Application of the Dresselhaus and HicksModel to WS2 based Thermoelectric Devices

Santiago GomezSupervisor: Dr. Yenny Hernandez

A thesis presented for the degree ofBachelor of Science in Physics

Supervisor’s signature

Student’s signature

Physics Department - Nanomaterials LaboratoryUniversidad de los Andes

Bogota- ColombiaJune 13, 2019

AcknowledgementsFirstly, I would like to thank my parents and closest friends for their love, support andaffection. The time we have spent together I cherish deeply; my appreciation for all of yougoes beyond words. Secondly, I eagerly thank Dr. Yenny Hernandez for her knowledge,active involvement and keen interest in my project. Working with her has strengthened mypassion for science, deepened my interest in physics and taught me invaluable lessons thatallowed me to grow as a person and as a future scientist. Thirdly, I am grateful for thelifelong lessons that my professors and fellow students have taught me at this university, atruly unique experience which ends with the completion of this project. Finally, I would liketo express my gratitude towards everyone who helped or contributed with this work, suchas Maria Cristina Navarrete, Daniel Olaya and all of the people working at the laboratory.Without you, none of this would have been possible.

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AbstractThe quest for high performance thermoelectric devices has made nano structured materialsan attractive and viable alternative. Here we study tungsten disulfide/graphene heterostruc-tures as a possible thermoelectric material from a theoretical and experimental point ofview. Theoretically speaking, tungsten disulfide, o WS2, should be a high temperature (∼1000 K) good thermoelectric, as predicted by the constant relaxation time approximation.Furthermore, its thermoelectric efficiency, given by the dimentionless ZT coefficient, may beoptimally maximized through doping. By modelling the heterostructure between tungstendisulfide and graphene as a Kronning Penney type potential, we find that conduction elec-trons may move across the interphase in a quasi free like fashion. This lead us to believethat the main benefit from nano structures is phononic rather than electronic. From anexperimental point of view, we fabricated thermoelectric devices based on WS2 and elec-trochemically exfoliated graphene. Although SEM and AFM show a poor surface coverageof WS2, Harman measurements demonstrate that such devices have a good thermoelectricperformance, reaching ZT values as high as 1.75. However, power factors were found to beon the order of 5-18µW/mK2, which is low. This means that such high ZT values must comefrom a reduction of the thermal conductivity. We suspect this happens because of phononblocking due to scattering at the interphase.

ResumenLa busqueda de materiales termoelectricos de alta eficiencia ha hecho de los materiales

nano estructurados una alternativa atractiva para dispositivos de conversion termoelectrica.En este trabajo se estudiaron las heteroestructuras de grafeno y disulfuro de tungsteno, oWS2, desde un punto de vista teorico y experimental. Desde la perspectica teorica, la aprox-imacion del tiempo de relajacion predice que el disulfuro de tungsteno deberıa ser un buentermoelectrico de alta temperatura (∼ 1000 K). Ası mismo, su eficiencia termoelectrica, dadapor el coeficiente ZT, se puede optimizar mediante dopaje. Al modelar la heteroestructuraentre disulfuro de tungsteno y grafeno con un potencial tipo Kronnig Penney, encontramosque los electrones de conduccion se mueven cuasi libremente a traves de la interfase. Esto noslleva a pensar que la contribucion de la nano estructuracion es fononica mas que electronica.En el ambito experimental se fabricaron dichas heteroestructuras a partir de WS2 y grafenoexfoliado electroquımicamente. A pesar de que SEM y AFM muestran una pobre coberturasuperficial del disulfuro de tungsteno, las medidas de Harman mostraron un buen desempenotermoelectrico, con valores de ZT de hasta 1.75. No obstante, los bajos valores del factor depotencia (5-18µW/mK2) permiten deducir que el buen rendimiento termoelectrico es por unareduccion sustancial de la conductividad termica. Sospechamos que esto se debe al bloquede fonones debido a su dispersion en la interfase.

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Contents

1 Introduction 6

2 Theoretical Considerations 102.1 Relaxation Time Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Calculation of Transport Properties with the Dresselhaus and Hicks Model . 14

2.2.1 DC Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . 142.2.2 Seebeck Coefficient and Thermal Conductivity . . . . . . . . . . . . . 182.2.3 Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Broido’s Extension and Out of Plane Properties . . . . . . . . . . . . . . . . 22

3 Experimental Work 253.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.2 Harman Characterization . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 AFM and SEM characterization . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Characterization by Harman Transient Method . . . . . . . . . . . . . . . . 30

4 Conclusions and Future Work 33

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List of Figures1 Schematic representation of the Thermoelectric effect, in which a temperature

gradient induces an electrical current (left) or vice versa (right). Taken fromZhang et al[22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Most relevant reported figures of merit from different authors since 1960. Ma-terials reported after 1990 involve some form of nano structuring. Taken fromVineis et al [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Schematic representation of the band structure of multilayered Bi2Te3/Sb2Te3lattices, which form a multiple quantum well structure. Taken from Venkata-subramanian et al[19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Schematic representation of a single layer of a transition metal dichalcogenide,where yellow atoms represent the chalcogen and the black atoms are the tran-sition metal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Schematic representation of a Harman transient measurement by with a singlecurrent pulse. Taken from Venkatasubramanian et al [19] . . . . . . . . . . . 9

6 Diagram of temporal evolution of electrons in the phase space . . . . . . . . 117 Comparison between experimental data of WS2 doped with Carbon Nanotubes

and relaxation time approximation for electrical DC conductivity . . . . . . 178 Seebeck coefficients obtained a) experimentally by Suh et al and b) under the

relaxation time approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 209 In plane figure of merit (x direction) as a function of electron carrier concentra-

tion calculated with a) Density functional theory simulations [5] b) RelaxationTime approximation c) Experimentally Measured [18] . . . . . . . . . . . . . 21

10 Dispersion relation of graphene/WS2 heterostructures using the Kronnig Pen-ney model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11 Comparison between Broido’s dispersion relation and free electron dispersionrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

12 Fabrication of graphene solution by electrochemical exfoliation . . . . . . . . 2513 Fabricated devices based on Graphene/WS2 heterostructures. EEG stands for

Electrochemically Exfoliated graphene . . . . . . . . . . . . . . . . . . . . . 2614 Diagram of the experimental setup for Harman measurements, which ideally

should by adiabatically sealed. . . . . . . . . . . . . . . . . . . . . . . . . . . 2615 SEM images of WS2 deposited on graphene coated Si (GWG devices) with

different magnification, 5µm and 100µm . . . . . . . . . . . . . . . . . . . . 2716 SEM images of WS2 deposited on bare Si (WGW devices) with different mag-

nification, 20µm and 100µm . . . . . . . . . . . . . . . . . . . . . . . . . . . 2717 Example of compositional analysis (EDS) on a GWG device. . . . . . . . . . 2818 a) AFM image of WS2 deposited on graphene coated Si b) Height profiles for

the selected grains, labelled as 1, 2 or 3 . . . . . . . . . . . . . . . . . . . . . 2819 a) AFM image of WS2 deposited on bare Si b) Height profiles for the selected

grains, labelled as 1, 2 or 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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20 Examples of the collected data a) before and b) after data processing andregression. The red dotted lines show the fitted curve in the linear (Ohmic)and exponential (Thermoelectric) regimes. The continuous black line marksthe Seebeck Voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

21 Experimental results for the out of plane figure of merit and electrical conduc-tivity for the fabricated devices at different temperatures. . . . . . . . . . . . 31

22 Experimental results for the out of plane figure of merit and electrical conduc-tivity for the fabricated devices at different temperatures. The data point ofGWG device at 50°C is an outlier and hence it is not shown. Error bars arenot visible at this scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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1 IntroductionThermoelectric materials, which allow the direct conversion between heat and electric current,may hold the promise of sustainable and efficient energy conversion. Thus the quest for highperformance thermoelectrics has been at the center of intensive research. The thermoelectriceffect was first described by Peltier and Seebeck in the early XIX century[22, 20]. Theyobserved the appearence of an electromotive force in a material under the influence of atemperature gradient (Seebeck) or, conversely, the formation of a temperature gradient whenan external voltage is applied (Peltier).

