPdRe samples,[73] which are shown in the main frame. Since small changes in the chemical composition of icosahedral samples give rise to a relative shift of their corresponding Fermi level positions, the purported dependence of thermoelectric power factor with e=a is clearly indicating the signi cant role of electronic structure effects in their thermoelectric response. In fact, the high temperature dependence of the Seebeck coef cient of i-AlPdRe samples as a function of e=a was recently discussed by Kirihara and Kimura, concluding that the variation of the S(e=a) curve arises from a combined effect involving a Fermi level shift and a deepening of the DOS pseudogap due to covalent bond formation.[45]

2.4.

Improving the thermoelectrical performance

The analytical results obtained in the previous section assume that the ratio kl =ke is almost negligible in Eq.(41) in the considered temperature range, which, in turn, leads to the symmetric ZT ( 1 ) = ZT ( 1 ) curve given by Eq.(44) (Figure7 inset). In Figure 9 we plot the ZT curve as a function of the phenomenological coef cient 1 value at different temperatures, as derived from Eq.(19) for a suitable choice of the model

0 .0

F

-0 .5

0 . 000

0 . 005

0 . 010

ξ

1

< 0

E

E (e V)

ξ

1

= 0

0 .5

22

( cm ) σ( E ) Ω -1

6 0 0 .0 0 -6 0

0 .0 1

0 .0 2

0 .0 3

0 .0 4

0 .0 5

0 .0 6

-4 0

-2 0

ξ1

0

-1

(e V )

2 0

4 0

3 0 0 K

4 0 0 K

5 0 0 K

Figure 8. Sketch illustrating the relationship between the slope of the DOS close to the Fermi level and the main ZT curve features.

Z T

Figure 9. Dependence of the thermoelectric gure of merit as a function of the phenomenological coef cient 1 at T = 300 K (solid line); T = 400 K (dashed line), and T = 500 K (dot-dashed line) for an icosahedral AlPdRe sample with = 30 ( cm) 1 and l = 1:1 Wm 1 K 1 (Adapted from ref.[73]).

23

40 ξ1

0

ZT ( ξ1 )

-40 -60

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Room T

-20

ξ1

0.00 -60

0.02

0.04

0.06

0.08

0.10

0.12

-40

20

-20

0

20

40

60

60


s (ξ1 )

Figure 10. Room temperature dependence of the compatibility factor (main frame) and the thermoelectric gure of merit (inset) as a function of the phenomenological coef cient 1 for i-AlPdRe (dashed line), and i-AlPdMn (solid line).

parameters. As we can see, the inclusion of a non-vanishing thermal conductivity lattice contribution results in a clearly asymmetric ZT ( 1 ) curve, which exhibits a deep minimum anked by two maxima at about 1 ' 25 (eV) 1 and 1 ' +40 (eV) 1 . Note, however, that the minimum of the ZT ( 1 ) curve does not coincide with 1 = 0 in this case. As the Fermi level progressively shifts from the pseudogap's minimum (due to a systematic change in the sample stoichiometry, for instance), the ZT values progressively increase attaining well de ned maxima as it occurred in the symmetric case, but now these maxima reach different peak values depending on the sign of 1 . Accordingly, we conclude that best thermoelectric performances will be expected for those stoichiometries able to locate the Fermi level below the minimum of the pseudogap, and that the deeper the pseudogap the larger the resulting gure of merit at a given temperature (see Figure8). Finally, we observe that as temperature increases the ZT ( 1 ) maximum below (above) the Fermi level progressively increases (decreases) and shifts towards (away from) the value 1 = 0: This behavior indicates that the precise stoichiometry to get an optimal thermoelectric performance will depend in general on the working temperature for the considered sample. Similar results are obtained for the compatibility factor obtained from Eq.(3). In Figure10 we compare the room temperature compatibility factors of i-AlPdRe ( = 0:7 Wm 1 K 1 ;[45], 0 = 30 ( cm) 1 ;[73]) and i-AlPdMn ( = 1:6 Wm 1 K 1 , 0 = 740 ( cm) 1 ;[70]) as a function of 1 :The difference between the AlPdRe and AlPdMn s( 1 ) curves is due to the signi cant difference between their respective residual conductivities. In the inset of Figure 10 we plot the corresponding ZT ( 1 ) curves, which exhibit a deep minimum, anked by two maxima. Both ZT and s vanish (see Eq.(3)) at 01 = +5:76

24 (eV) 1 : Consequently, QCs can exhibit p-type ( 1 < 01 ) or n-type ( 1 > 01 ) thermopowers depending on the 1 value which, according to Eq.(39), is very sensitive to the sample's electronic structure near EF . Since the 1 value can be controlled by changing the sample stoichiometry, hence shifting EF in a scale of a few meV, we can con dently expect that larger values of the room temperature compatibility factor, close to s = 2:0 V 1 , may be attained in AlPdMn QCs with 1 ' 15 (eV) 1 . Consequently, it seems reasonable to expect that, by a judicious choice of both sample composition and processing and annealing conditions, higher FOM values, comparable to the gures currently obtained for good TEMs (or even better) may be ultimately reached by following a systematic research. Although such a possibility is just a tentative one, we deem it is based on ground physical basis, particularly if one keeps in mind the purported sensitivity of the relevant transport coef cients of QCs and related phases to different preparation techniques.[70]

3. 3.1.

DNA based thermoelectric devices Physical motivations

In this Section we present a prospective study on the possible design of DNA-based Peltier nano-cells. In order to properly substantiate this proposal in the rst place we shall brie y summarize most basic properties of polymeric materials of thermoelectric interest. These materials are usually prepared as thin lms or pellets which are sandwiched between metallic plates. Afterwards, we will turn our attention to some recent experiments measuring thermoelectricity in molecular junctions at the nano-scale. From the basic knowledge gained from these general studies we will then describe the main features corresponding to the DNA case. 3.1.1.

Thermoelectric properties of polymer materials

In contrast to metals, polymers are typically insulators, which according to Eq.(2) prevent them from thermoelectric applications, though polymers inherently possess a very low thermal conductivity, within the range ' 0:1 0:5 Wm 1 K 1 at room temperature.[74] Fortunately enough, during the last two decades new classes of polymers have been synthesized that are capable of carrying unusually high electrical currents. For instance, a doped form of the polyacetylene molecule (as a consequence of an acidic treatment, for instance) was reported to have an electrical conductivity comparable to that of certain semiconducting 1 cm 1 ).[75] Other polymers such as polyaniline, polypyrmaterials (i.e., 6 104 role, or polythiophene were found to have high electrical conductivities when chemically doped in a proper way as well.[76, 77, 78] Along with the electrical conductivity, Seebeck coef cient values have also been reported in the literature for polyacetylene, polyanilines, polypyrroles, and polythiophenes (Table 8). The best conjugated polymer up to now is polyacetylene (ZT ' 0:6 at room temperature) due to its good electrical conductivity and relatively high Seebeck coef cient. Nevertheless it has poor stability in the (required) doped state, even in an inert atmosphere. The other considered polymers show better stability in the doped state, but lower Seebeck coef cient and electrical conductivity, which leads to small estimated FOM values.

Novel Thermoelectric Materials: From Quasicrystal to DNA Sample ( polyacetylene polyaniline polythiophene polypyrrole

1 cm 1 )

4

104 7000 100 200

S ( V K 1) +16 +7 +22 +7

P ( Wm 1 K 103 34 4:8 0:01

2)

25

ZT (estimated) 0:6 0:02 1 10 3 6 10 6

Table 8. Room temperature values of the transport coef cients, power factor and FOM for different polymer materials. (After Ref. [79]) The FOM value is estimated by assuming ' 0:5 Wm 1 K 1 as a representative gure for these materials. Positive values of the Seebeck coef cient are obtained for all considered (doped) samples, indicating that the charge carriers are holes.

Figure 11. Structure of some polycarbazole derivatives proposed as potential TEMs. (Adapted from Refs. [80] and [81])

On the basis of theoretical band structure calculations, indicating the presence of both at and dispersive bands close to the Fermi level (see Section 1.3), some conducting polymers, like polythiophene, polyaminosquarine and polycarbazole doped derivatives have been recently considered as potential TEMs (Figure 11).[15, 80, 81] These polymeric materials were either synthesized as thin lms or pressed into a pellet and cut as a rectangle. The samples were then sandwiched between copper plates covered with thin gold sheets for a simultaneous measurement of the Seebeck coef cient and the electrical resistivity (twoprobe) in air at room temperature. The experimentally measured thermoelectric parameters are listed in Table 9. By comparing the data listed in Tables 8 and 9 we see that, though polycarbazole derivatives exhibit signi cantly larger S values, their electrical conductivity is far too small to yield ZT gures of practical interest.

