Towards Holographic Hydrodynamics in Strange Metal Cuprates

Towards Holographic Hydrodynamics

in Strange Metal Cuprates

THESIS

submitted in partial fulfillment of the requirements for the degree of

BACHELOR OF SCIENCE

in

PHYSICS& ASTRONOMY

Author : Frans van Die

Student ID : s1824392

Supervisor : Remko Fermin Msc.

Prof. Dr. Jan Aarts

Towards Holographic Hydrodynamics

in Strange Metal Cuprates

Frans van Die

Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands

July 10, 2020

Abstract

Unconventional superconductivity comes in a variety of forms, of which the high-Tc cuprates are one with a complex phase diagram containing rich physics.

Over certain doping range, strange metal behaviour is seen, for which a theo-retical explanation is still lacking. This thesis reviews the theotheo-retical advances in strange metal research and provides an overview of the experimental tools available. The experimental suggestions are based on the predictions of the AdS-CFT correspondence as a possible explanation for the strange metal pecu-liarities. This theory of holographic duality is characterised by the prediction of unconventional hydrodynamics in the strange metal cuprates. Therefore, several suggestions for hydrodynamic experimentation are given, firstly on the boundary effects for the electron current flow profile and secondly on viscous electron backflow after a nano-sized constriction. These are phenomena only yet observed in graphene. Thirdly, the sign-reversing Hall effect as seen in graphene possibly points towards unconventional hydrodynamics in the cuprates as well. This is followed by the description of preparatory experiments conducted on exfoliated flakes of Bi2Sr2CuO6+x, with a determined Tc of 10.0 K ± 0.3 K.

These strange metal experiments confirm literature values and provide a solid platform for conducting future experiments on possible unconventional electron hydrodynamics in this material.

Contents

1 Introduction 7

2 Conventional Electrodynamics in Condensed Matter 9

2.1 Regimes of Electron Flow . . . 9

2.2 Superconductivity . . . 12

2.3 Unconventional Superconductivity . . . 16

2.4 Hall Effect . . . 17

3 Bridging the Gap - The Phase Diagram 21 3.1 Antiferromagnetic Mott Insulator . . . 22

3.2 Pseudogap . . . 23

3.3 Fermi liquid . . . 24

3.4 Strange Metal . . . 26

3.5 Quantum Critical Point . . . 27

4 Unconventional Hydrodynamics 29 4.1 Holography . . . 29

4.2 Towards Unconventional Hydrodynamics . . . 32

4.3 Hydrodynamic Framework . . . 33

4.4 Probing Hydrodynamics . . . 35

4.4.1 Boundary Effects . . . 36

4.4.2 Viscous Backflow . . . 38

6 CONTENTS

5 BSCCO as a Strange Metal 41

5.1 The Structure of BSCCO . . . 42

5.2 Sample Fabrication . . . 43 5.2.1 Substrate Cleaning . . . 44 5.2.2 Exfoliation . . . 44 5.2.3 Lithography . . . 45 5.2.4 Sputtering . . . 48 5.2.5 Lift-Off . . . 48 5.2.6 Structuring . . . 49 6 BSCO Experimentation 51 6.1 Contact Testing . . . 53

6.2 Hall Bar Structuring and Contact Set-Up . . . 53

6.3 Resistivity Measurements . . . 54

6.4 Hall Effect Measurements . . . 57

7 Conclusion and Outlook 61

Acknowledgments 63

Bibliography 65

Chapter

1

Introduction

Ever since their discovery, various types of high-temperature cuprate su-perconductors have been found not following the theoretical framework of BCS theory. The cuprates furthermore exhibit multiple other remark-able phenomena that are not explained in the framework of conventional condensed matter physics, with the strange metal phase, characterised by T-linear resistivity excessing the Mott-Ioffe-Regel limit, as most mysteri-ous one. This thesis aims to examine these strange phenomena and sug-gests experiments for showing possible electrohydrodynamic behaviour as an experimental signature for theories based on holography.

This thesis commences with discussing conventional theories of con-densed matter electron transport in Ch. 2. Furthermore, the theoretical development of superconductivity is followed chronologically up to the unconventional superconductivity in cuprates. In Ch. 3, the phase dia-gram for the cuprates is introduced, and each of the phases is expanded upon, with the focus on the strange metal phase. Holography is further explained in Ch. 4, elaborating on why hydrodynamics might provide experimental evidence for a holography-based theory. In addition, three types of experiments are suggested for demonstrating hydrodynamic be-haviour in strange metal cuprates: on boundary effects, on viscous back-flow and on the sign-reversing Hall effect.