Figure 1: Schematic representation of the Thermoelectric effect, in which a temperaturegradient induces an electrical current (left) or vice versa (right). Taken from Zhang et al[22]

Figure 1 shows how thermoelectric materials convert between heat and electric potential.Mathematically, this phenomenon may be described by the Seebeck Law:

~E = S∇T, (1)

where ~E is the electric field, S is the Seebeck coefficient and T is the temperature. Further-more, the thermoelectric efficiency of a material may be quantified through the dimensionlessfigure of merit or ZT[19, 5], defined as

ZT = S2σ

κT, (2)

where σ and κ are the electrical and thermal conductivities, respectively. A material musthave a ZT greater than 3 in order to compete with traditional energy conversion systems[20].

Although the transport properties involved in equation 2 may be changed over a widerange of orders of magnitude, their interdependence preclude the figure of merit from beingsignificantly altered. Such limitation is exemplified by the Wiedemann Franz law [20]:

κ

Tσ= π

3

(kBe

)2, (3)

where kB is the Boltzmann constant and e is the elementary charge. Since the left handside of the equation is composed of fundamental constants, the quotient κ/σ is also constantat a fixed temperature T. Therefore, any enhancement in the electrical conductivity willresult in a similar change in thermal conductivity, thereby leaving ZT virtually unchanged.

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Consequently, increasing ZT above unity in bulk materials proved to be a rather difficulttask[20]. From the discovery of BiTe alloys in the 1960’s as a thermoelectric with ZT ≈ 1near room temperature[19, 18], little progress was made despite intensive research[6, 10].

In 1993, Dresselhaus and Hicks theorized that nano materials arranged in multilayered su-perlattices would exhibit a higher thermoelectric performance than its bulk counterparts[10].Such enhancement would occur because of the quantization of electronic states in multiplequantum wells and the reduction of the lattice thermal conductivity due to the scatteringof phonons at the interfaces. Such concept motivated the research of thermoelectric nanodevices, resulting in a drastic increase in the reported figures of merit as shown in figure 2.

Figure 2: Most relevant reported figures of merit from different authors since 1960. Materialsreported after 1990 involve some form of nano structuring. Taken from Vineis et al [20]

Figure 2 illustrates how, since Dresselhaus and Hicks study was published in the early1990s, nano structuring allowed to surpass the unity limit in bulk materials, reaching figuresof merit of values as high as 2.5[20]. It is worth noting that the highest ZT in figure 2 closelyfollow the idea proposed by Dresselhaus and Hicks: Venkatasubramanian et al engineeredmultiple quantum well structures by using Bi2Te3/Sb2Te3 superlattices [19]. Since the bot-tom of the conduction band of Sb2Te3 is higher than that of Bi2Te3, conduction electronsexperience a multiple quantum well like potential, as shown in figure 3. Other studies suchas those published by Hicks et al[9] and Harman et al [7] follow a similar procedure.

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Figure 3: Schematic representation of the band structure of multilayered Bi2Te3/Sb2Te3lattices, which form a multiple quantum well structure. Taken from Venkatasubramanian etal[19]

Despite the important breakthroughs in thermoelectricity, several limitations and ques-tions remain. On one hand, the reason why nano structuration enhances thermoelectricperformance is still a question of debate [20]: the nano structures mentioned above mainlybenefited from the phonon blocking[20, 19] effect rather than from quantization of electrons.The exact role of the latter in the increase of the figure of merit is still unclear.

On the other hand, one must notice how most materials shown in figure 2 use elementssuch as Pb, Sb, Bi or Te. These elements are not only rare earth and therefore of limitedavailability, they are also highly toxic to humans[20]. Consequently, recent research focuseson new materials based on more available and less toxic elements in order to facilitate largescale production. Given these requirements, transition metal dichalcogenide monolayers havegained interest within the thermoelectric community.

A transition metal dichalcogenide monolayer (TMD) is a three atom thick layer composedof transition metal atoms that are bonded to two chalcogen atoms, as shown in figure 4.

Figure 4: Schematic representation of a single layer of a transition metal dichalcogenide,where yellow atoms represent the chalcogen and the black atoms are the transition metal.

TMDs such as tungsten disulfide and tungsten molybdenum are attractive for thermo-electric applications because they are non toxic and have a higher availability than othermaterials. Also, they possess a tunable band gap and a moderate thermal conductivity,which makes them a suitable candidate for thermoelectric devices unlike other nano materi-als such as graphene[11]. However, pristine TMDs by themselves are not good thermoelectricsdue to their low electrical conductivity and Seebeck power[18]. Hence, research on TMDsfor thermoelectricity has focused on possible modifications in order to increase their figureof merit.

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On one hand, experimental studies have shown that the thermoelectric performance ofWS2 can be drastically enhanced by doping with titanium or carbon nanotubes [18, 11], whichincrease substantially the electrical conductivity while reducing thermal conductivity. Thesestudies have shown a maximum ZT of 0.22[18, 11]. On the other hand, theoretical studieshave proven that TMDs have great potential for thermoelectric applications; Sadeghi et altheoretically proved that MoS2/Graphene heterostructures would have a high ZT since thestructure would preclude the transmission of high energy phonons, hence reducing drasticallythe lattice thermal conductivity[16]. Similarly, Gandi et al used Density Functional Theory(DFT) simulations to show that WS2 would theoretically have a good performance at hightemperatures, near 1000K[5]. Such studies predicted an optimized ZT of the order of 2.8 and0.9, respectively [5, 16].

Another important contribution to the thermoelectric community was done by Harmanet al[7], who proposed a novel and more accurate way of characterizing the thermoelectricproperties of thin film devices. Determination of figure of merit by separate measurementof transport properties results in gross experimental errors due to parasitic currents or heatleaks[15]. Such a direct and more precise technique, known as the Harman transient method,consists on applying a square pulsed voltage on to the material or device, which is also underthe influence of a known temperature gradient. The voltage drop across the sample is thenmeasured as a function of time, as shown below.