26 Sample ( 27PCT2 PCDT PCDBT PCDTBT

1 cm 1 )

0:3 23 87 160

S ( V K 1) +71 +53 +40 +34

P ( Wm 1 K 0:15 6:5 14 19

2)

ZT (estimated) 10 4 0:004 0:008 0:011

Table 9. Room temperature values of the transport coef cients, power factor and FOM for different polycarbazole derivatives whose chemical structure is shown in Fig. 11. The 27PCT2 compound was prepared in form of a pellet whereas the remaining samples were prepared as thin lms. The FOM value is estimated by assuming ' 0:5 Wm 1 K 1 as a representative gure for these materials. Sample BDT 2,5 dimethyl-BDT 4F-BDT 4Cl-BDT B-CN DBDT TBDT TPT

Ref. [21] [21] [21] [21] [21] [20] [20] [22]

S ( V K 1) +7:0 0:2 +8:3 0:3 +5:4 0:4 +4:0 0:6 (1:3 0:5) +12:9 2:2 +14:2 3:2 +16:9 1:4

Table 10. Room temperature (T = 293 K) thermopower for different molecules whose chemical structure is shown in Fig. 12. 3.1.2.

Thermoelectricity in molecular junctions

Several studies have demonstrated that structural parameters can in uence the charge transport properties in conjugated polymers. For instance, relative orientation and solid-state spatial organization of the polymer network may play a crucial role due to strong interchain interactions facilitating the transport of charge carriers through the polymeric structure. In order to gain information about the intrinsic transport properties of the molecules out of which a polymeric material is made it is convenient to study single-molecules trapped between appropriate contacts, thereby forming a molecular junction. In a series of recent experiments the thermoelectric properties of molecular junctions containing different benzene related moieties chemically bond to gold nanocontacts with thiol end groups have been investigated with a suitably modi ed scanning tunneling microscope (Figure 12) The experimentally measured thermoelectric parameters (in air at ambient conditions) are listed in Table 10. A clear increase of the Seebeck coef cient with the molecule length is appreciated in contrast with the measurements performed in polymer lms, were no signi cant dependence was appreciated among polycarbazole derivatives.[80] Positive values of the Seebeck coef cient are obtained for all considered molecules when contacted through thiol groups, indicating that the charge carriers are holes in this case. On the contrary, a negative value is obtained for a benzene molecule contacted to gold electrodes with cyanide-groups.


27

Figure 12. a) Single molecule molecular junction setup; b) structure of the benzene derivatives; c) structure of the polybenzo-dithiol and TPT derivatives. (Adapted from Refs. [20]-[22])

Thus, end-groups are key to controlling the very nature of charge carriers. In addition, by properly varying end-roups and molecular junction constituents one can engineer metalmolecule heterostructures with targeted thermoelectric properties. 3.1.3.

The DNA case

In this section we shall consider the physical motivations inspiring the nano-Peltier cell sketched in Figure 13. As we will see in more detail through the following sections these motivations are basically two-fold [82]. In fact, experimental current-voltage curves show that: periodically ordered, synthetic DNA chains like polyG-polyC or polyA-polyT exhibit a semiconducting behavior, double-stranded poly(dA)-poly(dT) chains behave as n-type semiconductors, whereas poly(dG)-poly(dC) ones behave as p-type semiconductors.[83] Thus, charge transfer mainly proceeds via hole (electron) propagation through the purine (pyrimidine) bases, where the HOMO (LUMO) carriers are respectively located in polyGpolyC (polyA-polyT) chains. Accordingly, these synthetic DNAs may provide the basic building blocks necessary to construct a nanoscale thermoelectric cell, where the DNA chains will play the role of

28

Figure 13. Sketch illustrating the basic features of a nanoscale DNA based Peltier cell. A polyA-polyT (polyG-polyC) oligonucleotide, playing the role of n-type, left (p-type, right) semiconductor legs, are connected to organic wires (light boxes) deposited onto ceramic heat sinks (dark boxes).

semiconducting legs in standard Peltier cells. From an experimental viewpoint the possible use of DNA-related molecules in the design of nanoscale thermoelectric devices was opened up by the measurement of an appreciable thermoelectric power (+18 V K 1 at room temperature) over guanine molecules adsorbed on a graphite substrate using a STM tip.[84] This gure is larger than those reported for benzene-dithiol derivatives in Table 10, albeit guanine molecules were deposited onto a substrate (physorption) rather than being chemically connected to it as in the case of molecular junction measurements. In any event there certainly exists a very long way from thermoelectric measurements performed at the single nucleotide scale to the full- edged helicoidal structure of duplex DNA chains we are interested in (Figure 13). In the next Section we further elaborate on these physical motivations by brie y reviewing most relevant features of both the electronic structure and transport properties of synthetic nucleic acids.

3.2.

Electronic structure and transport properties of DNA molecules

A number of ab-initio calculations based on the density functional theory have been performed during the last decade. The cases of the homopolymers poly(dG)-poly(dC) and poly(dA)-poly(dT) have been extensively considered,[85, 86, 87, 88, 90] along with some related structures like poly(GC)-poly(CG).[89] In order to reduce the computational effort earlier calculations did not explicitly take into account either the water shell or the cations around the sugar-phosphate backbone. Accordingly, these preliminary works focused on the dry A-DNA electronic structure. Close to the Fermi level it shows well de ned, narrow bands separated by a broad gap (2-3 eV). The valence bands in A-poly(dG)-poly(dC) and


29

Figure 14. (a,b) Energy bands close to the Fermi level as a function of the wave vector k of a polyG-polyC molecule in dry conditions. In the plot results obtained from ab-initio calculations (dots) are compared to those derived from a one-dimensional tight-binding model with one orbital per unit cell (curve). indicates the HOMO-LUMO gap, i the gap between closest orbitals in the guanine system (relevant to optical transitions), and WH(L) are the HOMO (LUMO) bandwidths, respectively.[85] (c)Surfaces of constant charge density for the states corresponding to the lowest unoccupied band (red) and highest occupied band (blue) of a polyG-polyC molecule in the A form in dry conditions.[85] (Courtesy of Emilio Artacho).

A-poly(dA)-poly(dT) consist of 11 states, that is, one per base pair in the unit cell. In the case of poly(dG)-poly(dC) the topmost valence band has a very small bandwidth (Figure 14a). This band is associated with the -like HOMO of the guanines. The charge density of the states associated with this band appears almost exclusively on the guanines, with negligible weight either in the backbones or in the cytosines (Figure 14c). The lowest conduction band is signi cantly broader and it is made of the LUMO of the cytosines. Similar results are obtained for A-poly(dA)-poly(dT) chains, where the charge density appears concentrated on the HOMO orbitals of the adenines and exhibit a broader valence band width ( 0:25 eV).[87] A remarkable characteristic of the doping behavior of the polycarbazole molecules discussed in the previous section is that the nitrogen atom is oxidized prior to the molecular skeleton. In this way, the charge is very localized, which results in the formation of narrow impurity bands leading to a relatively large Seebeck coef cient. However, this can also represent a problem due to charge-pinning effects, which adversely affect the conductivity. The goal is to get a balance in the level of oxidation of the polymer such that a certain degree of localization is maintained while still allowing for a signi cant charge carriers mobility. Quite interestingly, an analogous situation occurs in the DNA case, where we have a number of localized states in the HOMO-LUMO gap due to the presence of Na-water ions around phosphate groups (Figure 15). In fact, the phosphate groups of the DNA molecule are negatively charged. Hence positive protons or metal cations (usually referred to as coun-

30

Figure 15. Schematic energy level diagram around the Fermi level of a fully hydrated double-stranded polyG-polyC molecule in the Z conformation. The Fermi level positioned in the middle of the gap has been chosen as the zero of energy. The HOMO (located at about -0.6 eV) is associated to guanines. The states immediately below the top of the valence band are also related to G. The rst C localized state is located at 0.78 eV below the top. The bottom of the conduction band is a charge transfer state related to Na+ counterions and PO4 groups. The rst excited state with a strong C base character ( ) is located at 2.85 eV above the Fermi level and the rst G state is at 3.18 eV. The ! gap is 3.94 eV for cytosine and 3.82 eV for guanine bases. (Reprinted with permission from Gervasio F, Carloni P and Parrinello M 2002 Phys. Rev. Lett. 89 108102 c 2002 by the American Physical Society).