In Ch. 5, the production and contacting of thin flakes of Bi2Sr2CuO6+x

(BSCO), one type of strange metal cuprates, are discussed. Ch. 6 contains results gathered on experiments conducted on BSCO, in preparation for future experiments targeted more directly to hydrodynamic effects.

Chapter

2

Conventional Electrodynamics in

Condensed Matter

In order to obtain an understanding of the theory and the experiments on strange metals in later chapters, it is useful to discuss the conventional physics of electrodynamics in condensed matter. This chapter will discuss different regimes of electron transport, in addition to two phenomena that are important for the research that is discussed in this thesis: supercon-ductivity and the Hall effect.

2.1

Regimes of Electron Flow

In the first place, one should define what electronic transport means. Here, we use the definition used by S. Hartnoll in one of his lectures [1]:

Transport is the dynamics related to moving around conserved densities, like charge, energy and momentum.

In conventional metals, the way of describing transport mechanisms is by considering quasiparticles as the carriers of these conserved densities. The term quasiparticles is a quite poor description of the subject matter since there is nothing ’quasi’ about them. It is a name for the collective excitations of a certain microscopically complicated system. Quasiparti-cles are emergent phenomena in these systems, governed by weak inter-actions between the particles of the system. Due to these weak interac-tions, the collective excitations are allowed to be treated mathematically as though they were individual particles, because the weak interactions

10 Conventional Electrodynamics in Condensed Matter

allow us to determine single-particle states, without considering the colli-sions and possible entangled states with other particles. As an example of quasiparticles, both electrons and electron holes are treated as quasiparti-cles in electron band theory.

In general, we distinguish between two [2] or three [3] regimes of elec-tron transport in solids. These regimes are defined based on l, the so-called mean free path length of the quasiparticles. This l is related to the lifetime of the quasiparticle excitations in the following way:

l =vF·τqp (2.1)

where τqpis the lifetime of the flowing quasiparticle, and vF the Fermi

ve-locity of the system. In general, it is to say that the higher τqp, the longer

the quasiparticles live, and thus the higher the material’s conductivity. l represents the length that a quasiparticle typically travels in the medium before it is scattered. This scattering can be done either while conserving momentum, or relaxing momentum. The standard example of a momen-tum conserving scattering is the scattering between two electrons, com-monly written as e-e scattering. Momentum relaxing scattering can take up multiple forms. Electrons scattering in material impurities or defects, as well as electrons scattering with phonons (e-p scattering), are common examples of scattering events where momentum is not conserved. Since these two scattering mechanisms take up different values for the mean free path length, we now distinguish them. lMC is defined as being the mean

free path length of the momentum conserving scatterings, while lMRis the

same for the momentum relaxing scatterings.

Based on the relative values for lMC and lMR, different regimes of

elec-tron transport are distinguished [2][3]. In general, the scattering process that has the smallest value for l dominates the system’s behaviour, since the quasiparticle will more likely experience a scattering event for smaller length scales. We furthermore define the width of our system as W. This W is at the same time also a characteristic length scale of the system. The regimes then are as follows:

2.1 Regimes of Electron Flow 11

(a)

(b)

Figure 2.1: Cartoon depiction of the difference between regimes 1 and 2. In (a), the electrons flow independently. Transport is dominated by momentum relaxing scattering events with the impurities in the material, leading to Ohmic Knudsen flow. In (b), however, the electron flow is dominated by the momentum conserv-ing interparticle scatterconserv-ing events which leads to hydrodynamic Poisseuille flow. From [2].

1. In the first case, lMR << lMC << W, meaning that the electron

quasiparticle transport through the system is dominated by the mo-mentum relaxing scattering events, such as e-p scattering or the scat-tering of electrons with material impurities. An electron loses its mo-mentum due to these scatterings before it meets another electron [2]. The flow that results from this scattering behaviour is called Knud-sen flow and is depicted in fig. 2.1(a). This regime of electron trans-port is the most common and best-understood regime of all. It is also called the Ohmic regime, and this is where Drude theory ap-plies. Since in the Ohmic regime, the transport is governed by weak interactions, here a quasiparticle description has meaning. We can also look at this in a different way than by considering the length scales. The quasiparticle description makes sense due to the fact that they interact weakly and decay on much larger timescales than the thermal timescale. This eventually leads to the famous Boltzmann equation [1].