Figure 5: Schematic representation of a Harman transient measurement by with a singlecurrent pulse. Taken from Venkatasubramanian et al [19]

The total voltage VT will have ohmic (VOhm) and Seebeck contributions, the latter ofwhich is given by V0 on figure 5. Under adiabatic operation, the figure of merit will simplyby given by the ratio of these two contributions:

ZT = V0

VOhm= V0

VT − V0. (4)

Thus, despite the recent advances mentioned before there is still room for improvementwith TMDs as thermoelectric nano materials; experimental results such as those publishedby Suh et al or Huang et al[18, 11] are far beneath both the theoretical prediction of Sadeghiand Gandi [5, 16] and the minimum ZT in order to compete with traditional energy con-version systems [20]. It is, thus, the main goal of this work to study the relevant theory ofnano structured thermoelectric devices, apply it to WS2 based devices and propose possiblestrategies of thermoelectric performance optimization.

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2 Theoretical Considerations

2.1 Relaxation Time ApproximationMost theoretical models for thermoelectricity are based on the relaxation time approximation,a semi classical treatment in which electrons are governed by the following semi classicalequations of motion[12, 1, 23]:

~v(~k)n = 1h

∂εn

∂~k, (5a)

h~k = −e( ~E + ~vn × ~B), (5b)

for some energy dispersion relation εn, electric and magnetic fields ~E and ~B, velocity ~v andwavevector ~k. The model assumes the band index n to be a constant of motion, meaning thatno interband transitions are considered[12]. The goal of the relaxation time approximationis to estimate the non equilibrium electronic distribution, g. This is equivalent to find asolution to the Boltzmann transport equation,

∂g

∂t+ ~v(~k) · ∂g

∂~r+ ∂~k

∂t· ∂g∂~k

= ∂g

∂t col, (6)

where g is the distribution to be calculated, ~r is the position vector, t is time and ∂g∂t col

is therate of change of the distribution due to collisions. At this point there are several definitionsof the relaxation time approximation. Some authors[1, 23] start by writing the right handside of equation 6 as

∂g

∂t col= g0 − g

τn(~k), (7)

where τn(~k) is the so called relaxation time and g0 is the equilibrium distribution, which maybe Fermi-Dirac, Bose-Einstein or Maxwell-Boltzmann. Equation 7 states that the systemshould return to equilibrium uniformly, implying that the distribution does not differ signif-icantly from its equilibrium form. However, the derivation of relaxation time approximationdone by Ashcroft and Mermin [12] is much more rigorous and physically insightful. Thereforewe shall follow their reasoning and then prove that both approaches are equivalent.

The relaxation time approximation is based upon two main assumptions[12]. We definedgn(~k, ~r, t) as the distribution of electrons which suffer a collision and, as a consequence,emerge in a region near (~k, ~r) in the phase space. The first assumption states that suchterm will not depend on the distribution just before the collision takes place. Secondly, it isassumed that in those regions of the phase space where there is local equilibrium, collisionsare responsible of maintaining such equilibrium. We also define the relaxation time τn(~k, ~r)to be given by the probability p of an electron colliding between times t and t+ dt:

p(t, t+ dt) = dt

τn(~k). (8)

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Both postulates completely determine the form of dgn(~k, ~r, t)[12]. Consider a region ofthe phase space where there is local equilibrium, which means that g is exactly given by theFermi-Dirac distribution:

gFD = 1eε−µkBT + 1

, (9)

for energy ε and chemical potential µ. Given the second postulate, collisions should not alterthe form of this distribution. Equation 8 is the fraction of electrons which suffer a collisionaround a time differential dt. Therefore, the product of equations 8 and 9 gives the numberof electrons which suffer a collision and, consequently, leave this region of the phase space.In order for the form of the distribution to remain unchanged, the electrons emerging in thissame region of space, dgn(~k, ~r, t) , should exactly compensate such loss. Hence, we can write:

dgn(~k, ~r, t) = gFDdt

τn(~k), (10)

Furthermore, the first postulate states that dgn(~k, ~r, t) should not depend on the distri-bution prior to the collision, so equation 10 is valid for all of the phase space.

Now, it is possible to determine the explicit form of the non equilibrium distribution g.Consider two points phase space (~k, ~r) and (~k′, ~r′), corresponding to two different times tand t′ respectively. We assert that both points of phase space are solutions to equation 5.This means that, in the absence of collisions, electrons should move from (~k′, ~r′) to (~k, ~r).However, some electrons will experience at least one collision in this time interval. This isdepicted in figure 6:

Figure 6: Diagram of temporal evolution of electrons in the phase space

Let us remember that the number of electrons dN around a certain differential hypervol-ume of the phase space for a non degenerate system drdk is given by

dN = gdrdk

4π3 . (11)

From equation 10, the distribution of electrons that emerged as a consequence of collisionsaround (~k′, ~r′) is known. If the probability P (t, t′) is defined as the probability of not colliding

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between times t′ and t, then the number of electrons that arrive to (~k, ~r) will be given byP (t, t′)dgn(~k, ~r, t). The total number of electrons in the vicinity of (~k, ~r) is then given by theintegration of all times prior to t:

dN =∫ t

−∞P (t, t′) gFD

τn(~k)dt′drdk

4π3 . (12)

One might argue that not all electrons are taken into account, only those which appearedaround (~k′, ~r′) prior to a collision. There are, in fact, electrons which were initially at (~k′, ~r′)and were able to reach (~k, ~r) without any further collision. The integration over all timesallows us to solve this problem: by requiring that P (t, t′) vanishes in the limit t′→∞, we statethat no electron shall survive for an infinitely long time without colliding at least once. Byintegrating over all times prior to t, we account for all the electrons in the region of interest.

By comparing equations 11 and 12 one can see that a non equilibrium distribution hasbeen found in terms of P (t, t′). Such probability may be calculated from the definition ofthe relaxation time, given by equation 8. Consider now a differential time interval dt′. Inorder for an electron not to collide between t′ and t, it should not experience any collisionbetween t′ and t′ + dt′ nor between t′ + dt′ and t. Since we know the probability of collidingin a differential time interval is given by equation 8, one can write the former statement as

P (t, t′) = P (t′ + dt, t′)P (t, t′ + dt′) =(

1− dt′

τn(~k)

)P (t, t′ + dt′). (13)

By re arranging such equation appropriately and taking dt′→0, we arrive to the followingdifferential equation

∂P (t, t′)∂t′

= P (t, t′)τn(~k)

, (14)

whose solution is of course an exponential one:

P (t, t′) = e∫ tt′dt′τ . (15)

At this point it is worth asking about the dependence of τ . Ashcroft and Mermin [12]argue that, in the relaxation time approximation, relaxation time should only depend on ~kthrough ε[12]. Dresselhaus and Hicks, whose result we are trying to reproduce and extend,have gone so far as to assume a constant relaxation time [10]. Such assumption is alsomade by other authors when modelling the material of interests tungsten disulfide, withDFT simulations, obtaining results in excellent agreement with experimental data[5]. Otherauthors use a more general approach in which the relaxation time does depend on ε througha power law [14]:

τ = τ0εl, (16)

for some constants τ0 and l. In any case, literature suggests that a constant relaxation timeis a suitable approximation for the thermoelectric material of interest[5, 10]. Hence, we shallassume so in the rest of the calculation. Even if it was not, τ would only depend on ε, which