terions) are necessary to neutralize and stabilize DNA in physiological conditions. Water also plays a crucial role to this end. Hydrophobic forces compel DNA to adopt the B-form, and the polarity of the water molecules helps to screen DNA's charges. The comparison of the electronic structures corresponding to dry DNA structures with those obtained for wet conditions shows that the LUMO location is quite sensitive to the environment conditions. Thus the inclusion of Na+ cations evenly distributed through the backbone gives rise to the presence of a band related to the Na-phosphate groups between the -electron bands of the base molecules, so that the LUMO moves from cytosines to the phosphate-cations system when in presence of Na+ for both A-poly(dA)-poly(dT) and A-poly(dG)-poly(dC).[87] Accordingly, the water shell and the counterions can lead to the presence of a number of states in the main energy gap (which can be regarded as impurity states), hence effectively doping the DNA molecule. Nevertheless, the mobility of the charge carriers, proceeding through the overlapping of the orbitals of consecutive base pairs along the helical axis, is not appreciably affected by the presence of these states. Accordingly, the presence of very at bands can be fully exploited to improve the Seebeck coef cient in this case (see Section 1.3). The main features of the electronic structure obtained from numerical results have been experimentally con rmed by means of some spectroscopic techniques.[91, 92, 93] In particular, it has been con rmed that the HOMO originates in the DNA bases, in agreement with numerical calculations, for both polyG-polyC and polyA-polyT duplexes forming a mixture of A- and B-DNA forms.[91] It has been also demonstrated that when holes are


31

doped in polyG-polyC by chemical oxidation the doped hole charge is localized on G, but not on cytosine, deoxyribose, or phosphates.[93] At the same time, a number of experimental, systematic measurements aimed to directly probe the electric current as a function of the potential applied across the DNA molecules have been reported during the last few years.[94, 95, 96, 97, 98] Two representative experimental layouts are shown in Figure 16. A nanoelectronic platform based on single-walled carbon nanotubes was fabricated for measuring electrical transport in single-molecule single-strand DNA and double-strand DNA samples of a 80 base pair long DNA fragment.[98] To enhance the contact ef ciency a covalent bonding between an amine-terminated DNA molecule and a carboxyl-functionalized carbon nanotube was established and the DNA molecule was suspended over a nanotrench in order to mitigate the problem of compression-induced perturbation on the charge transport. A nonlinear I-V characteristic curve was observed indicating a semiconducting behavior (gap width 1 eV, p-type conduction) in both aqueous (sodium acetate buffer) and vacuum (10 5 torr) conditions. A current of about a 25-40 pA (0.5-1.5 pA) at 1V bias was measured for double-strand DNA (single-strand DNA) duplexes, respectively, at ambient conditions. The resistance increased in both cases in vacuum, presumably due to the depletion of water molecules in the hydration shell surrounding DNA (as well as possible conformational changes in the double helix at high vacuum). From basic principles it is expected that a single-strand DNA molecule will carry only a feeble current due to lack of structural integrity. This has been experimentally con rmed in a systematic way by comparing the single-molecule conductance of short thiolated single-strand DNA and double-strand DNA homopolymers in aqueous solution (sodium phosphate) at room temperature. In this way, it has been reported that the conductance measured for 5'-C6 S-(dG)15 -(dC)15 -C6 S-3' duplexes (G = 1:4 10 6 G0 ) between gold metallic contacts compares well with that measured for 5'-C6 S-(dG)7 -C6 S-3' single-stranded chains (G = 1:6 10 6 G0 ) at 0.2 V bias potential. Accordingly, the conductance of the double-stranded structures is about an order of magnitude higher than that of single-stranded ones with similar number of bases (such a conductivity difference is signi cantly greater for oligo-dC, oligo-dT, and oligo dA chains). This observation clearly demonstrates that the interactions between the base pairs and stacking effects play a vital role in charge transport through DNA.[97] Making use of the experimental set-up shown in Figure 16d measurements were performed on DNA duplexes of the form 5'-CGCG(AT)m CGCG-3', where some GC base pairs are replaced by AT ones, in order to analyze sequences effects on the transport properties. The conductance data can be described by an expression of the form G = Ae L ; where L is the length of the AT bridge, with A = (1:3 0:1) 10 3 G0 and = 0:43 0:01 Å 1 .[96] These ndings are consistent with a tunneling process across AT regions between the GC domains, in good agreement with the idea that HOMO guanine orbitals favour charge migration, whereas short A-T sequences create a tunneling barrier for charge hopping through guanines along the DNA stack. In summary, the reported experiments demonstrate the high sensitivity of DNA electrical conductivity to several factors. Firstly, we have the structural complexity of nucleic acids, which is signi cantly in uenced by its close surrounding chemical environment (humidity degree, counterions distribution) affecting the integrity of the base-pair stack, as well as by the unavoidable presence of thermal uctuations. Secondly, the kind of order present in the DNA macromolecule plays an important role in determining its transport character-

32

Figure 16. Schematics illustrating a method to chemically attach duplex DNA strands with molecular nanocontacts. (a) Functionalized point contacts made through the oxidative cutting of a single walled nanotube wired into a device (b) bridging by functionalization of both strands with amine functionality (c) bridging by functionalization of one strand with amines on either end. (Reprinted with permission from Nacmillan Publishers Ltd.: Nature Nanotechnology 3 163 c 2008). (d) The contact is formed through a thiolated chemical bond between the electrode (Au) and the DNA molecule, whose 3' end has been modi ed with a C3 H6 SH linker. In the same buffer solution a gold STM tip, which is covered with an insulating layer over most of the tip surface except for its end, is brought into contact. Once contact is formed the tip is pulled backwards and the resulting current is monitored with a piezoelectric transducer.(Reprinted with permission from ref.[96]. Copyright (2004) American Chemical Society).


33

istics: periodically ordered polyG-polyC chains exhibit semiconducting behavior, whereas biological -phage chains are more insulating. Finally, measuring charge transport in a DNA chain is strongly biased by the invasive role of contacts, the charge injection mechanism, the quality of the DNA-electrode interface, and the possible interaction with some inorganic substrate, or other components of the experimental layout.

3.3. 3.3.1.

Thermopower of single-stranded oligonucleotides Analytical expressions

Let us consider an experimental layout similar to that shown in Figure 12a where the benzene derivatives are replaced by properly functionalized (v.g., thioled) nucleobases adenine, guanine, cytosine, thymine and uracil or short oligonucleotides made from different combinations of these bases. As a rst approximation, the resulting contact-molecule-contact arrangement can be described within the mathematical framework introduced in Sec.1.2.2 in terms of the Hamiltonian given by Eq.(20). Single nucleobase In the case of a single nucleobase the transmission coef cient given by Eq.(26) reads,[100] T (E) = 1 +

2

W

1

(E)(E

)2

1

(45)

;

2 where = fG,A,C,Tg labels the considered nucleobase, 1, with =tM ; measuring the coupling strength between the nucleobase and the leads, is a coupling factor which vanishes when = tM (since tM > ) > 0) , W (E) (E E )(E+ E); with E = "M 2tM ; de nes the allowed spectral window as determined by the leads bandwidth, and 2 " "M : (46)

de nes a resonance energy satisfying the full transmission property T ( ) = 1. According to Eq.(45) the transmission amplitude is modulated by the lead-nucleobase coupling strength through the factor : The particular case = 0 (T (E) = 1) corresponds to metallic conduction over the molecule. In that case, a very small thermoelectric voltage is expected after Eq.(28). Consequently, we will consider the general case 6= 0. Making use of Eq.(45) into Eq.(28) we obtain,[100] S (T ) = 2jejL0 T

1+

ba cd

a 2 cd + a2

2

;

(47)

where a EF ; b EF "M ; c E+ EF ; and d = EF E : As expected, in the cases = 0 or EF = (i.e., a = 0) we have a vanishing thermopower. Conversely, the Seebeck coef cient asymptotically diverges as EF approaches the band edges (i.e., c = 0 or d = 0): Therefore, very large thermopower values can be eventually reached by properly shifting the Fermi level through the electronic structure of the system.

