On a phenomenological Ginzburg-Landau theory of charge density waves in dichalocogenides

Transcription

1 On a phenomenological Ginzburg-Landau theory of charge density waves in dichalocogenides Danielle Schaper Spring, 2014 Contents 1 Introduction 2 2 Transition metal dichalcogenides 3 3 Order Parameter and Broken Symmetry 4 4 Charge Density Waves 5 5 Free energy of Dichalcogenides 6 6 Euler Lagrange Formalism 9 Abstract This paper presents a phenomonological Ginzburg-Landau free energy theory tailored to describe the behavior of charge density waves in transition metal dichalcogenides, namely TaS 2 and TaSe 2. In the commensurate phase, we show that the theory admits topological defects in the order parameter field in three different, distinct directions and attempt to extract observable quantities from the parameters present in the proposed GL free energy expression.we compare our results from the proposed theory to experimental scanning tunneling microscopy data on TaSe 2. 1

2 1 Introduction In condensed matter physics, strongly correlated materials can give rise to a plethora of novel phases. Unlike band insulators or electron gas, strongly correlated materials that exhibit properties such as the quantum hall effect, heavy fermion systems, geometrically frustrated magnetism, and superconductivity defy simple analysis in terms of independent electrons due to the very strong interactions between electrons in the system. Sometimes accompanying the phenomenon of superconductivity is the existence of a static, periodic modulation electron density in the material. This state, which we will refer to the charge density wave (CDW) phase, describes the superlattice modulation of the electronic charge on top of the atomic crystal structure. This type of phase was first proposed by Rudolph E. Peierls and was postulated to occur for one dimensional systems, though such modulations are known to occur in electronic systems of higher dimensions. Charge density waves are found to exhibit many interesting phenomena due to their electronic condensation leading to materials exhibiting desirable properties such as enhanced electric conduction, plasmon depinning, and collective excitation of the free charge carriers present in the material.[1] This paper strives to better understand and interpret the phenomenological expression for the Ginzburg-Landau (GL) theory for free energy put forth by William McMillan in the late 1970 s. This is done by defining a general order parameter for the electron density and expanding McMillan s theory using the Euler-Lagrange formalism, focusing especially on the gradient term present in the expansion: F 3 = n [ d 2 r e ( q n i q n q n ) Ψ n ( r) 2 + f q n Ψ n ( r) 2] (1) We will then compare the theoretical results to data that was taken using scanning tunneling microscope (STM) techniques to see what empirical correlations can be made in an attempt to bridge theory and experiment. In this paper, an overview of transition metal dichalcogenides is given, followed by an introduction to the concept of order parameters and symmetry breaking. An example of the Ginzburg Landau expansion for a simple magnet system is given to draw an analogy to the CDW system, which is discussed in the following section, where we discuss McMillan s GL free energy expansion for CDWs for transition metal dichalcogenides. We then derive a somewhat coarse formulation of the GL expansion, and then we derive it using the Euler- Lagrange formalism to find an acceptable partial differential equation governing the order parameter. 2