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is of course conserved. Cases in which the relaxation time depends on other variables, suchas the wave vector, may not be well modelled by the relaxation time approximation [12].Hence,

P (t, t′) = et−t′τ . (17)

Finally, one can explicitly write the non equilibrium electronic distribution, but we willnot use the explicit form of P (t, t′), equation 17, yet. From equations 11, 12 and 14

g =∫ t

∞P (t, t′)gFD

τdt′ =

∫ t

∞

∂P (t, t′)∂t′

gFDdt′ = gFD −

∫ t

−∞

dgFDdt′

P (t, t′)dt′, (18)

where we have used integration by parts in the last step. Several remarks can be donefrom this last equation: first, notice that the non equilibrium distribution is simply a Fermi-Dirac distribution plus some correction. This is extremely important, as the relaxation timeapproximation is only valid when the distribution does not differ strongly from its equilibriumform[1, 23]. Second, we can reconcile Ashcroft and Mermin’s definition of relaxation timeapproximation with the definition found elsewhere[1, 23]. Let us differentiate both sides ofequation with respect to t, considering only the collision related terms. We will denote suchoperation with the subtext “col” as done before:

∂g

∂tcoll= ∂gFD∂tcoll

− ∂

∂tcoll

∫ t

−∞

dgFDdt′

P (t, t′)dt′. (19)

Since the first term of the right hand side is an equilibrium distribution, such derivativeshould be zero. Furthermore, the integrand is with respect to t′ and not t, so it may commutewith the derivative. Since the effect of collisions is contained in P (t, t′), the derivative onlyacts on this term:

∂g

∂tcoll= −

∫ t

∞

dgFDdt′

∂P (t, t′)∂tcoll

dt′ = −∫ t

−∞

dgFDdt′

P (t, t′)τ

dt′, (20)

where we have used equation 14 in the last step. Replacing equation 18 in this last equa-tion, we arrive to equation 7. We have thus shown the equivalence of the relaxation timeapproximation approach of Ashcroft and Mermin[12] with that of other sources[1, 23].

We return to the expression 18. The exact time derivative of the equilibrium distributioncan be expanded by using a exact differential. Given the form of such distribution, explicitlystated in equation 9, it only depends explicitly on ε, T and µ. We write:

dgFD = ∂gFD∂ε

dε+ ∂gFD∂T

dT + ∂gFD∂µ

dµ

= ∂gFD∂ε

∂ε

∂~k· d~k + ∂gFD

∂T

∂T

∂~r· d~r + ∂gFD

∂µ

∂µ

∂~r· d~r.

(21)

Let us notice how a the derivatives of the distribution function can be written in termsof a single partial derivative. It is convenient to write the derivatives with respect to T andµ as a function of the derivative with respect to energy. This is because, for temperatureslower than the Fermi temperature, such derivative will become a Dirac delta function. This

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will make integration easier. It is clear that the µ derivative will be simply the negative ofthe derivative with respect to energy. This can be easily deducted from equation 9. Thetemperature derivative is slightly lengthier to compute:

∂gFD∂T

= (ε− µ)eε−µkBT

kBT 2(eε−µkBT + 1)2

= −ε− µT

∂gFD∂ε

. (22)

Returning to equation 21, we divide by dt and replace the equations of motion. We obtain

dgFDdt

= −∂gFD∂E

~v ·(e ~E + ε− µ

T∇T +∇µ

). (23)

Replacing this into the distribution, the final form of the non equilibrium distribution canbe written explicitly, recognizing that the derivative with respect to ~r is simply the nablaoperator:

g = gFD +∫ t

−∞P (t, t′)∂gFD

∂ε~v ·(e ~E + ε− µ

T∇T +∇µ

)= gFD +

∫ t

−∞et−t′τ∂gFD∂ε

~v ·(e ~E + ε− µ

T∇T +∇µ

).

(24)

Several remarks may be done from equation 24. First, it is clear that the relaxation timeapproximation is a linear theory: the final distribution is linear in all of the terms insidethe integral. Therefore, higher order effects such as Joule heating are not contemplated inthe approximation[12]. Second, the derivative within the integrand with respect to ε impliesthat only the energy states near the Fermi Energy contribute to non equilibrium properties,given that only in these states have non zero derivative. This is to be expected: given thePauli Exclusion principle, only electrons near the Fermi surface may experience a change intheir wave vector, as they are near available k states. In a semi classical picture this meansonly these electrons may be accelerated. Finally, notice that this distribution is valid even inthe presence of a magnetic field, since the magnetic component of the Lorentz force is alwaysorthogonal to ~v, and thus the term vanishes when taking the dot product.

2.2 Calculation of Transport Properties with the Dresselhaus andHicks Model

2.2.1 DC Electrical Conductivity

We proceed to calculate the relevant transport properties for thermoelectric materials, namelythermal conductivity, electrical conductivity and Seebeck coefficient. For a DC electric fieldin absence of thermal or chemical potential gradient, equation 24 may be further simplified ifwe consider velocity to be approximately time independent. Such approximation is extremelyfeasible since most materials, WS2 included, have an incredibly short relaxation time[5]. Thismeans that collisions will balance out the acceleration due to the electric field. Such claimcan also be easily verified: when a voltage is applied to a material, electrons do not accelerateinfinitely, but reach a rather small final velocity instead. Similar approximation can be donefor temperature and chemical potential gradient [12].

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Under such approximations, the final equilibrium distribution may be written as

g = gFD + eτ ~E · ~v∂gFD∂ε

. (25)

From such expression it is possible to calculate a current density:

~je = e∫~vg

d~k

4π2 = e2τ∫~v~v∂gFD∂ε

d~k

4π3~E. (26)

Notice how only the correction plays an important role in the calculation of the electriccurrent. Physically, this makes sense, as in an equilibrium condition there is no net chargemovement. Mathematically, this may be justified by noticing from the equations of motion(equation 5) that the equilibrium function is a derivative of the dispersion relation, whichshould be symmetric in ~k in a classical picture. Hence, the derivative should be anti symmetricand g should be symmetric, so the integrand vanishes.

From equation 26 we may recognize Ohm’s law, where the electrical conductivity is givenby the term multiplying the electric field on the right hand side of the equation. The product~v~v is a dyadic product, so such factor is a tensor quantity, as expected:

σ = e2τ∫~v~v∂gFD∂ε

d~k

4π3 . (27)

Such integrand will of course depend on the form of the velocity, which in turn dependson the dispersion relation ε(~k). Dresselhaus and Hicks, whose model we will now reproduce,[10] consider two different cases: the dispersion relation for a free electron and for an electronconfined in the z direction to a potential well but free in the xy plane. The dispersion relationfor each of these situations is:

ε(kx, ky, kz) = h2k2x

2mx

+h2k2

y

2my

+ h2k2z

2mz

, (28)

for a completely free electron, where mi and ki are the ith components of the mass tensorand wave vector, respectively. Similarly,

ε(kx, ky) = h2k2x

2mx

+h2k2

y

2my

+ h2π2

2mza2 = h2k2x

2mx

+h2k2

y

2my

+ ε0, (29)

for an electron confined by an infinite square potential of length a in the z direction but freein the xy plane, where ε0 is the base energy of the square infinite potential well. We havetherefore assumed that only the base state is significantly populated; such claim is valid ifa << Lx,y, as there are a lot of available low energy states in the n = 1 configuration.