34 Dimer nucleobase In the case of a dimer oligonucleotide we have nucleobases of energies "n and "m ; respectively coupled with a hopping term tnm between them. Making use of Eq.(26) one obtains, 2 Tnm (E) = 1 + qnm +

2 nm W

1

(E)Q2nm (E)

1

(48)

;

where, qnm

j"n "m j ; 2tnm

is a measure of the dimer's chemical diversity, Qnm (E) = (

4

2 2 nm )tM

+ (E

(49) 2

nm

"n )(E

1 , and tnm

2

"m )

(E

"nm )(E

"M );

(50)

where nm tnm =tM measures the coupling strength between the bases in units of the lead bandwidth, and "nm ("n + "m )=2. Thus, in the case of homomers (i.e., GG, AA, CC or TT) we have qnm = 0 and, according to Eq.(48), we get full transmission (T = 1) under resonance conditions corresponding to those energies satisfying Qnm (E) = 0: Conversely, 2 ) 1 < 1; under the same when both bases are different in nature we get Tmax = (1 + qnm conditions. For a given qnm value the transmission amplitude is modulated by the nm ratio factor, involving contributions from both contact-molecule coupling strength ( ) and orbital overlapping between both bases (tnm ). At variance with the single base case, in the dimer case it is not possible to get nm = 0; so that the transmission through a dimer base will be in general lower than that corresponding to a single base. Making use of Eq.(48) into Eq.(28) we get the Seebeck coef cient corresponding to the dimer case as Snm (T ) = 2jejL0 T 0

where Qnm (EF )

2 nm

0

cdQnm (EF ) + bQnm (EF ) Qnm (EF )Tnm (EF ) ; (cd)2

(51)

(dQ=dE)EF .

Trimer nucleobase In the case of a trimer oligonucleotide we have three nucleobases of energies "k ; "n and "m (the base labeled n occupying the central position and those labeled k and m located at the left and right sides, respectively, see Figure 1), coupled with hopping terms tkn and tnm tkn ; and the transmission coef cient takes the form, Tknm (E) = 1 + 4t2M

2 kn W

1

(E)[Pknm + f Qknm ]2

1

;

(52)

2 where kn ( kn ) 2 1 , Pknm = 4 xn + Kknm f Q+ knm , Kknm = xk Hnm xm kn , 2 2 2 2 Qknm (Jkn Hnm )=2; with Jkn 4xk xn kn and Hnm = 4xn xm kn , and we have introduced the auxiliary variables 2x = (E " )=tM and 2f = (E "M )=tM . In this case we have three possible resonance energies given by the relationship Pknm + f Qknm = 0. Certain symmetry relations follow from Eq.(52), which further simplify the thermopower expression. Thus, for homomers of the general form XXX, where X stands for a given 2 , arbitrary base, we have xk = xn = xm x, = 1, and one gets Jnn = Hnn = 4x2 nn 2 2 ), so that the transmission coef cient Q+ Jnn , Qnnn = 0, and Knnn = x(Jnn nnn = nn simpli es to 1 2 Tnnn (E) = 1 + 4t2M 2nn W 1 (E)Pnnn (E) ; (53)


35

where Pnnn (E) = 4x3 4 2 x2 f + ( 4 2 2nn )x + 2 2nn f . The roots of polynomial Pnnn (E) determine the energy values of the full transmission peaks. By plugging Eq.(53) into Eq.(28) we obtain Tnnn (EF )Pnnn (EF ) [cdVnnn (EF ) + tM bPnnn (EF )] ; (cd)2 (54) 2 2 where Vnnn 4(3 )x 8 2 xf + 4 2 2nn + 2 2nn . As we see, the thermopower value ispmodulated by the nn value, so that the condition nn > 1 leads to the relation tM > t in order to obtain a good thermopower signal. The Seebeck coef cient also depends on the the transmission coef cient value, and identically vanishes when Tnnn = 1, since Pnnn = 0 in this case (see Eq.(53)). In a similar way, for trimers of the general form XYX we have xk = xm x, = 1, 2 2 , Q+ and one gets Jnn = Hnn = 4x2 xn Jkn , Qknk = 0, and Knnn = kn knk = 2 x(Jkn nn ), so that the transmission coef cient takes the same functional form as that given by Eq.(53) by simply replacing Pnnn (E) by Pknk (E) = 4x2 xn 4 2 xxn f + 4 xn 2 2 x + 2 2kn f , and the Seebeck coef cient takes the form Snnn (EF ) = 4jejL0 T tM

2 nn

Sknk (EF ) = 4jejL0 T tM

2 kn

where Uknk 3.3.2.

4x(x + 2xn

Tknk (EF )Pknk (EF ) [cdUknk (EF ) + tM bPknk (EF )] ; (cd)2 (55) 2 xn ) 4 2 (x + xn )f + 4 2 2kn + 2 2kn .

Transport curves

Single nucleobase case The resulting thermopower curves are shown in Figure 17. In the inset of Figure 17 we compare the transmission coef cient of different nucleobases as a function of the Fermi energy position. The curves are very similar in shape, exhibiting a well de ned maximum at the resonance energy E = "M + 4 2 =(" "M ), which corresponds to S = 0 in the thermopower curve. In fact, as the Fermi level approaches this resonance energy from above we get positive values of the Seebeck coef cient, which vanishes at E = , and changes its sign as the Fermi level further shifts towards the spectral window bottom edge. The location of the contact's band center "M marks two distinct behaviors. For EF > "M , all the considered bases have very similar values of thermopower (S ' 5 VK 1 ) over a wide energy range up to E ' 5 eV, where the thermopower curve suddenly rises as EF progressively approaches the spectral window upper edge. On the contrary, due to resonance effects involving the molecular orbitals of the bases, in the energy range 11 . E . 9, the thermopower curves corresponding to each nucleobase are clearly discernible from each other, and relatively large (negative) values of thermopower can be reached when EF ! E . We have checked that the thermopower curves are very sensitive to the adopted coupling strength value, exhibiting the following trend: As the contact-molecule coupling weakens the transmission curves progressively narrow and their resonant peaks shift towards the "M reference value. In addition, the thermopower curve develops two well-de ned features (see Figure 18) around the resonant energy , peaking at values within the range 30 40 VK 1 in absolute value.

-6 -7 -9 -10

-20 -11

-10

0

10

20

T = 300 K

-8

E

0.0

0.5

1.0

T (E)

εM

-11

-10

E

-9

-5

-8

εM

-7

-4

36

S ( µV K ) -1

Figure 17. Room temperature dependence of the Seebeck coef cient as a function of the Fermi energy for the nucleobases G, A, C and T (from right to left) with = 1:0 eV, tM = 2:0 eV, and "M = 7:58 eV. (inset) The transmission coef cient as a function of the Fermi energy for he nucleobases G, A, C and T (from right to left). The vertical dashed lines indicate the position of the contact's band center "M .

37

-11

-40

-20

0

20

40

-10

-9

-8

E

εM

-7

-6

-5

T = 300 K

-4


S ( µV K ) -1

Figure 18. Room temperature dependence of the Seebeck coef cient as a function of the Fermi energy for the nucleobases G, A, C and T (from right to left) with = 0:5 eV, tM = 2:0 eV, and "M = 7:58 eV. The vertical dashed line indicates the position of the contact's band center "M .

E -11

-8 -9 -10 -11

T = 300 K -30

-15

0

T 15

30

0.0

0.5

T (E)

1.0

E

-7

T

-6

-10

T-T

T-T

-5

-9

-4

38

S ( µV K ) -1

Figure 19. Room temperature dependence of the Seebeck coef cient as a function of the Fermi energy for TT and T nucleobases with = 1:0 eV, tm = 2:0 eV, tT T = 0:4 eV, and "m = 7:58 eV. (inset) Transmission coef cient as a function of the Fermi energy for T and TT bases. The vertical arrows indicate the location of the resonance energy values (inset) and the related peaks in the thermopower curve (main frame).