2. The second case is characterised by lMC << W << lMR. This

regime, which only quite recently has sparked the interest in con-densed matter physicists, is called the hydrodynamic regime. In this regime, the momentum conserving scattering dominates. In this case, this means that the interactions between the electrons are way

12 Conventional Electrodynamics in Condensed Matter

more important for the transport behaviour than the Ohmic Knud-sen flow of impurity and phonon scattering. This will result in an electron-electron fluid [2] that reaches equilibrium by the dominant e-e interactions before impurities are encountered by the electrons. This regime of hydrodynamic transport is sometimes also called Poisseuille flow. Fig. 2.1(b) shows this behaviour schematically. In this regime, the thermal timescale will be smaller or of the same or-der of magnitude as the local equilibration time of the electrons [1]. Due to the fact that quasiparticles as single state excitations are not a possibility for the description of transport, one needs to consider the transport of the conserved quantities of the system as a whole. 3. A third regime, which might be seen as a somewhat trivial one, is the

regime where W << lMC, lMR. This is the ballistic regime, in which

transport is dominated by the scatterings with the boundaries of the sample only.

So in these different regimes, the electron flow is characterised by dif-ferent length scales. The mean free path of the electron is the average distance it travels between two consecutive scattering events. As seen above, the mean free path l will always be of the same order of magni-tude as the smallest value of lMC, lMR and W, which dominates the

trans-port behaviour. An imtrans-portant thing to consider is that hydrodynamic be-haviour can also occur in conventional materials. When looking at ex-tremely large timescales, the Boltzmann equation even displays hydro-dynamic behaviour [1]. A very clean Fermi liquid metal (i.e. a normal, ’good’, metal, see Ch. 3) might also show hydrodynamics [4][5]. In this thesis, we are interested in unconventional metals where hydrodynamics might appear. The specific peculiarities of hydrodynamic behaviour in metals will be elaborated on in Ch. 4.

2.2

Superconductivity

The concept of superconductivity was discovered by the Leiden physi-cist Kamerlingh Onnes in 1911 [6]. The remarkable observation was that in some materials electrical resistivity seems to disappear completely if cooled to a temperature low enough. The temperature at which the resis-tivity of the material vanishes, is called the critical temperature Tc. In the

1930s, Meissner et al. [7] found that also magnetic fields are expelled from the core of the material.

2.2 Superconductivity 13

Both findings were new and striking experimental facts, that were awaiting a theoretical explanation. The theory for the expulsion of the magnetic fields in superconducting materials was found by F. and H. Lon-don in 1935 [8]. They proposed a model in which the superconducting material is equipped with a thin layer, of thickness λ, called the penetra-tion depth. In these edges of the material, the superconducting current flows with no resistance, thereby shielding the inner core of the material from magnetic fields. For high enough magnetic fields, however, this ex-pulsion of magnetic fields breaks, breaking the entire superconductivity of the material at the same time. The field at which this happens is called the critical field Bc.

The first theory that explained the absent resistance in superconduc-tors, however, only appeared in the 1950s. The Ginzburg-Landau theory [9] explained in a phenomenological way the macroscopic observations in superconducting materials. It states that superconductivity is nothing less than a thermodynamic state. As it turned out, this theory also predicted the existence of two types of superconducting materials, distinguished by the relation between the prementioned λ parameter and the ξ parameter, which one might interpret as the ’strength’ of the superconducting state. These two types of superconductivity are:

1. Type I superconductors are characterised by the fact that ξ &λ. This

means that the superconducting state is generally strong enough to prevent magnetic penetration in the material. Eventually, at some Bc,

the resistance will succumb and superconductivity is broken.

2. Type II superconductors typically have ξ . λ. Above a first critical

field Bc,1 some channels appear in the material through which the

magnetic field lines propagate. These channels are called Abrikosov vortices [10] and by the superconductive current around them, the material outside the vortices is still shielded, meaning the supercon-ductive state is not broken yet. Increasing the field further also in-creases the number of vortices. Eventually, once the field reaches a second critical value Bc,2, superconductivity is destroyed. The

differ-ence between the two types of superconductors can be seen in fig. 2.2.