Clearly, equations 28 and 29 imply that there is equivalence in the x, y and, in thethree dimensional case, z directions. Therefore, we have implicitly assumed the modelledsystem to be isotropic in these directions, which means that the conductivity tensor will bediagonal[12]. In this specific case, this is easily verifiable by noticing that, when performingthe integration in equation 27, it will be convenient to use polar or spherical coordinates.Therefore, off diagonal components will take the form of an integral of the product of sines

15

and cosines, which of course is zero. Such symmetry simplifies enormously the determinationof σ; it suffices to simply calculate one of the diagonal elements.

For the 3D case, integration is performed using spherical coordinates. Since the compo-nents of the mass tensor may not be equal, we perform the following substitution in order tosimplify calculations:

ki = √miki. (30)We choose the z component of the tensor for simplicity:

σzz = −e2τ∫ (

k2z h

2

m2z

)∂gFD∂ε

d~k

4π3

= −e2τ√mxmymz

∫ (k2 cos2(θ)h2

mz

)∂gFD∂ε

k2 sin(θ)dkdθdφ4π3

=−e2τ

√mxmymz

3π2mz

∫ (2εh2

)3/2∂gFD∂ε

dε.

(31)

As we can see, spherical coordinates simplify greatly the integration task. In the last stepwe have used the dispersion relation to write the radial integral as an energy integral. Nowwe proceed with integration by parts

∫ ∞0

ε3/2∂gFD∂ε

dε = ε3/2gFD

∣∣∣∣∣∞

0− 3

2

∫ ∞0

ε1/2gFD = −32(kBT )3/2F1/2(ζ), (32)

where we have made the replacements x = ε/kBT and ζ = µ/kBT and we have defined Fi(ζ),commonly known as the Fermi function or Fermi integral, as

Fi(ζ) =∫ ∞

0

xi

ex−ζ + 1 . (33)

Putting everything together, we arrive to the final result:

σzz =e2τ√mxmymz(2kBT )3/2

2π2mzh3 F1/2(ζ) =

eµz√mxmymz(2kBT )3/2

2π2h3 F1/2(ζ), (34)

where we have replaced the relaxation time in terms of the electron mobility along the zdirection µz, a more experimentally accessible quantity:

µz = eτ

mz

. (35)

The calculation for the in plane conductivity is completely analogous: it consist in usingpolar coordinates to reduce the triple integral to a single radial integral, which may be writtenas an energy integral. It is clear that such integrals may only be analytically solved in limitedcases. Namely, when the topology of the Fermi surface, given by the dispersion relation, maybe easily handled and integrated over analytically.

The calculation for the conductivity for the confined electrons is analogous; using polarcoordinates to reduce the integral to a single energy integral and then use integration byparts. We shall only write the most relevant steps:

16

σxx = −e2τ∫ (

k2xh

2

m2x

)∂gFD∂ε

d~k

4π3

= −e2τ√mxmy

∫ (k2 cos2(θ)h2

mx

)∂gFD∂ε

kdkdθ

4π3

=−e2τ

√mxmy

2π2mx

∫(ε− ε0)∂gFD

∂εdε

= 12πa

(2kBTh2

)F0(ζ∗)eµx,

(36)

where now the chemical potential is measured from the lowest energy level of the well:

ζ∗ = ζ − ε0

kBT. (37)

In order to validate the approximation, we compare the in plane electric transport prop-erties predicted by equation 36 with experimental data reported from literature for dopedtungsten disulfide. We first estimate the relationship between the electron concentrationand the chemical potential µ. We assume that the correction to the electronic distributionin equation 24 is small enough compared to the equilibrium distribution so that it can beneglected. Recalling that the relaxation time approximation is a non degenerate theory:

N

A=∫ ∞

0gd~k

4π2 ≈∫ ∞

0gFD

kdkdθ

4π2

=√mxmy

2π2h2

∫ ∞0

gFDdE =2πkBT

√mxmy

h2 ln(eζ∗ + 1).(38)

We can then plot in plane conductivity as a function of carrier concentration

(a) Experimental data by Suh et al [18] (b) Relaxation time approximation

Figure 7: Comparison between experimental data of WS2 doped with Carbon Nanotubesand relaxation time approximation for electrical DC conductivity

17

In figure 7 we observe that the relaxation time approximation correctly predicts thebehavior of electrical conductivity for tungsten disulfide as a function of temperature fordifferent carrier concentrations; such parameter rises sharply at low temperatures and thenstabilizes asymptotically at higher temperatures. We also observe that the relaxation timeapproximation correctly predicts the rise in conductivity for higher electron concentrations.However, it is also noticeable that at lower temperatures the relaxation time approximationfails to predict the correct order of magnitude. This may be because Suh et al [18] used carbonnanotubes as dopant, which are known for being incredibly good conductors. Therefore, themeasured conductivity has contributions of both tungsten disulfide and carbon nanotubes.Also, the calculation was done assuming a single TMD monolayer, which is not the case forthe experimental study which we compared with [18]. Similarly, the developed theory is nondegenerate[12, 10], so degeneracy might play a role in the discrepancies.

2.2.2 Seebeck Coefficient and Thermal Conductivity

So far we have calculated electrical conductivities. Now we proceed to calculate the thermalconductivity and the Seebeck power, which is a lengthier calculation. We begin with the firstlaw of thermodynamics for open systems in the absence of work:

δQ = dε− µdN, (39)which may be written in terms of energy and electron currents, in analogy with equation 26:

~jQ = ~jE − µ~jN =∫~vn(ε− µ)g d

~k

4π3 , (40)

were we have defined ~jn in the same way as in 26 and ~jE is:

~jE =∫~vnεg

d~k

4π3 . (41)

In order to express these currents in terms of the non equilibrium distribution in a morecompact way, we define the following tensor

Li = e2τ∫~v~v(ε− µ)i∂gfD

∂ε

d~k

4π3 . (42)

We then replace equation 24 in the definition of jN and jQ , and express them in termsof such tensor, obtaining

~jn = L0( ~E + 1e∇µ)− 1

eTL1(−∇T ), (43)

~jQ = −1eL1( ~E + 1

e∇µ) + 1

e2TL2(−∇T ). (44)

Since both the Seebeck power and the thermal conductivity are measured at zero electriccurrent[7, 15, 17], the right hand side of equation 43 is set to zero. Therefore,

18

~E + 1e∇µ = 1

eTL−1

0 L1(−∇T ), (45)

for equation 43. Notice how equation 45 relates linearly an effective electric field to a tem-perature gradient. This is nothing else than Seebeck’s law, equation 1 , in tensor form.Further simplification may be done by recalling that we are dealing with diagonal tensors,so the multiplication by the inverse L−1

0 is nothing more than a division. Hence, the Seebeckcoefficient tensor S may be written as

S = 1eTL−1

0 L1 = L1

eTL0. (46)

Furthermore, by replacing equation 45 in 44, we obtain

~jQ = 1e2T

(L2 − L1L−10 L1)(−∇T ). (47)