Dimer nucleobase case In this case we are mainly interested in the dimerization effects on the transport properties. In Table 11 we list some representative tnm values derived from quantum chemistry/molecular dynamics calculations along with the corresponding values for the chemical diversity parameters qnm and "nm . By inspecting this Table we note that i) the tnm values for identical nucleobases (XX) are larger for TT and GG than for AA and CC, and ii) among heterodimers the largest transfer integral is obtained for GT, which also has the largest chemical diversity qnm value. For the sake of illustration, in the main frame of Figure19 we compare the monomer T and dimer TT thermopower curves for = 1:0 eV. Similar curves are obtained for the homomers GG, AA and CC. An overall increase of the dimer thermopower is clearly appreciated over a wide energy range, in agreement with experiments (see Table 10),[21] and exhibits a systematic behavior, increasing as the Fermi level shifts from E ' 5:5 eV towards lower energies, reaching a well-de ned peak (S ' 22 VK 1 ) at E ' 9:2 eV. A similar trend is observed for the other bases as well. In the inset of Figure19, we compare the transmission coef cients for T and TT bases, respectively. As we can see, orbital overlapping effects in the thymine dimer result in the presence of two resonance energies and the overall narrowing of the transmission coef cient, which is centered around E ' 10:2 eV. As a consequence of this narrowing effect the T (E) gets steeper, which according to Eq.(28) ultimately leads to a thermopower enhancement


39

close to the transmission resonant peaks. In addition the formation of a dimer nucleobase results in an electrical conductance reduction through dimers as compared to that observed in monomers, and this effect is signi cantly enhanced as the orbital overlapping decreases leading to smaller interbase hopping terms. DIMER TT GG AA CC GT GC GA CT AT AC

tnm (eV) 0:158 0:084 0:070 0:041 0:137 0:110 0:089 0:100 0:105 0:061

qnm 0 0 0 0 4: 67 3: 59 3: 03 2: 45 3: 52 2: 05

(eV) "T "G "A "C 9: 220 8: 975 8: 850 9: 615 9: 490 9: 245

nm

Table 11. Transfer integral values (tnm ) after Ref.[101] along with the chemical diversity parameter (qnm ) derived from Eq.(49), and mean on-site energies ( nm ) for the different nucleobase dimers.( ) Mean value [102]. A similar qualitative and quantitative behavior is observed for the heteromers (GC, GT, GA, AC, AT, and TA) as well. The general trend is that both the conductance and thermopower curves of a given heterodimer, say XY, are intermediate between those corresponding to the homomers XX and YY, respectively. This property is illustrated in Figure 20.

3.4.

Thermoelectric codon sequencing

There currently exists a growing interest in the search for new sequencing methods entirely based on physical principles able to allow for non invasive analysis of a huge number of nucleotides along the DNA strands. In this regard, scanning tunnel spectroscopy, which directly detects the molecular levels of single DNA bases, has been exploited during the last few years. In fact, nucleobase-modi ed tip STM measurements demonstrate the ability to identify the different DNA nucleobases due to selective chemical interactions, although it remains a chemically based rather than a purely physically based technique.[103] In this Section, we will consider the possibility of sequencing short DNA fragments by employing thermoelectric measurements. To this end, we shall analyze the thermoelectric performance of short DNA chains connected between metallic contacts at different temperatures in order to estimate the possibility of directly sensing triplet nucleobases associations (including codons in codifying regions) via their thermoelectric signature. In the rst place we shall consider the transport properties corresponding to GGG, AAA, CCC, and TTT codon trimers as derived from Eq.(53). In Figs.21 and 22 we respectively show the homomers transmission and thermopower curves for a suitable choice of the model parameters. The Snnn (E) curves corresponding to the GGG, AAA and CCC (not shown) trimers are characterized by two peaks and a crossing point which de nes two different regimes ex-

-10

-7 -8

0.00

-60

-40

-20

0

20

40

-11

-10

T-T

-9

G -T

E F (eV)

-11

G- T 0.25

0.50

T (E)

εM

-6

E

-9

G- G

T-T

G- G

-5

-8

εM

-7

-4

40

S ( µV K ) -1

Figure 20. Room temperature dependence of the Seebeck coef cient as a function of the Fermi energy for TT, GG, and GT nucleobases with = 1:0 eV, tm = 2:0 eV, tT T = 0:158 eV, tGG = 0:08 eV, tGT = 0:137 eV, and "m = 7:58 eV. (inset) Transmission coef cient as a function of the Fermi energy for TT, GG, and GT bases. The vertical dashed lines indicate the position of the reference contact's band center "M .

-9.5 E -9.0 F

CCC TTT

-10.0

AAA 0.5

0.0

GGG T (E)

1.0

41

-8.5


Figure 21. Transmission coef cient as a function of the Fermi energy for TTT, CCC, AAA and GGG trimers with = 0:5 eV, tM = 2:0 eV, and "M = 7:58 eV.

hibiting p-type or n-type thermopower, respectively. The Seebeck coef cient value attained at the peaks is signi cantly large (200 300 VK 1 in absolute value, the peak values for the CCC trimer are about one order of magnitude larger) and compare well with values reported for benchmark TEMs. The Seebeck coef cient of the TTT trimer exhibits two additional minor peaks between the larger ones. These additional features stem from a stronger coupling between neighboring T-T bases in the trimer (see Table 11) which gives rise to a characteristic triplet structure in the transmission coef cient shown in Figure 21. In fact, such a triplet structure is not well resolved for the other trimers due to the signi cantly weaker value of their molecular orbital overlapping. By inspecting Figure 22 two main conclusions can be drawn. First, the homomers thermopower curves peak at characteristic energy values, which are different enough to allow for a reliable energy resolution among different XXX moieties. Second, the thermopower curves corresponding to certain trimers display complementary responses among them. Thus, the AAA thermopower curve peaks within the energy range 9:1 . E . 8:6 showing a p-type behavior, whereas the GGG curve takes large negative values instead in the same energy interval. The same occurs for CCC and AAA (not shown), or TTT and AAA in the energy windows 9:3 . E . 9:1, and 9:7 . E . 9:1, respectively. In addition the ST T T (E) curve exhibits two subsidiary (positive) peaks located at an energy interval where the other three trimers take on negative values. The presence of these characteristic peaks provide an additional criterion highlighting the presence of TTT trimers. By properly combining all these characteristic features one can con dently discriminate among different homomers present in a test sample from thermopower measurements.

-9.0 -300

-200

-100

0

100

200

300

-10.5

TTT

-10.0

-9.5

AAA

EF

-8.5

-8.0

GGG

-7.5

42

S ( µV K ) -1

EF

-9.0

-8.5 EF

-10.0

-1200

0.0 -9.5

-9.5

-9.0

AGA 0.5

-800

-400

0

1.0

AAA

GGG

AAA 400

T (E)

800

-8.5

AGA

GGG

-8.0

-7.5

Figure 22. Dependence of the room temperature thermopower as a function of the Fermi level energy for TTT, AAA and GGG trimers with = 0:5 eV, tM = 2:0 eV, and "M = 7:58 eV.

S ( µV K ) -1

Figure 23. Dependence of the room temperature thermopower as a function of the Fermi level energy for AGA, AAA and GGG trimers with = 0:5 eV, tM = 2:0 eV, and "M = 7:58 eV.


43

Now we consider the thermopower curve corresponding to trimers of the form XYX as compared to those previously obtained for XXX trimers. For the sake of illustration, in Figure23 we compare the thermopower curves for the AGA, AAA and GGG trimers as a function of the Fermi energy. The presence of three crossing points satisfying SAGA = 0 indicates that the equation PAGA = 0 has three real solutions. Below the point at E = 8:55 eV the main features of the AGA thermopower curve bear some resemblance with those corresponding to the AAA curve, with the peak located at about E = 9:3 eV signi cantly reduced in the AGA case due to the broadening of the resonance feature at E = 9:18 in the TAGA (E) curve (shown in the inset). Above the crossing point we have a very pronounced peak, located close to the GGG curve peak but much more enhanced due to the extremely narrow resonant feature in the AGA transmission coef cient. On the other hand, the overall curve for GAG is completely analogous to that obtained for AGA with the role of A-related and G-related resonances interchanged. This complementary behavior is a general feature of the thermopower curve corresponding to all the trimers of the form XYX. In summary, the presence of resonance effects among electronic levels in oligonucleotides composed of three nucleobases leads to a signi cant enhancement of the thermoelectric signal. By comparing the transport curves corresponding to different types of trimers we see that a characteristic thermoelectric signature can be used to identify the XYX type codons from XXX homonucleotide ones on the basis of their different thermoelectric responses.