3 2 Transition metal dichalcogenides A simple realization of charge density wave occurs in materials known as the transition-metal dichalcogenides (TMDs). TMDs consist only of two different chemical elements, MX 2 where M is a transition metal and X=S, Se, Te is a chalgogen anion. Most of the TMDs exhibit a layered structure (alternating planes consisting of a single atom type) with strong in-plane bonding and weak out-of-plane interactions. These materials are desirable candidates for study for many reasons; the fortuitous electron/hole combination of the metal and the chalcogen results in an absence of dangling bonds at a cleaved surface, allowing the surface to remain rather inert and not react with any other species present in the environment. Even more importantly, the bonding between the layers is due to weak Van der Waals interactions, and it is this weak interlayer bonding that allows us to cleave the material easily and get a clean surface for study, thereby making them ideal for studying surface properties with techniques such as atomic force microscopy or scanning tunneling microscopy [8]. In this paper, we are specifically interested in the TMDs TaS 2 and TaSe 2. These materials consist of a hexagonally-packed layer of metal Ta atoms sandwiched between two layers of chalcogen (S or Se) atoms with a trigonal prismatic structure. However, the purposes rest of the paper, we will restrict our analysis to the two-dimensional properties of a single layer encountered at the cleave surface of these materials. The lattice superstructure below the critical temperature T CDW is formed by the triple-q modulations of the ionic positions along the three symmetry axes of the hexagonal 2H-phase. The q-wavevector characterizing the charge density wave can be incommensurate. However, in the case at hand the q wavevector remains locks itself to k/3 (here k are the primitive lattice vector. In terms of the lattice, the periodic lattice is described as R nm = n a 1 + m a 2 The reciprocal lattice vectors are a 1 = aê x (2) [ ( π ) ( π ) ] a 2 = a cos ê x + sin ê y (3) 3 3 (4) k1 = 2π a êy (5) k2 = 2π [ ( π ) ( π ) ] cos ê x sin ê y (6) a 6 6 It will be useful to introduce a third wavevector [ ( k 3 = 2π a cos π ) 6 êx sin ( ) ] π 6 êy. For our purpose it will be useful to introduce the unit reciprocal vectors ˆq n ˆq n = k n k (7) 3

4 (a) Lattice structure of TaSe 2 (b) Location of Ta atoms in STM topograph (c) Location of Se atoms in STM topograph Figure 1: Location and orientation of atoms in materials [8] where n = {1, 2, 3}. The period of the electronic density is commensurate with the atomic crystal structure and form a superlattice. Since the period of the electronic modulation is larger than the crystalline order, it implies an arbitrariness to the phase of the charge density wave. therefore, the charge density wave must be a broken symmetry of the system. 3 Order Parameter and Broken Symmetry In order to explain the idea of broken symmetry and order parameter, let s take a detour into the field of emerging behavior. In condensed matter physics, we can try to understand the behavior of large systems of particles by studying their emerging behavior that is, try to understand why messy combinations of zillions of electrons and nuclei do such interesting simple things; the fundamental desire is not to discover the underlying quantum mechanical laws, but to understand and explain the new laws that emerge when many particles interact. Although the laws that govern the behavior of a single electron are well known after all, electrons follow the laws of quantum mechanics with a dynamic dictated by the laws of electromagnetism the emerging behavior of many electrons can reveal many surprises. On a practical level, it is impossible to keep track of so many individual particles in a system; however, one can describe their behavior using an order parameter, the behavior of which can break the symmetry of a system and describe the state or behavior of the system at hand. For instance, in the case of a magnet, the order parameter is the magnetization that is, the average local magnetic moment of the electrons. Magnets have a broken rotational symmetry, i.e. the lowest energy state of the system is independent of the direction of the order parameter; the local magnetization is arbitrary. This follows from the fact that the electronic dipole moment prefers to point in a direction that is parallel to its neighbor. This means that the energy of any spin configuration depends solely on the relative angle between two adjacent spins and is independent on the overall direction of the magnetization. As a 4

5 result the free energy of a magnet is of the form [ K F = d 3 r 2 m( r) 2 + µ 2 m 2 + g ] 4 m 4 where K, µ, g are constant (although they can depend on temperature). The important thing to note here is that the free energy is invariant under rotation. The free energy can be thought as a Taylor expansion of a coarse grained field; that is, when the value of the field is small, the free energy expansion is solely dictated by symmetries of the order parameter. 1 It is apparent that when µ changes signs the free energy exhibits a discontinuity. Indeed, when µ > 0, the free energy is minimized when the order parameter vanishes m = 0 whereas when µ < 0, the free energy is minimized when 6µ m = g If µ = a(t T c ), it implies that the magnetization follows the form 6a m = T Tc g below the critical temperature T c. Therefore, this qualitatively correct description of the magnetization near the critical temperature was based on very general properties of the order parameter, its symmetry and the free energy of the system. In much the same manner, we can characterize the charge density wave system using an order parameter (the electronic density at the surface of the crystal), and use this GL expansion to characterize the free energy of the system. 4 Charge Density Waves In this work, we will study the behavior of charge density waves. CDWs correspond to a periodic modulation of the electronic density, i.e. regions of rarefactions and compression of the electronic density, that also correspond to a associated modulation in the underlying atomic lattice positions. CDWs are caused by an instability in the metallic Fermi surface involving the electron-phonon interaction, resulting in energy gaps at the Fermi surface with a wavelength of λ CDW = π k F, where k F is the Fermi wavevector. How does this happen? In a normal metal, the electron distribution is fairly uniform due to the Coulomb repulsion between electrons. However, below some critical temperature, T CDW, an appropriate modulation of the lattice atoms would produce gaps in the Fermi energy at ±k F, and the cost in elastic energy to distort the lattice is less than the gain in conduction energy and overcomes 1 The particular form that this free energy takes is known as a scalar φ 4 field, and one most extensively studied systems in field theory. (8) 5