Again, we may recognize Fourier’s heat conduction law from equation 47, as it expressesa linear relationship between a heat current and a temperature gradient. As before, thecases considered by Dresselhaus and Hicks are isotropic, so tensors may be treated as scalars.However, such thermal conductivity is only the electronic contribution to the total thermalconductivity. The lattice contribution, mediated by phonons, and its theoretical explanationis outside the scope of this work. That being said, we can write

κe = 1e2T

(L2 − L1L−10 L1). (48)

Using the dispersion relations from equations 29 and 28, such tensors may be explicitlycalculated. It is obvious that L0 = σ For a completely free electron, we have

L1,zz = eτ√mxmymz

∫ (k2z h

2

mz

)(ε− µ)∂gFD

∂ε

kdkdθ

4π3

=e2τ√mxmymz

3π2mzh3

∫(2ε)3/2(ε− µ)∂gFD

∂εdε

= eµz2√2mxmymz

3π2h3 (kBT )5/2(− 5

2F3/2(ζ) + ζ32F1/2(ζ)

).

(49)

Hence, the Seebeck coefficient will be

S3D,zz = −kBe

(53F3/2(ζ)F1/2(ζ) − ζ

). (50)

Similarly, one can write the other tensor and, as a result, the thermal conductivity

L2,zz =e2τ√mxmymz

3π2mzh3

∫(2ε)3/2(ε− µ)2∂gFD

∂εdε

=e2τ√mxmymz

3π2mzh3 (kBT )5/2

(− 7

2F5/2(ζ) + 5ζF3/2(ζ)− ζ2 32F1/2(ζ)

).

(51)

Therefore, the electronic contribution to the thermal conductivity will be

19

κezz = τkB(2kBT )5/2

6π2h3

(mxmy

mz

)1/2(72F5/2(ζ)−

25F 23/2(ζ)

6F1/2(ζ)

). (52)

A similar procedure is followed in order to obtain the thermoelectric coefficient and thethermal conductivity in the case of electrons confined to a two dimensional plane. Sincethe mathematical calculation is analogous, with the only difference that integration is doneover a two dimensional polar coordinates and not three dimensional spherical coordinates,we shall briefly quote the result[10]:

S2D,xx = −kBe

(2F1(ζ∗)F0(ζ∗) − ζ

∗), (53)

κe2D,xx = τh2

4πa

(2kBTh2

)2(my

mx

)1/2kB

(3F2(ζ∗)− 4F 2

1 (ζ∗)F0(ζ∗)

). (54)

We conclude this subsection by comparing the prediction of Seebeck coefficient to ex-perimentally acquired data. We calculate the electron concentration in accordance withequation38. We do not compare thermal conductivities as phonons have not been consideredin this work.

(a) Experimental Result[18] (b) Relaxation time approximation

Figure 8: Seebeck coefficients obtained a) experimentally by Suh et al and b) under therelaxation time approximation

Figure 8 allows us to compare experimental data of Suh et al [18] with the relaxation timeapproximation. As observed, the semi classical model correctly predicts the order of magni-tude, as well as the near constant tendency for increasing temperature. This is to be expectedfrom equation 53: for temperatures significantly lower than the Fermi temperature, the ratioof Fermi integrals becomes the reduced Fermi energy, which is approximately constant. Un-like the case of electrical conductivity, the relaxation time approximation constitutes a moreaccurate estimation for the Seebeck coefficient.

20

2.2.3 Figure of Merit

The mathematical treatment done above finally allows us to write the theoretical predictionfor the figure of merit in the relaxation time approximation. For a bulk material in whichelectrons move freely[10],

ZT3D =32

(5F3/2(ζ)3F1/2(ζ) − ζ

)2F1/2(ζ)

B−1 + 72F5/2(ζ)−

25F 23/2(ζ)

6F1/2(ζ)

, (55)

where B is a constant related to the phononic thermal conductivity κp:

B =√mxmymz

3π2

(2kBTh

)3/2k2BTµzeκp

. (56)

Similarly for a two dimensional quantum well[10],

ZT2D =

(2F1(ζ∗)F0(ζ∗) − ζ

∗)2F0(ζ∗)

B−1 + 3F2(ζ∗)− 4F 21 (ζ∗)

F0(ζ∗)

, (57)

where

B =√mxmy

2πa

(2kBTh

)k2BTµzeκp

. (58)

As before, we compare the in plane prediction of the figure of merit, given by equation57, with two different authors, who calculated the same quantity by experimentation [18] orby Density Functional Theory (DFT) simulations[5]. The phononic contribution is given asexperimental input [11].

(a) Density Functional Theory (b) Relaxation time approximation (c) Experimental data[18]

Figure 9: In plane figure of merit (x direction) as a function of electron carrier concentrationcalculated with a) Density functional theory simulations [5] b) Relaxation Time approxima-tion c) Experimentally Measured [18]

Several remarks may be done from figure 9. Firstly, the relaxation time approximationcorrectly predicts the existence of a maximum figure of merit which can be achieved through

21

doping, in agreement with both DFT simulations and experimental data. The ZT fallsmonotonically to zero on both sides of the maximum: for lower electron concentrations, thereis a limited number of carriers so the electrical conductivity is reduced. For a higher numberof charge carriers, the thermal conductivity increases, which negatively impacts the ZT.Similarly, the relaxation time approximation also predicts the rise in ZT with temperature,which happens because of more available charge carriers in the conduction band and becauselattice thermal transport is hampered, as the mean free path of phonons is shortened [5].

However, several limitations arise in the relaxation time approximation; on one hand,the curves in figure 9b are much more narrow than that of figure 9a. Furthermore, therelaxation time approximation overestimates the figure of merit: while DFT simulationspredict a maximum of 0.9, figure 9b reaches values as high as 2. The main reason for thisis because, as we will discuss below, the assumption that the potential barrier is infinite isunrealistic, affecting the accuracy of the model. Other reasons for the discrepancy mightinclude the disregard for degeneracy or higher order effects.

2.3 Broido’s Extension and Out of Plane PropertiesSo far we have only estimated in plane transport properties by assuming an infinite potentialwell like structure. However, the calculation of out of plane properties using the dispersionrelation in equation 29 is not possible due to the fact that it does not depend on kz, so itsderivative is zero. This is understandable since, if the potential barrier was truly infinite, noelectron could tunnel through it, so the net electrical conductivity is zero. However, severalstudies [13, 5] suggest that the out of plane direction may be advantageous. Therefore, wewish to extend the model in order to estimate the out of plane transport properties.

Broido et al [2] argue that the infinite well model is unrealistic, and therefore accountsfor limitations of the model developed by Dresselhaus and Hicks. Therefore, they refined thecalculation by considering a periodic square potential of finite height V0. This is known asthe Kronnig-Penney model[3]. Broido proposed that for weak coupling between the finitepotential well (i.e a high potential barrier or significantly separated potential structures), thedispersion relation for a specific subband in the z direction may be written as[2]:

εz(kz) = εz,0 + ∆(1± cos(kzd)), (59)where ∆ is some constant, d is the superlattice period and εz,0 is the bottom of the band.However, the authors assumed that the coupling between barrier was weak in order to opti-mize in plane figure of merit, rather than out of plane thermoelectric transport. Consequently,we need to determine how weak the coupling among a WS2 based superlattice would be.