3.5.

Duplex DNA model

As we have seen in Section 3.2, quantum chemical and ab-initio band structure calculations reveal the existence of different subsystems in DNA, each one of them characterized by its own energy scale. Thus, the description of the electronic energetics of realistic doublestranded DNA chains, as that shown in Figure 24a, must take into account three different contributions stemming from (i) the nucleobase system, (ii) the backbone system and (iii) the environment, as it is sketched in Figure 24b. In the nucleobase system one must include the HOMOs associated to one of the strands along with the LUMOs associated to the complementary one, as well as the Watson-Crick H-bonding interactions. Attending to the energies involved in the different interactions, the resulting energy network can be hierarchically arranged, starting from high energy values related to the on-site energies of the bases and sugar-phosphate groups (8 12 eV),[104, 105, 106] passing through intermediate energy values related to the hydrogen bonding between Watson-Crick pairs ( 0:5 eV),[104, 107] and the coupling between the bases and the sugar moiety ( 1 eV),[105] and ending up with the aromatic base stacking low energies (0:01 0:2 eV).[104, 108, 109, 102] The energy scale of environmental effects (1 5 eV) is related to the presence of counterions and water molecules, interacting with the nucleobases and the backbone by means of hydration, solvation and charge transfer processes. In Figure25a we introduce a tight-binding ladder model for a double stranded polyGACT-polyCTGA unit cell including four different nucleotides. This unit cell provides a basis for both periodic and aperiodic longer DNA chains, where "j , with j = fG,C,A,T}, are the on-site energies of the bases, tj is the hopping integral between the sugar's oxygen

44

Figure 24. (a) Quantum chemical description of a realistic double-stranded DNA chain including the sugar-phosphate backbone and the presence of solvated counterions. The isosurface plot of the HOMO of B-poly(dG)-poly(dC) derived from ab-initio calculations is shown in side-view. The dark and gray surfaces show positive and negative isovalues, respectively. (Adapted from ref.[87]). (b) Diagram illustrating the overall energetics of a double-stranded DNA model including the different parameters considered in the DNA tight-binding model described in the text.


45

Figure 25. Sketch illustrating the two step renormalization process mapping a double-strand DNA chain into a linear diatomic lattice. a) Starting effective tight-binding model for the polyGACT-polyCTGA unit cell. b) renormalized model after the rst decimation step. c) renormalized model after the second decimation step.The particular cases polyG-polyC and polyA-polyT are described in terms of effective monoatomic chains.

atom and the base's nitrogen atom, and tGC (tAT ) respectively describe the hydrogen bonding between complementary bases. The backbone's contribution is described by means of the on-site energies j ; introduced in Figure 25b. In general, j will depend on the nature of the neighboring base as well as the presence of water molecules and/or counterions attached to the backbone, according to the overall scheme illustrated in Figure 24b. In order to obtain a simple mathematical description, containing most of the relevant physical information, we will map the tight-binding model shown in Figure 25a into the equivalent binary lattice model shown in Figure 25c. To this end, the Watson-Crick base pairs are rst renormalized to obtain the branched tight-binding model shown in Figure 25b. This mapping allows one to write the corresponding Hamiltonian in a form completely analogous to that of Eq.(20). The rst term now accounts for the charge carrier propagation through the DNA chain, where the hopping integral tn;n+1 t0 describes the aromatic base stacking between adjacent nucleotides, and the renormalized ”atoms” correspond to the Watson-Crick complementary pairs in the DNA molecule with on-site energies "n (E) = f (E); (E)g given by,[110] ;

=t

;

+

2 (E G;A

(E

C;T )

+

G;A )(E

2 (E C;T C;T )

G;A )

;

(56)

where t tCG ; t tAT , and n = tn + "n (E n )=tn . In this way, this approach provides a realistic description, including 15 physical parameters, {"j ; tj ; j ; tGC ; tAT ; t0 },

46 fully describing the energetics of the DNA molecule in terms of just three variables (i.e., ; ; t0 ) in a uni ed way. By considering the values listed in Table 12 the location of the Model Hamiltonian parameters (eV) "G = 7:77 "C = 8:87 "A = 8:25 "T = 9:13 = 12:27 t = 1:5 tGC = 0:90 tAT = 0:34 t0 = 0:15 Table 12. Parameters adopted for the effective Hamiltonian considered arranged by decreasing energies in order to illustrate the different energy scales of relevance in the DNA system. different allowed bands and their respective bandwidths are listed in Table 13.[110] Band center (eV) E1 = 14:209 E2 = 0:423 E3 = +6:440 E4 = +11:595

Band width (meV) W1 = 269 W2 = 120 W3 = 29 W4 = 177

Gap width (eV) 12 = 13:591 = 6:788 34 = 5:052

Table 13. Locations of the allowed bands centers (Ei ), bandwidhts (Wi ), and gap widths ( ij ) in the energy spectrum corresponding to the polyGACT-polyCTGA chain. Assuming, as it is usual, that each base pair contributes one free charge carrier, the HOMO band is centered at E = 0:423 eV, yielding an HOMO-LUMO gap width = 6:79 eV. This gure occupies an intermediate position between numerically obtained values for polyGpolyC chains (7:4 7:8 eV),[111] and photoemission spectroscopy measurements (4:5 5:0 eV) performed in polyG-polyC and polyA-polyT chains.[91, 92] As we see, the energy spectrum consists of four narrow bands separated by wide gaps. The wide separation among the different allowed bands stems from hybridization effects between the nucleobase system and the sugar-phosphate backbone.[100, 106] We note that the obtained bandwidths compare well with the values reported for short (5-12 base pair) polyG-polyC and polyA-polyT chains from rst principles band structure calculations (HOMO bandwidths ' 50 400 meV; LUMO bandwidths ' 100 300 meV, see Section 3.2).

3.6.

Thermopower of double-stranded DNA chains

In this Section, we will analyze the thermoelectric response of more realistic doublestranded DNA chains, in order to estimate the potential of synthetic DNA chains as thermoelectric materials. In order to substantiate this proposal, one must consider the energy dependence of Seebeck coef cient and thermoelectric power factor of polyG-polyC and polyA-polyT chains at room temperature. The duplex DNA molecules are modeled in terms of the renormalized one-dimensional effective Hamiltonian described in Section 3.5.

Novel Thermoelectric Materials: From Quasicrystal to DNA 3.6.1.

47

Analytical expressions

The Schrödinger equation of a renormalized polyG-polyC chain can be expressed in the form, [100]

N +1 N

=

1

2x

0

1

1

2x 1

0

0

1 0

N 2

2x 1

0

0

1

;

(57)

0

where n is the wavefunction amplitude for the energy E at site n; N is the number of base pairs, 2x (E (E))=t0 describes the DNA energetics, where (E) is given by Eq.(56), and the ratio 0 =t0 measures the DNA-lead coupling strength (the Schrödinger equation for polyA-polyT is obtained by just replacing (E) $ (E)). The transmission coef cient at zero bias as a function of energy is derived by embedding the chain between two semi-in nite leads (see Section 1.2.2) ant it is given by, [100] TN (E) = [1 + W

1

(E) h2 (E)]

1

(58)

;

where h(E) (E "M )UN 1 (UN + 20 UN 2 ), tM = 0 , and Uk (x) are Chebyshev polynomials of the second kind. Making use of Eq.(58) into Eq.(28) one gets,[82] ( ) @ ln h(E) SN (EF ; T ) = S0 (T ) G(EF ) B(EF ) + ; (59) @E EF where S0 (T ) = 2jejL0 T , B(EF ) b(cd) 1 , and G(EF ) 1 GN =G0 = 1 TN (EF ). The Seebeck coef cient is then expressed as a product involving three contributions. The factor S0 sets the thermovoltage scale (in VK 1 eV units) and accounts for the linear temperature dependence of SN . The factor G links the thermopower magnitude to the conductance properties of the chain, so that the Seebeck coef cient progressively decreases (increases) as the conductance increases (decreases), vanishing when TN = 1. The last factor in Eq.(59) depends on two additive contributions in turn. The value of B(EF ) depends on the relative position of the Fermi level with respect to both the band center, "M ; and the band edges, E ; of the contacts. Thus, its contribution vanishes when EF ! "M , whereas B (and consequently SN ) asymptotically diverges as the Fermi level approaches the spectral window edges (i.e., EF ! E ): Finally, the logarithmic derivative term in Eq.(59) contains most physically relevant information, accounting for (i) contact effects (related to the coupling constants 0 and ); (ii) size effects (described by the N parameter dependence), and (iii) resonance effects related to the DNA energetics by means of the Chebyshev polynomials' argument (see Eq.(56)) x(EF )