14 Conventional Electrodynamics in Condensed Matter

B

T

Bc(T) Bc(0) 0 _T c

B

T

Bc,2(T) Bc,2(0) 0 _T c Bc,1(0) Bc,1(T)

Type I

Type II

Vortices Meissner Normal Normal Meissner

Figure 2.2: The two types of superconductivity. In type I, below Bc(T)the

mag-netic fields are expelled via the Meissner effect. Above Bc(T)superconductivity

is broken. For type II superconductors, there is an intermediate phase, where the magnetic field lines penetrate the materials at some places in vortices. Once the magnetic field becomes higher than Bc,2(T), superconductivity is broken after all.

Inset drawings taken from [11].

Somewhat later in the 1950s, Bardeen, Cooper and Schrieffer [12] pre-sented a microscopic model for understanding superconductivity. This Nobel prize-winning model has become known under the name ’BCS The-ory’. This model explains superconductivity by stating that when the ther-mal energy of the surroundings is low enough, electrons tend to form elec-tron pairs due to the interaction of the flowing elecelec-trons with the phonons in the lattice. These so-called Cooper pairs are bosons, making it pos-sible for the system to form a Bose-Einstein condensate. This condensate would then explain the superconducting phenomena of vanishing resistiv-ity since all electron pairs are collapsed into a single quantum state with-out any interactions with the lattice. Individual electrons that could scatter on the lattice are non-existent in the superconducting state. Once tem-perature, current or internal field exceeds one of their critical values, the Cooper pairs are destroyed, thereby stopping superconductive behaviour. It is also at this time that we can get an intuitive feeling for what the pre-mentioned ξ parameter means. This ξ can namely be thought of as a mea-sure of the distance between two electrons forming a Cooper pair in the superconductive state or, put differently, the length scale over which the magnitude of the condensate changes.

2.2 Superconductivity 15

The theoretical definition of the critical temperature is the temperature at which a material becomes superconducting. However, experimentally, this distinction often cannot be made this sharply. Often the supercon-ducting transition is observed to be more gradual in a certain temperature range, making it difficult to pinpoint Tc to a certain value. In order to

de-fine a proper measurable Tc, it is good to regard the critical current. An

IV-characteristic is a graph of voltage versus a constant DC current. For a normal Ohmic resistor, this will be a straight line. The IV graph that results from a superconductor, will ideally look like fig. 2.3.

Figure 2.3:Ideal form of an experimental IV on some material in superconducting state. At currents higher than Ic, superconductive behaviour is broken and Ohmic

behaviour is restored.

At currents higher than Ic, the voltage over the superconductor will

be non-zero and the slope will be finite. Between −Ic and Ic, however,

the material will be superconducting and therefore there is a zero voltage difference between the contacts. When this IV-measurement is done at dif-ferent temperatures, one will find difdif-ferent values for Icas well. For higher

temperatures, superconductivity will be broken more easily than for lower temperatures, resulting in a smaller value for Ic. The temperature at which

Ic becomes equal to zero we define as the critical temperature Tc. At this

16 Conventional Electrodynamics in Condensed Matter

2.3

Unconventional Superconductivity

Since the Cooper pair is formed by two fermions, the overall wavefunc-tion of the Cooper pair needs to be antisymmetric, following the Pauli principle. This can be done using the three different variables on which the wavefunction depends. The wavefunction has to be antisymmetric in a combination of the spin, time and spatial component of the wavefunc-tion. By the concept of the Fourier Transform, one can respectively also speak about the frequency and the momentum component instead of the time and spatial component, as is shown in fig. 2.4.

2

12

2. P

AIRING SYMMETRY

Figure 2.2: The four classes of Cooper pair symmetry, as allowed by the Pauli principle. This classification is based on three independent components which determine the overall pairing symmetry of the super-conducting wavefunction: spin, frequency and momentum. The drawings in the right panel represent the allowed orbital symmetries for each category. The black wavy line represents odd frequency.