We now wish to apply the Kronnig Penney model to such structures. Said task requiresfinding the dispersion relation ε(~k), which is given by the solutions to the following equation:

Q2 −K2

2KQ sinh(Qb) sin(Ka) + cosh(Qb) cos(Ka) = cos(kzd), (60)

where

22

Q =√

2mz(V0 − ε(k))h2 , (61)

and

K =√

2mzε(k)h2 , (62)

and a and b are the widths of the well and the barrier (d = a+b). Clearly equation 60 cannotbe solved analytically for a finite V0, so we solve it numerically instead.

The heterostructures which were studied in this work consist of graphene and WS2, sowe will assume a periodic intercalation of a monolayer of these materials for simplicity. Aexperimental study performed by Yamaguchi et al [21] suggests that such configuration doesexhibit tunneling properties, where tungsten disulfide acts as a barrier with a height of 0.37eV.

Figure 10: Dispersion relation of graphene/WS2 heterostructures using the Kronnig Penneymodel

Figure 10 shows that the dispersion relation for a TDMCs structure, far from having aweak coupling between the wells, assimilates that of a quasi free electron in the presence ofa small, perturbative potential. In order to confirm this, we take the lowest energy band andadjust it to both equation 59 and to a free electron dispersion, given by the z component ofequation 28.

23

Figure 11: Comparison between Broido’s dispersion relation and free electron dispersionrelation

In figure 11 it is clear that the electrons behave as quasi free electrons, as the numericalsolution for the Kronnig Penney model matches exactly the parabolic form of free electronsexcept at the borders of the First Brillouin zone. Therefore, all of the transport propertiesestimated above would be valid for such a structure. However, it is worth asking if nanostructuration would serve any meaningful purpose. After all, according to figure 11 electronsmove freely like in a normal three dimensional material. The fact is that the main impact ofnanometric heterostructures in the thermoelectric structure is on the phononic rather thanelectronic transport[20]. Heterostructures such as graphene/MoS2 of thin layers have beenfound to act as phonon filtering structures in which high energy phonons are not transmitted,hence reducing the lattice thermal conductivity and improving the figure of merit [16, 13].If this is true for the fabricated structures based on graphene and WS2, then a stacking ofmonolayers of both materials may decouple phonons from electrons: it allows electrons tomove freely while blocking phonon transport. The verification of such claim, or the theoreticalstudy of phonon transmission and transport, is outside the scope of this thesis, and shouldbe investigated in future works.

24

3 Experimental Work

3.1 Procedure

3.1.1 Device Fabrication

The experimental method consisted on fabrication of Graphene/WS2 heterostructures de-posited on Si substrates, which where then characterized by AFM microscopy, SEM mi-croscopy and Harman transient measurements. Graphene was produced by electrochemicalexfoliation[8]; a 3cm x 4cm piece of graphite paper is electrochemically exfoliated into 60mlof H2SO4 0.1M solution with an applied voltage of 8.5V until the sheet is completely de-tached from the electrode. Then the resulting product is filtered several times with a pumpin order to remove the acid. The product is then dispersed in distilled water and centrifugedat 3600rpm for two hours. This procedure is shown below

Figure 12: Fabrication of graphene solution by electrochemical exfoliation

Tungsten disulfide was fabricated by micro-tip sonication in N-Methyl-2-pyrrolidone forfour hours. The material was then filtered and dispersed in water via ultrasonication[18].Both solutions where spray coated onto 1cm x 1cm, 0.5mm thick Silicon substrate. Byvarying the order of deposition, two types of devices where made, which are shown below infigure 13.

25

(a) WGW Device (b) GWG Device

Figure 13: Fabricated devices based on Graphene/WS2 heterostructures. EEG stands forElectrochemically Exfoliated graphene

Two replicates of each type of device where made, for a total of four devices. For clarity,we will refer to devices in figure 13a as WGW and devices in figure 13b as GWG. For theWGW devices, the silicon substrate was activated through oxygen plasma for ten minutesbefore depositing the first layer. Before the devices were completed, the layer of WS2 ongraphene and on silicon was observed through SEM and AFM microscopes. After depositingall the layers, silver contacts were deposited via thermal evaporation.

3.1.2 Harman Characterization

Figure 14: Diagram of the experimental setup for Harman measurements, which ideallyshould by adiabatically sealed.

In figure 14 the experimental setup for Harman characterization is shown. The sampleswhere contacted using silver paint to four wires which where then conected to a source meter.Similarly, the device was placed on a heating plates whose temperature could be controlled.Thermocouples on the hot and cold side of the sample where placed for monitoring. Pulsedpotential differences where sent via the source meter in order to do the characterization.

26

Measurements were taken between 20°C and 65 °C. For each temperature, the sample wassubjected to five on-off cycles. We report the statistical average.

3.2 AFM and SEM characterization

(a) (b)

Figure 15: SEM images of WS2 deposited on graphene coated Si (GWG devices) with differentmagnification, 5µm and 100µm

(a) (b)

Figure 16: SEM images of WS2 deposited on bare Si (WGW devices) with different magni-fication, 20µm and 100µm

27

(a) Analyzed areas

(b) Results of spectrum 1 (c) Results of spectrum 3

Figure 17: Example of compositional analysis (EDS) on a GWG device.

(a) (b)

Figure 18: a) AFM image of WS2 deposited on graphene coated Si b) Height profiles for theselected grains, labelled as 1, 2 or 3

28

(a) (b)

Figure 19: a) AFM image of WS2 deposited on bare Si b) Height profiles for the selectedgrains, labelled as 1, 2 or 3

Figures 15 and 16 show the SEM images for WS2 deposited onto graphene coated and baresilicon, respectively. Similarly, figure 17 shows the EDS compositional analysis performed tothe samples. Such analysis confirms that on both samples the white gray crystals correspondto Tungsten disulfide, as seen in figure 17b. Also, figure 15b shows a significantly darker areaon the center, which occupies a significant portion of such figure. EDS revealed high contentsof carbon in said area, which allows us to recognize the layered graphene (figure 17c) on GWGdevices. Figures 15a and 16a that the deposition technique produced relatively large WS2monocrystals, whose size is in the order of micrometers, regardless of the substrate. However,it is clear from these images that tungsten disulfide, unlike graphene, does not form thin films,as it tends to form clusters instead. Even with more sophisticated deposition techniques suchas magnetron enhanced sputtering[4], the formation of WS2 uniform films is experimentallydifficult. This will constitute a major difference between the theoretically predicted, whichassumes uniform superlattice structures, and the experimentally measured figure of merit.