1 2t0

0

+(

1

1)EF +

2t2 EF

;

(60)

where 0 t + 4" 2"2 =t2 . Since we are mainly interested in the 1 , and 1 study of the intrinsic transport properties of DNA chains, we will minimize contact effects by adopting tM = t0 = henceforth, so that 0 = 1 and = tM . Thus, making use of Eq.(27) we can rewrite Eqs.(58) and (59) in the form GN (EF ) =

G0 2 1 + C(EF )UN

; 1

(61)

48 where C(EF )

[ (EF )

"M ]2 =W (EF ); and "

P2 (EF ) + SN (EF ; T ) = S0 (T ) G(EF ) B(EF ) + EF where P2 (EF )

1 (EF

1 (EF )2 + (a0

)2 2t2 "M )(EF

@ ln UN @E

1

) + 2t2

EF

:

#

;

(62)

(63)

By comparing Eqs.(59) and (62) we see that the logarithmic derivative in Eq.(59) has been split into two separate contributions. The rst one includes sugar-phosphate backbone effects through the parameter dependence. In particular, since P2 ( ) = 1; we realize that SN asymptotically diverges as the Fermi level approaches the backbone onsite energy (i.e., EF ! ). In general, the value will depend on the chemical nature of the nucleotides, as well as the possible presence of water molecules and/or counterions attached to the backbone.[106] Accordingly, this resonant enhancement of thermoelectric power strongly depends on environmental conditions affecting the DNA electronic structure. Finally, the Chebyshev polynomial logarithmic derivative appearing in Eq.(62) describes size effects in the thermoelectric response for DNA chains of different length. 3.6.2.

Transport curves

In order to reasonably ful ll the condition = tM = t0 ; we shall assume a contact geometry corresponding to a DNA chain connected to guanine wires at both sides. In this way, the spectral window is given by the energy interval [ 0:3; 0:3] eV, where the origin of energies is set at the guanine contact level (i.e., "M = "G 0): Note that the resulting contact bandwidth (4tM = 0:6 eV) compares well with the HOMO bandwidths reported for periodic guanosine stacked ribbons from rst-principle studies.[112] In Figure 26 we plot the thermopower and electrical conductance curves as a function of the Fermi energy obtained from Eqs.(61) and (62) for both G-C and A-T complementary pairs (N = 1) connected to guanine wires at both ends. The S(E) curves exhibit typically metallic values (1 10 VK 1 ) over a broad energy interval around the guanine energy level in reasonable agreement with the experimental values presented in Section 3.1.1. As we can see, the thermoelectric response is very similar for both kinds of Watson-Crick pairs, though the Seebeck coef cient is somewhat larger for the A-T one, due to its smaller conductance value (shown in the inset). In this case (U0 = 1) the transmission coef cient reduces to T1 = (1 + C) 1 and the corresponding conductance curves attain the maximum G1 ' 3: 1 at the resonance energy E = 8: 64 10 2 (E = 5: 8 10 5 (G1 = 5: 182 7 10 6 ) 1 1 2 50 10 ) eV for G-C (A-T) base pairs, respectively. As the number of base pairs composing the DNA chain is progressively increased several topological features (i.e, maxima, minima, and crossing points) appear in the thermopower curves, as it is illustrated in Figure 27 for the case N = 5. As we see, the Seebeck coef cient is characterized by the presence of two peaks around a crossing point located at the energy E0 = 0:116 eV. The thermopower values attained at the peaks are signi cantly high, and compare well with the values reported for benchmark TEMs. Nevertheless, as the Fermi level shifts away from the resonance energy, the Seebeck coef cient

49

0.05

0 .0

0.00

E (eV)

G 1 (E) / G0

-30 -0.10

-20

-10

0

10

G-C

-0.05

0 .0 -0 .3

0 .1

0 .2

0 .3

0 .4

0 .5

-0 .2

-0 .1

E (e V )

A-T

0 .1

0 .2

0 .3

0.10


S (µV K ) -1

Figure 26. Room temperature dependence of the Seebeck coef cient as a function of the Fermi energy for a G-C (solid curve) and A-T (dashed curve) Watson-Crick base pairs. Inset: The Landauer conductance as a function of the Fermi energy for the same base pairs. The origin of energies is set at the guanine contact level "M = "G 0 eV.

signi cantly decreases, clearly illustrating the ne tuning capabilities of thermopower measurements. On the other hand, according to Eq.(28) the main features of the polyG-polyC Seebeck coef cient shown in Figures 26 and 27 can be properly accounted for in terms of the conductance curves. In fact, when the Fermi level is located at the left (right) of the conductance peak the slope of the transmission coef cient curve TN (E) is positive (negative) leading to n-type (p-type) thermopower, respectively. In addition, the steeper the conductance curve the higher the thermopower value close to the resonance energy. Finally, we note that the crossover energy E0 de nes two different regimes where the polyG-polyC oligomer alternatively exhibits n-type or p-type thermopower. In this regard it is worth mentioning that when the Fermi level is located above E0 ; the Seebeck coef cient of each DNA chain exhibits contrary signs, so that the polyG-polyC chain behaves as a p-type material, while the polyA-polyT chain behaves like a n-type one, in agreement with previous experimental results.[83] By properly combining the previous results, making use of the typical values LN = 0:34 N nm for the length, and R = 1 nm for the radius of B-form DNA, we can determine 2 = G L S 2 =( R2 ) for the magnitude of the thermoelectric power factor PN = N SN N N N the considered samples. In the case N = 1 the power factor takes on relatively small values over a broad range of energies located around the conductance peak, but it signi cantly increases as the Fermi level approaches the band edges, as it was previously discussed. In the case N = 5; in addition to this general behavior the power factor also attains signi cantly large values close to the resonance energy of the polyG-polyC chain due to the presence of

-0.15 -600

-400

-200

0

200

400

-0.25

-0.20

E (eV)

E0

-0.10

-0.05

50

S (µV K ) -1

Figure 27. Room temperature dependence of the Seebeck coef cient as a function of the Fermi energy for a polyG-polyC (solid curve) and a polyA-polyT (dashed curve) oligomer with N = 5 base pairs. The vertical dashed line separates the energy regions exhibiting n-type and p-type thermopowers, respectively.


51

Figure 28. Room temperature Seebeck coef cient as a function of the Fermi energy for a polyG-polyC oligomer with N = 5 base pairs and = 4:5 eV (solid curve), = 4:0 eV (dashed curve), and = 3:0 eV (dotted curve) with = tM = 0:15 eV, and "M = 0 eV. Inset: Landauer conductance as a function of the Fermi level energy for the same samples shown in the main frame. (Reprinted with permission from Maciá E 2007 Phys. Rev. B 75 035130 c 2007 by the American Physical Society).

the above mentioned Seebeck coef cient peaks. The values of the power factor maxima attained in this case (P5 = 1:5 3 mWm 1 K 2 ) nicely t with those reported for benchmark thermoelectric materials (P = 2:5 3:5 mWm 1 K 2 ) at high temperatures.[113] Up to now, we have neglected the possible in uence of environmental effects, keeping a xed value for the backbone related on-site energy : However, more realistic treatments should take into account the presence of a number of counterions located along the DNA sugar-phosphate backbone (mainly in the vicinity of negatively charged phosphates) as well as the grooves of the DNA helix (mainly near the nitrogen electronegative atoms of guanine and adenine). In Figure 28 we compare the Seebeck coef cient as a function of the energy for different values for a polyG-polyC chain with N = 5. By inspecting this plot we realize the remarkable role played by environmental effects on thermopower. In fact, by systematically varying the on-site energy parameter from = 4:5 eV (no environmental effects) to = 3:0 eV, the thermoelectric response of the DNA chain can be modulated from typically semiconducting values to typically metallic ones. As expected from Eq.(59), this thermopower reduction is related to a progressive enhancement of the DNA conductance. This result is shown in the inset of Figure 28, where we plot the systematic variation of the polyG-polyC oligomer conductance as is progressively decreased. In summary we conclude that the thermoelectric response of short double-strand DNA chains strongly depends on the relative position between the contacts Fermi level and the DNA molecular levels. Thus, while the thermoelectric power of polyA-polyT oligomers is quite insensitive to the number of base pairs composing the chain, polyG-polyC oligomers exhibit a strong dependence on the chain length. Accordingly, we can ef ciently optimize