The available symmetries for spin triplet Cooper pairs are represented in the third

and fourth classes of Figure

2.2

: even-frequency with odd-parity; and odd-frequency

even-parity respectively. The former category was first discovered in the p-wave

su-perfluid that forms in

3

He [

7

], and is also the proposed symmetry for the

supercon-ducting phase that occurs in Sr

2

RuO

4

below 1.5 K.

The last category of triplets was intitially introduced by in 1974 by Berezinski˘ı [

8

] as a

proposal for superfluidity in

3

He, which later was found to be p-wave instead. While

an odd-frequency triplet state has so far never been observed by itself in nature,

it was found that its pairing amplitude can be generated at (carefully engineered)

superconductor-ferromagnet (S-F) interfaces [

9

,

10

]. The triplet correlations studied

in our S-F hybrids correspond to odd-frequency with s-wave symmetry.

Odd-frequency triplet pairing can be realised with a simple s-wave gap. This has a

profound consequence on the survival of these correlations in a diffusive

environ-ment, where strong scattering leads to the mixing of different k states, see Figure

2.3

.

A p-wave gap, characterised by a k-dependent phase, would be fully suppressed

un-der strong averaging in k-space. The s-wave gap on the other hand, is protected

from scattering by its k-independent phase. As a consequence, the (odd-frequency)

s-wave triplet pairing can be realised in a variety of diffusive S-F hybrids, made from

a wide range of materials. In contrast, odd-parity (e.g. p-wave) triplet correlations

are characterised by the clean limit (i.e. non-diffusive), and are restricted to a rather

small number of materials, amongst which, Sr

2

RuO

4

is one of the most prominent

candidates (for a review see Refs. [

11

]). In this particular case, the order parameter

Figure 2.4: The different ways in which the wavefunction can be antisymmetric (odd), either in one of the components of spin, frequency and momentum or in all of them. The drawings on the right represent the orbital symmetry of the super-conducting gap function using the letters borrowed from atomic and molecular physics. The black wiggly line represents the odd frequency. Taken from [13], after [14], [15] and [16].

The singlet (odd spin component, meaning pairing between two oppo-site spin electrons) state can only be antisymmetric if both frequency and momentum are even, or when they both are odd. For the triplet (even in spin) state, the frequency and momentum component have to be of op-16

2.4 Hall Effect 17

posite parity. Based on the fact that the gap functions as displayed in fig. 2.4 look like orbitals, it is convention to use the letters from atomic and molecular physics to describe the pairing symmetry. In this way, we see that s- and d-wave superconductors are momentum-symmetric, while p-and f-wave superconductors are odd in the momentum component.

Now as it turns out that the BCS theory for the origin of the Cooper pairs only holds for singlet s-wave superconductors. There the electron-phonon interaction is the driving mechanism behind the formation of the Cooper pairs. In the 1980s, however, more superconducting materials were discovered that are not s-wave. The origin behind the formation of Cooper pairs in these so-called ’unconventional superconductors’ remains largely unknown. Many different types of unconventional superconduct-ing materials have since been found. For instance, there are the high Tc

cuprates, as discovered by Bednorz and M ¨uller in 1986 [17]. These su-perconductors are singlet d-wave symmetric [18]. Together with the BCS superconductors, these cuprates make up the vast majority of all currently known superconductors. Next to these there are also the iron-pnictides [19], ruthenates [20] and organic superconductors [21]. In this thesis, we will focus on the cuprate superconductivity.

As is the case for several other types of unconventional superconduc-tors, superconducting cuprates are all characterised by the periodically layered form of the lattice. The complex interactions between the super-conducting CuO2 layers and the other layers with different dopants

ac-cord for the intrinsic behaviour of high-temperature superconductivity in the cuprates. Apart from this, the layered structure causes a distinction between electronic behaviour in the planar direction with respect to the orthogonal direction. In Ch. 5 and 6, we will experimentally look at a well-known cuprate: Bi2Sr2Can−1CunO2n+4+x.