6 Figure 2: Scanning tunneling image of TaSe2 at temperature of 77K. the Coulomb repulsion, causing the CDW state to be the preferred state. 2 This second-order phase transition that occurs between the metallic state and the CDW state is known as the Peierls transition. [9] As seen in Figure 2, the material shows a CDW present at the cleave surface of a TaSe 2 crystal. The charge density wave was measured using a scanning tunneling microscope (STM), which relies on the quantum tunneling effect of electrons from the surface of the system to the tip of the STM. Since quantum tunneling is exponentially dependent upon the distance between the sample and the tip, it provides an atomically sensitive way to measure the electron density. The topograph shows the electronic density as represented via a color scheme. The unit cell is illustrated on the figure as well as the superlattice structure of the charge density wave. As you can readily observed, there are three charge density waves along the three ˆq n directions. In 2H-TaSe 2 the incommensurate charge density appears below a temperature of 122K. Lowering further the temperature, it is found that the charge density waves locks in to the crystal structure to form a commensurate charge density wave below temperature of 90 K. 5 Free energy of Dichalcogenides In his seminal work, William McMillan proposed that the free energy of the CDWs as seen in TMDs could be expressed as [6] F = F 1 + F 2 + F 3 (9) 2 At temperatures above T CDW, the electronic energy gain is reduced by the thermal excitation of electrons across the gap, causing the metallic state to be the more energetically stable. 6

7 y a 2 a 1 x Figure 3: Triangular lattice with the primitive unit vectors a 1 and a 2.. where F 1 is the homogeneous contribution to the free energy, F 2 are contributions to the free energy due to disorder and impurities present in the crystal lattice and F 3 are gradient terms. The Ginzburg-Landau free energy is written in terms of a power series expansion and gradient terms of the relevant order parameter, which McMillan proposed can be described by a complex order parameter, Ψ( r). The homogeneous term, F 1 is [ F 1 = d 2 r aα 2 ( r) bα 3 ( r) + cα 4 ( r) + d ( Ψ 1 ( r)ψ 2 ( r) 2 + Ψ 2 ( r)ψ 3 ( r) 2 + Ψ 3 ( r)ψ 1 ( r) 2)] (10) where α( r) = R [Ψ 1 ( r) + Ψ 2 ( r) + Ψ 3 ( r)] is the modulation of the electron density and Ψ n ( r) are the complex order parameter associated with the CDW along the 1, 2, 3 directions. The ˆq n direction are 120 o apart. The integral is performed over the surface of one the layer. The term F 2 includes the random potential U( r) due to impurities or defect in the crystal lattice. F 2 = d 2 rgu( r)α( r) (11) The gradient term: F 3 = [ d 2 r e ( q n i q n q n ) Ψ n ( r) 2 + f q n Ψ n ( r) 2] (12) n 7