Figures 18 and 19 show the AFM Height retrace of the WS2 deposited on bare Siliconand Graphene coated silicon, respectively. AFM measurements show that the height of WS2grains on both substrates is relatively large, reaching values of up to 500nm-900nm accordingto figures 18b and 19b. This, from an experimental point of view, may suggest that tryingto deposit a higher quantity of Tungsten Disulfide may be useless or even counterproductive:instead of increasing the coverage, the deposited substance will just form bigger and higherclusters. As we have seen on the previous section, Tungsten disulfide acts as an electronpotential barrier. Also, it is a well reported fact that undoped WS2 has a low electricalconductivity[18]. Hence, bigger grains may have a negative impact in the out of plane figureof merit. Additionally, such huge grains imply that a large quantity of TMD monolayers arestacked, meaning that the quasi free electron model discussed before might not be suitablefor the modelling of these devices.

Finally, AFM measurements show an average roughness of 35nm for the graphene coatedsubstrate and of 0.8nm for the bare silicon. Such roughness was calculated in areas were

29

no WS2 clusters were observed. The higher roughness is of course due to the underlyinggraphene, and the fact that the difference is two orders of magnitude suggests that a goodsurface coverage was achieved with graphene deposition.

3.3 Characterization by Harman Transient Method

(a) Sample raw data (b) Sample Processed Data

Figure 20: Examples of the collected data a) before and b) after data processing and re-gression. The red dotted lines show the fitted curve in the linear (Ohmic) and exponential(Thermoelectric) regimes. The continuous black line marks the Seebeck Voltage.

Figure 20 shows an example of the collected data. Each raw data consists of several pulses ofpotential difference, as shown in figure 20a. For each of the pulses, a linear and exponentialfit was made to the corresponding regimes, as illustrated in figure 20b. This permits thedetermination of the Seebeck voltage, which in turn permits the calculation of the figure ofmerit, Seebeck coefficient and electrical conductivity. The measurements were repeated atdifferent temperatures as stated above. This also allows us to determine the power factor α,which is defined as:

α = S2σ. (63)

30

(a) Figure of merit (b) Electrical conductivity

Figure 21: Experimental results for the out of plane figure of merit and electrical conductivityfor the fabricated devices at different temperatures.

(a) Seebeck Coefficient (b) Power Factor

Figure 22: Experimental results for the out of plane figure of merit and electrical conductivityfor the fabricated devices at different temperatures. The data point of GWG device at 50°Cis an outlier and hence it is not shown. Error bars are not visible at this scale.

Figures 21 and 22 show the figure of merit, electrical conductivity, Seebeck coefficient andpower factor for the measured temperature range of the devices. The error bars account forthe uncertainty arising from the measurement of multiple cycles. In figure 21a we can observethat the obtained ZT values are very high, ranging from 0.5 at near ambient temperatureto 1.3-1.75 at 50 - 65 °C. These values constitute a dramatic increase from the reported ZTof pristine WS2 or graphene[18, 11]; the devices show a relatively high as a thermoelectric.However, the data should be interpreted with caution, as there are two main experimentalconsiderations to take into account. First, figure 13 shows that the contact deposition wasmade in such a way that the silicon substrate is also included in the measurements, soit contributes to the overall transport properties. Second, the experimental setup fails toensure fully adiabatic operation, so there might be an experimental error introduced by

31

minor heat leaks. Despite these limitations, the fabricated devices where found to be goodthermoelectrics at higher temperatures, as predicted by the relaxation time approximation.

Figure 21b show that the measured experimental conductivities are in an order of magni-tude of 1.0 S/m. These are very low values when compared to those reported elsewhere, whichmay be as high as 103 - 104 S/m [18, 11]. Nonetheless, this is to be expected from devicesmade of undoped tungsten disulfide and layered graphene, whose out of plane conductivityis known to be low. These values, despite being low, may be promising. If good ZT resultswere obtained from these devices, even higher thermoelectric performance could be achievedby increasing the electrical conductivity through doping, as predicted by the relaxation timeapproximation or by DFT simulations [5]. Finally, the figure shows that conductivity in-creases with temperature. This agrees with the relaxation time approximation and evincesa semiconductor like behavior, which is desirable for thermoelectric applications. This isespecially true for WGW devices, which experience a sharper increase in their conductivitythan their GWG counterparts. Such higher conductivities would explain why WGW deviceshave a better overall performance.

On the other hand, figure 22 illustrates the behavior of the Seebeck coefficient and thepower factor as a function of temperature. The achieved power factors for the fabricateddevices range from 5 to 20 µW/mK2 for most temperatures, but values of 70 µW/mK2 forGWG and 120 µW/mK2 for WGW were reached at 25 °C. These orders of magnitude arelow compared to other sources[18, 11] but expected from the fact that tungsten disulfide isundoped. On the other hand, the Seebeck coefficient ranges between 20-120mV/K, with itshighest value being at 25°C. Both the order of magnitude and the descending trend withincreasing temperature of the Seebeck coefficient is comparable to other reports in literature[13]. We conclude that the low power factors are due to low electrical conductivities. Ifthe power factor is low, then the high thermoelectric performance for the devices can onlybe attributed to significant reduction in thermal conductivity. We believe, as stated before,that the fabricated heterostructures act as high energy phonon filters[16], reducing the latticecontribution to the thermal conductivity that is in itself the predominant contribution [5].However, further work is needed in order to confirm this hypothesis.

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4 Conclusions and Future WorkThe nature of this work was, as planed, both theoretical and experimental. On the theoreti-cal aspect, we successfully reproduced Dresselhaus and Hicks calculations under the constantrelaxation time approximation, and used it to model thermoelectric transport properties. Wecompared and contrasted the predictions of this model with experimentally acquired data forin plane Seebeck coefficient, thermal conductivity, electrical conductivity, and figure of meritof tungsten disulfide. Such comparison demonstrated that the relaxation time approximationcorrectly describes the behavior of these variables in function of electron concentration andtemperature, but may not estimate well the orders of magnitude. In particular, we find thatthe semi classical model in question predicts a maximal ZT for an optimized concentration,in accordance to both DFT simulations and experimental data. The relaxation time approx-imation also shows that tungsten disulfide improves its thermoelectric performance at hightemperatures, in accordance with the reported findings in literature. Finally, we attepted touse Broido’s refinement to better estimate the thermoelectric properties of WS2/grapheneheterostructures. We found that electrons should behave in a quasi free fashion in thesenano structures, given the low potential barrier. As future theoretical work, a more completetheoretical model which includes phonon scattering, degeneracies and band structure shouldbe developed.

On the other hand, these heterostructures were fabricated by exfoliation of both grapheneand tungsten disulfide, followed by deposition via spray coating. Morfological characteriza-tion with SEM and AFM shows that, unlike graphene, WS2 does not form uniform layers, asit accumulates in large clusters instead. Despite this, Harman characterization shows thatthe devices exhibit good thermoelectric properties, with figures of merit reaching values ashigh as 1.75. The figure of merit was found to increase as the temperature rose, as expectedin the relaxation time approximation. Nonetheless, the power factor of these devices wasfound to be low, in the order of 5 -18µW/mK2. Such low power factor means that the goodthermoelectric performance is due to a significant reduction in thermal conductivity. Thislead us to believe that this is because of high energy phonon blocking effect, but such claimis outside the scope of this thesis. Experimentally, future work should focus on combiningthe fabrication of these heterostructures with optimal doping in order to further increase thefigure of merit. Also, further experimentation is needed in order to confirm this high figuresof merit and to confirm if the phonon blocking is responsible for such performance.

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