2.4

Hall Effect

In order to better understand the concept of the Hall effect, it is necessary to first describe how a material responds to an electric current. This re-sponse is entailed in the resistivity tensor, which is an intrinsic property of materials. In the case of an isotropic conductor without magnetic field applied on it, this follows Ohms law as defined as follows:

18 Conventional Electrodynamics in Condensed Matter

However, far more useful is to write the electric field and current density as vectors. Therefore the resistivity is described by a tensor:

~_E=↔_ρ~_{j (2.3)}

Here~E represents the electric field vector in the material,↔ρ is the

resistiv-ity tensor, and~j the electric current density vector. In matrix representa-tion, the resistivity tensor will take the following form:

  ρaa ρab ρac ρba ρbb ρbc ρca ρcb ρcc  

Here we have taken up the notation that is conventional for layered structures of the cuprates. The a, b-plane will represent the plane in which the different layers of the lattice are organised, while the c-axis is the or-thogonal direction to this plane. Because of this structure, we have that

ρaa =ρbb ≡ ρplane. The diagonal elements of the resistivity tensor display

the resistivity that the material inhibits in the longitudinal direction, par-allel to the current. Now when time-reversal symmetry is preserved, the off-diagonal components will absent. By adding a magnetic field into the system, however, time-reversal symmetry is broken and the off-diagonal components will become nonzero, describing the non-dissipative part of the resistivity tensor.

Now ideally the current flow only takes place in the a, b-plane. For the sake of convenience, we assume that the only non-zero component of~j is ja. Hence the current flows along the a-axis. Now following equation

2.3, we see that the electric field inside the material will not be parallel to the current, due to the fact that a magnetic field is present, creating a transverse electric field by the use of the Lorentz force. For this case, let us assume that the magnetic~B field is directed only along the c-axis. The transverse electric field caused by Lorentz effects will then be directed in the b-direction, resulting in a nonzero ρab, while ρba = −ρab, due to

the symmetric property of the tensor. The matrix representation of the resistivity tensor thus reduces to

  ρplane ρab 0 −ρab ρplane 0 0 0 ρcc  ,

thereby effectively describing a 2D Hall-effect in the a, b-plane. Now by considering equation 2.3 again, we see that the Hall resistivity ρab will

2.4 Hall Effect 19

be equal to Eb/ja. The strength of the Hall effect is measured using the

so-called Hall coefficient RH, which is derived from the combination of

Lorentz force and Drude theory, resulting in: RH ≡ Eb jaB = ρab B ∝ n −1 _(2.4)

Here it is assumed, as in our case that B is the magnitude of the magnetic field entirely pointed in the c-direction. n is the charge carrier density in the system. RH can than measured via the Hall resistivity ρab in the

following way: ρab= Eb ja → Z ρabdb= 1 ja Z Ebdb →ρabW = Vb ja = VbWT Ia →ρab= VbT Ia (2.5)

Here we first integrate both sides of the starting equation in the direction transverse to the current path, which is the b-axis. This results in the factor W appearing as the width of the current path. On the right hand Vb

ap-pears that represents the electric potential difference between both sides of the current path. Due to the conversion of current density to normal current Ia, also the thickness of the current path T appears in the formula.

By measuring the transverse voltage across the current path, it is thus pos-sible to obtain the Hall coefficient RH, which also can be used as a measure

Chapter

3

Bridging the Gap - The Phase

Diagram

Now that we have a thorough understanding of the conventional theory of electron transport in metals, the gap towards possible unconventional hydrodynamics in strange metals (Ch. 4) and strange metal experiments (Ch. 5 and 6) can be bridged. This chapter therefore discusses the phase diagram for the cuprates and the different types of electron transport that occurs in these phases.

The phase diagram for cuprates is shown in fig. 3.1. Increasing hole doping is often achieved by adding oxygen in the lattice of the cuprate crystal. The holes in the crystal can act as charge carriers, making the material conducting and for certain doping values even superconducting. This d-wave unconventional superconducting behaviour takes places un-der the superconducting dome, being bounded by the critical temperature Tc. The point where Tc is maximum is called optimal doping. Lower

dop-ing than this is called the underdoped regime, while a higher than optimal doping is called the overdoped regime.

Now in general, we can discern between four phases of different elec-tronic behaviour. The Antiferromagnetic (AFM) phase is sharply bounded by the N´eel temperature, while the bounds between the Pseudogap and Strange Metal phase and between the Strange Metal and Fermi liquid phase are less well-defined. These phase transitions happen more grad-ually. We will now describe the characteristics of each of the phases.