8 As usual, we make the assumptions that near the critical temperatures, the coefficients a, b, c, d, e, f, g are smooth functions of temperature. Since the magnitude of α depends critically of the sign of a, we assume that a = a 0 (T T c ) and that all the other coefficients are temperature independent near T c. For the purpose of this paper we will consider a clean metal that is U( r = 0) and consider the minimum free energy states, It might be useful to rewrite the CDW order parameter in terms of a slowly fluctuating field ψ n ( r) where Ψ n ( r) = e i qn r ψ n ( r). In this case the gradient term can be rewritten as F 3 = d 2 r n [ e ( q n ) ψ n ( r) 2 + f q n ψ n ( r) 2] (13) At this point it become apparent that the e and f terms described fluctuations of the ψ fields along and perpendicularly to the wavevector q n. The homogeneous term becomes [ F 1 = d 2 r aα 2 ( r) bα 3 ( r) + cα 4 ( r) + d ( Ψ 1 ( r)ψ 2 ( r) 2 + Ψ 2 ( r)ψ 3 ( r) 2 + Ψ 3 ( r)ψ 1 ( r) 2)] (14) where α( r) = Re [ Ψ n ( r)] = Re [ n ei qn r ψ n ( r) ] [ = 1 2 n ei qn r ψ n ( r) + e i qn r ψn( r) ], Given that ψ n ( r) are slow coarsed-grained oscillating fields, we can approximate d 2 raα 2 ( r) = d 2 r a ( ψn ( r)ψ m ( r)e i( qn+ qm) r + ψ 4 n( r)ψ m ( r)e i( qn+ qm) r n,m + ψ n ( r)ψ m( r)e i( qn qm) r + ψ n( r)ψ m( r)e i( qn qm) r) As you can see from the previous equation, the oscillatory terms will make a negligible contribution to the integrals unless either, q n + q m = 0 or q n q m = 0 is fullfilled. Only the second condition is possible, and occurs when for the case that n = m. As a result, the previous equation can be simplified to d 2 raα 2 ( r) d 2 r a ψ 2 n( r)ψ n ( r) (15) Similarly, for the cubic term, we find d 2 r ( bα 3 ( r) ) d 2 r b 8 (ψ 1( r)ψ 2( r)ψ 3( r) + ψ 1 ( r)ψ 2 ( r)ψ 3 ( r)) (16) where we have used the property that q 1 + q 2 + q 3 = 0, a reciprocal lattice vector. We note in passing that if the CDW is forming a superlattice that is commensurate with the lattice, an additional term 3 q n = K, where K is a reciprocal lattice vector, then additional umklapp terms are allowed, d 2 r ( bα 3 ( r) ) d 2 r b 8 (ψ 1( r)ψ 2( r)ψ 3( r) + ψ 1 ( r)ψ 2 ( r)ψ 3 ( r)) (17) + d 2 r b ( ψ 3 8 n ( r) + ψn 3 ( r) ) δ 3 qn K (18) n n 8

9 In the limit where only one CDW is present, we can say (without any loss of generality) that ψ 1 = ψ and ψ 2 = 0 and ψ 3 = 0. Then, the homogeneous term simplifies to [ a F 1 = d 2 r 2 ψ ψ + 6c ] 16 (ψ ψ) 2 (19) The net result is (writing q = q 1 ) [ a F = d 2 r 2 ψ ψ + 6c 16 (ψ ψ) 2 + e q 2 ψ( r) ] 2 + f q 2 ψ( r) 2 6 Euler Lagrange Formalism (20) Minimizing the free energy with respect to ψ, we find that the field must obey the differential equation, i.e. we evaluate the function using the Euler-Lagrange formalism i.e. F + F = F (21) r ψ r ψ r ψ r Using the free energy density for a single CDW, we arrive to a 2 ψ + 3c 4 ψ 2 ψ e q 2 2 ψ ( r) f q 2 2 ψ ( r) = 0 (22) This is equivalent to a complex wave z(x, y) of the kind a 3c 2 z(x, y) 2 z(x, y) z(x, y) z 2 z(x, y) m x x 2 m y y 2 = 0 (23) The anisotropy in the last two terms can be eliminated by introducing a judicious change of coordinates, The field equation becomes, x = m x x y = m y y a 2 z(x, y ) + 3c 4 z 2 z(x, y ) 2 z(x, y ) x 2 2 z(x, y ) y 2 = 0 Introducing polar coordinates (r, θ), where x = r cos θ y = r sin θ r 2 = x 2 + y 2 tan θ = y x 9