52 the power factor of DNA chains by properly shifting the Fermi level position close to the resonance energy, which plays the role of a tuning parameter. On the other hand, depending on the EF position, n-type and p-type thermoelectric responses can be simultaneously obtained for polyA-polyT and polyG-polyC DNA chains, respectively. This is a very convenient feature in order to design DNA based thermoelectric devices, where both oligomers would play the role that semiconducting materials legs usually play in standard Peltier cells (Figure 13). To this end, the relatively low value of the polyA-polyT chain Seebeck coef cient could be signi cantly improved by connecting it to adenine wires, rather than guanine ones, in order to get a proper alignment between the contacts Fermi level and the DNA molecular levels. As it was mentioned in Section 1.1 the thermoelectric quality of a material is expressed in terms of the dimensionless gure of merit given by Eq.(2). Therefore, the potential of DNA oligomers as thermoelectric materials will ultimately depend on their thermal transport properties, which have not been experimentally reported to date. It seems reasonable to expect that the thermal conduction is dominated by phonon transport in most organic compounds, leading to small thermal conductivities within the range = 0:25 0:50 Wm 1 K 1 at room temperature for different organic lms (see Section 3.1.1). Even smaller values are expected when considering the thermal conductance through a singlemolecule, as deduced from ash thermal conductance measurements yielding G ' 50 pWK 1 ( ' 0:05 Wm 1 K 1 for a nanometer sized molecule) recently reported for a selfassembled monolayer of alkane-thiol molecules (SH-(CH2 )n -CH3 ) absorbed on Au.[114] A suitable estimation of thermal conductivity for ideal coupling between a ballistic thermal conductor and the reservoirs relies on the quantum of thermal conductance g0 = 2 k 2 T =(3h) = 9:46 10 13 T WK 1 ; which represents the maximum possible value of B energy transported per phonon mode.[115] In the regime of low temperatures four main modes, arising from dilatational, torsional and exural degrees of freedom are expected for a quantum wire.[116] Therefore, the thermal conductivity of a DNA oligomer of length LN = 0:34N nm will be given by N ' 4g0 LN =( R2 ) = 0:02 Wm 1 K 1 (at T = 10 K) and N ' 0:6 Wm 1 K 1 (at room temperature) in optimal conditions. By taking the value ' 0:2 Wm 1 K 1 as a suitable estimation,[117] from the power factors values previously obtained, we get ZT ' 2 4:5 for polyG-polyC chains with ve base pairs at room temperature. These remarkably high gure of merit values must be properly balanced with the signi cant role played by unavoidable environmental effects, stemming from the presence of a cation/water molecules atmosphere around the DNA chain, on the actual thermoelectric ef ciency of DNA based nano-cells. In addition, the role of polarons (whose formation is a very common process for organic polymers with a exible backbone such as DNA) in the electrical transport ef ciency will deserve a closer scrutiny.[119, 120, 121, 122]

4.

Conclusion

In this Chapter we have considered the potential use of quasicrystalline alloys and DNA oligomers as termoelectric materials. Thogh these materials are quite different to each other their share some commnon features in their electronic properties endowing them with appealing transport properties. To illustrate this point, the room temperature electrical resistivities of several classes of materials are compared in Figure 29. It can be seen that


53

Figure 29. Room temperature electrical resistivity is compared for different materials of technological interest. Both quasicrystals and DNA are located at the border line between metals and semiconductors.

both quasicrystals and DNA ll the gap between metals and semiconductors, exhibiting electrical resistivity values comparable to those reported for doped conducting polymers, metallic macrocycles, and fullerenes.Quite interestingly their electronic structures satisfy these general ZT optimization requirements presented in Section 1.3 in a natural way, since their electronic structure is characterized by two main contributions, including in QCs : a pronounced pseudogap at the Fermi level (due to the Hume-Rothery stabilization mechanism) and several narrow spectral features in the DOS near the Fermi level (steeming for resonance effects between d-states belonging to the transition metal atoms in the polyhedral clusters). in DNA: a semiconducting gap (due to the H-bond interaction between complementary pairs in double-stranded DNA) and a series of narrow energy bands inside the gap (arising from solvated counterions and phosphate groups through the backbone). One of the main advantages of QCs is that one can ef ciently exploit the high sensitivity of their transport coef cients to stoichiometric changes in order to properly enhance their power factors without sacri cing their characteristic low thermal conductivity. The most promising QCs to date belong to the icosahedral AlPd(Mn,Re) family, exhibiting ZT ' 0:1 at room temperature and ZT ' 0:25 in the temperature range 475 550 K. Consequently, it seems reasonable to expect that, by a judicious choice of both sample composition and processing and annealing conditions, higher FOM values, comparable to the gures currently obtained for good TEMs (or even better) may be ultimately reached by following a systematic research. At this point some words are regarding the very possibility of tailoring the electronic structure properties of actual QCs and their related approximant phases in practice. As it is well-known no de nite guideline for the discovery of new QC systems has

54 been found and the rules for tuning their electronic properties are far from being properly understood. Notwithstanding this, very promising results have been recently reported on the basis of pseudogap tuning concepts.[123] Within this framework one starts by choosing an appropriate trial sample candidate (v.g., a polar intermetallic Zintl phase) taking into account its crystal symmetry (according to the group-subgroup relationships), the existence of structural clusters with the appropriate vefold symmetry, and the presence of a significant pseudogap in the DOS below the Fermi level. Then, the average electron per atom ratio is systematically changed by substituting metal cations by electron-richer elements (shifting the Fermi level towards the DOS minimum) of comparable ionic radius (hence preserving the structural network) having low-lying d orbitals (which favours the sp-d orbital mixing and bond formation). In this way, icosahedral QC and related approximants in the ScMgCuGa, and CaAuIn systems have been obtained from the parent cubic crystals Mg2 Cu6 Ga5 and Na2 Au6 In5 , respectively.[124] Since these compounds are synthesized following a well-de ned band structure engineering process from the very enjoining, it is reasonable to expect that the electronic structure re nement considered in this chapter may be attainable (at some degree at least) in the years to come. Regarding DNA transport properties the reported experiments demonstrate the high sensitivity of DNA electrical conductivity to the following factors: i) the structural complexity of nucleic acids, which is signi cantly in uenced by its close surrounding chemical environment (humidity degree, counterions distribution) affecting the integrity of the base-pair stack, ii) the kind of order present in the DNA macromolecule: periodically ordered polyG-polyC chains exhibit semiconducting behavior, whereas biological -phage chains are more insulating, iii) the role of contacts, the charge injection mechanism, the quality of the DNA-electrode interface, and the possible interaction with some inorganic substrate, or other components of the experimental layout. On the other hand, the presence of resonance effects among electronic levels in oligonucleotides composed of three nucleobases leads to a signi cant enhancement of the thermoelectric signal. By comparing the transport curves corresponding to different types of trimers we see that that relatively large thermopower values can indeed be obtained by properly locating the system's Fermi level. Thus, power factor values within the range 1:5 3 mWm 1 K 2 were obtained for short polyG-polyC chains containing ve base pairs, eventually leading to ZT ' 2 4:5 at room temperature. In addition, a characteristic thermoelectric signature can be used to identify the XYX type codons from XXX homonucleotide ones on the basis of their different thermoelectric responses, hence introducing a thermoelectric signature for different codons of biological interest, in close analogy with the transversal electronic signature recently proposed for single-stranded DNA chains.[125, 126] Accordingly, prospective studies on the thermoelectric properties of synthetic DNA oligonucleotides suggest that these materials are suitable candidates to be considered in the design of nano-scale sized thermoelectric cells. Experimental work aimed to test the actual capabilities of DNA based thermoelectric devices under different environmental conditions as well as to accurately determine the thermal transport properties of synthetic DNA samples, would be then very appealing.


55

Aknowledgements I thank M. Victoria Hernández for a critical reading of the manuscript. This work has been partially supported by the Universidad Complutense de Madrid and Banco Santander through project No. PR34/07-15824.

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