22 Bridging the Gap - The Phase Diagram

Figure 3.1:The phase diagram for cuprate hole doping, with TN the N´eel

temper-ature, and T∗ separating the pseudogap phase from the strange metal phase. Tc

is the critical temperature under which d-wave superconductivity appears. The phase diagram is often displayed as temperature vs. doping, but temperature vs. pressure would result in a similar picture. In this graph, hole doping is given as the number of holes per CuO2unit. Based on [22].

3.1

Antiferromagnetic Mott Insulator

In the extreme underdoped regime, cuprates are insulating antiferromag-nets. This can easily be understood by considering the case of zero doping. At that point, each lattice point is occupied by an electron, thereby result-ing in a half-filled band. Because of this, one might suspect the material to be conducting. However, due to strong electron-electron repulsion, it takes a large energy to remove an electron from one lattice site and add it to another site [23]. Since this prevents the movement of charge carriers through the lattice, this causes a ’traffic jam’ of electrons, causing the ma-terial to insulate. The fact that the cuprates are microscopically ordered antiferromagnetically, can be understood by considering the Pauli exclu-sion principle. Even though strong electron-electron repulexclu-sion prevents this, were one electron to ’hop’ from its lattice point to a neighbouring one, this would only be possible if the electrons had opposite spin. The 22

3.2 Pseudogap 23

possibility of so-called ’virtual hopping’ broadens the degrees of freedom and thus the energy of the state in which spins of neighbouring electrons anti-align, called anti-ferromagnetic ordering. Above the N´eel tempera-ture TN, thermal fluctuations break this microscopic ordering, making the

material paramagnetic.

3.2

Pseudogap

Together with the strange metal phase, the pseudogap is a less well un-derstood part of the phase diagram. It corresponds with a gap in its band structure at temperatures above the superconducting dome. This gap is not uniform over all energies though, as is the characteristic of a supercon-ducting material. Also the range in energy in which the gap exists contains only a few states, while a perfect gap would contain no gaps at all, making this a pseudogap. The clearest evidence that the pseudogap phase exists is based on angle-resolved photoemission spectroscopy (ARPES), which directly measures the density of states. Amongst other remarkable char-acteristic behaviours for underdoped materials is the fact that the in-plane resistivity ρplane depends linearly on temperature for temperatures above

T∗. This is in accordance with the strange metal phase. Cooling the ma-terial down further than T∗, it was observed that the resistivity is a lower than linear order function of temperature [24].

Multiple theories have been coined to theoretically explain what hap-pens at the pseudogap phase, none of which have been completely ver-ified yet. Some proposals suggest strong spin and charge fluctuations [25][26]. Others propose antiferromagnetic fluctuations for the pseudogap behaviour [27], which is a more logical approach given that the pseudo-gap phase arises from the antiferromagnetic Mott insulator by doping the material with holes. Both of these explanations, however, fail to predict the magnitude of the pseudogap correctly [28].

A more profound theory on the pseudogap is the theory of supercon-ducting fluctuations in the pseudogap phase. In this scenario, it is stated that already at a temperature of T∗, some of the electrons start to pre-form Cooper pairs [29]. The problem with this idea is that, although this doesn’t result in a macroscopic quantum object of a condensate, one would expect the resistance of the material to rise once the Cooper pairs between Tcand

T∗ are broken by inducing a magnetic field. This is not observed experi-mentally, however.

24 Bridging the Gap - The Phase Diagram

Finally, some theories require a Quantum Critical Point (QCP) in the phase diagram. From this perspective, the pseudogap is seen as a distinct phase. The possible existence of such a point has been the reason for a long-standing debate in the field of cuprate superconductivity. We will focus more on the QCP later on in this chapter.

3.3

Fermi liquid

In the highly overdoped regime of the phase diagram, the cuprate behaves as a Fermi liquid, also called a Landau-Fermi liquid or a ’good’ metal. This is a well-known and established theory of electron transport and it is widely observed in all sorts of metals. In this regime, due to the high amount of hole doping, the electron excitations are not as highly corre-lated as is the case for lower doping. A quasiparticle description thus makes sense in this regime. In contrast to the Fermi gas, the interactions between the electrons do not need to be absent, although they have to be weak. The quasiparticles can be thought of as the charge carrying elec-trons, ’dressed’ with the electron-electron interactions, that are included in the quasiparticle formalism itself. The quasiparticles responsible for the transport characteristics propagate freely through a medium defined by the lattice and the rest of the electrons [23]. This effectively means that the Fermi liquid has a one-to-one correspondence to the Fermi gas. Only by replacing the free electrons in the gas with quasiparticles, the Fermi liquid automatically appears [30]. This correspondence only holds within cer-tain boundaries, as one can easily understand by considering the bound-ary case of the superconductive state, that has no analogue for the Fermi gas model.