10 a 3c z(r, θ) z 2 z(r, θ) 2 z(r, θ) r 2 1 z(r, θ) 1 2 z(r, θ) r r r 2 θ 2 = 0 (24) To solve this partial differential equation we use the technique of separation of variables, z(r, θ) = R(r)Θ(θ), d 2 Θ(θ) dθ 2 = m 2 Θ(θ) (25) Given that the solution must satisfy the condition Θ(θ + 2π) = Θ(θ), then the solution is Θ(θ) = e imθ where m is an integer. The radial portion of the differential equation becomes, a 3c R(r) R3 (r) 2 R(r) r 2 1 R(r) + m 2 1 R(r) = 0 (26) r r r2 This non-linear differential equation can not be solved analytically but can be solved numerically. We are interested in a topological solution for which m = 1 and the amplitude of the CDW vanish at the origin, R(0) = 0 but reach 2a its steady value at infinity, R( ) = 3c. if cz 2 (x)is small then a 3c z(x) z3 (x) m d2 z(x) dx 2 = 0 (27) a 2 z(x) z(x) md2 dx 2 = 0 (28) In the limit of c being small it becomes an exponential decaying function. ( z(x) = Const exp ( ) ma/2)x (29) The length scale for the decay is ξ = 2m a 2e q 2 2f q a, and ξ = 2 a. In our case we have two correlation lengths, ξ = In the case Although this is not the main subject of this thesis, we note in passing that the nature of the charge density wave has not been determined and remains unsolved since their discoveries in the 70 s. A popular scenario attributes the charges density to an electronic instability associated with nesting of the Fermi surface, i.e. two Fermi surfaces would be separated by a constant wavevector Q. In the near future, comparing the expected behavior from this expansion to the data that has been taken for this project is expected. 10

11 References [1] Y. I. Latyshev, O. Laborde, P. Monceau, and S. Klaumnzer, Phys. Rev. Lett. 78, 5, (1997). [2] M. Tinkham, Introduction to Superconductivity, (McGraw-Hill, New York, 1975). [3] R. E. Peierls, Theory of Solids (Oxford University Press, London, 1955). [4] W. L. McMillan, Phys. Rev. B 16, 643 (1977). [5] W. L. McMillan, Phys. Rev. B 12, 1197 (1975). [6] W. L. McMillan, Phys. Rev. B 12, 1187 (1975). [7] W. L. McMillan, Phys. Rev. B 14, 1496 (1976). [8] Manish Chhowalla, Hyeon Suk Shin, Goki Eda, Lain-Jong Li, Kian Ping Loh, and Hua Zhang. The chemistry of two-dimensional layered transition metal dichalcogenide nanosheets. doi: /NCHEM.1589 [9] Robert E. Thorne. Charge Density Wave Conductors. Physics Today,

12 y a 3 a 2 a 1 x Figure 4: Triangular lattice with the primitive unit vectors a 1 and a 2. We have indicated a third lattice vector a 3 that will be useful for our purposes. Here, the stripes of larger dots show a line of maxima on a CDW; note how the CDW period is about three times that of the atomic lattice. 12

13 y k1 x k3 k2 Figure 5: Reciprocal lattice vector: In the reciprocal vector space, (a) Atomic Lattice (b) CDW along q 1, q 2 and q 3 Figure 6: Atomic Lattice and Superposition of charge density along q 1, q 2 and q 3 13

14 (a) CDW along q 1 (b) CDW along q 2 (c) CDW along q 3 Figure 7: Charge density along the three directions (a) CDW along q 1 and q 2 (b) CDW along q 2 and q 3 (c) CDW along q 1 and q 3 Figure 8: Superposition of charge density along the two directions (a) Charge displacement (b) Charge density wave Figure 9: Two pictures 14

15 (a) Charge displacement (b) Charge density wave Figure 10: Two pictures (a) Charge displacement (b) Charge density wave Figure 11: Two pictures 15

16 (a) Charge displacement (b) Charge density wave Figure 12: Two pictures 16