For a metal to behave as a Fermi liquid, the quasiparticle states should have a sufficiently long-lived lifetime [30]. More precisely, the quasipar-ticle lifetime τqp should be larger than the thermal timescale τth = ¯h/kBT

[1]. This ensures that the transport carriers are only weakly interacting. It is the Fermi liquid phase that can be considered as being the ’con-ventional’ metal phase since the discussion of conventional electron flow as discussed in Ch. 2 is applicable for the cuprates in this phase. Impor-tant to note is that a Fermi liquid metal, when extremely clean, meaning there are very few lattice defects and negligible Umklapp scattering, even conventional hydrodynamic behaviour can appear, which has also been experimentally found [4][5].

3.3 Fermi liquid 25

The typical characteristic of a Fermi liquid metal is that the resistivity

ρ scales with T2, with T the temperature. Apart from this, there is the

Mott Ioffe Regel (MIR) limit that puts a maximum on the resistivity of the Fermi liquid material. This limit can be easily understood following the argumentation given in [1]. It is based on the electrical conductivity that arises from the Drude theory:

σ= ne 2

τqp

m (3.1)

Here σ is the electrical conductivity, n is the charge carrier density. These charge carriers are the quasiparticle electron excitations in this case. m is the mass of a single quasiparticle. Now from Fermi liquid theory it follows that n ∼ kd_F, with d the dimensionality of the system and kF the Fermi

momentum defined by mvF = ¯hkF. Combining this with the fact that the

mean free path length l is equal to vFτqp, in accordance with equation 2.1,

this gives that:

σ ∼ e 2

¯hk

d−2

F (kFl) (3.2)

Now since the Fermi liquid only holds when it makes sense to use quasi-particles for the model of transport, we can say that the momentum of the individual quasiparticles k is larger than the uncertainty ∆k, in order to distinguish individual excitations. Since k is approximately of the same order as kF, we can thus say that:

kFl>∆kl ≥1 (3.3)

This last step is based on Heisenberg’s uncertainty principle. Hence we end up with a minimal conductivity and hence a maximal resistivity:

σ& e 2 ¯h k d−2 F ≡σmin → ρ. ¯h e2k 2−d F ≡ρmax (3.4)

This bound on the resistivity is observed widely amongst the ’good’ met-als, not limited to cuprates. [31].

26 Bridging the Gap - The Phase Diagram

3.4

Strange Metal

The quasiparticle description fails most abruptly in the strange metal regime [23], which is located in the phase diagram in a fan above opti-mal doping. In this regime, the single electron excitations decay to quickly to describe the resistivity behaviour of the system in terms of Drude be-haviour of the individual excitations. Sometimes this phase is also called the non-Fermi liquid phase or a ’bad’ metal since it doesn’t follow the Fermi liquid theory. The most important characteristic of the strange metal phase is that ρ ∝ T for temperatures as high as room temperature and above [32]. By exhibiting this behaviour, a strange metal is also not bound by the MIR limit [31]. This is not specific to strange metal cuprates, though, as also multiple other unconventional superconductors are found that cross the MIR limit [31][33].

Furthermore, because of the strong interactions and the absence of quasiparticles, it becomes difficult to identify the charge carriers in the system. In most cases, it is the holes introduced by doping that can be thought of as the charge carriers. However multiple other strange metal cuprates exhibit a combination of electrons and holes carrying the electric charge. Consequently, it is not easy to describe how the Hall effect might work in strange metals. As has been observed experimentally, the Hall coefficient is temperature-dependent for strange metals [23][34], such that the Hall charge carrier density is reduced for increasing temperatures [35]. As this is something that is not predicted by the Fermi liquid theory for ’good’ metals, this phenomena is another characteristic feature of strange metal behaviour. On top of that, multiple experiments have shown that the Hall-effect in strange metals even appears to change its sign in certain temperature and field domains [36].

Explanations for this peculiar behaviour of the Hall effect have been sought for extensively. As such, multiple options are given. It has been ar-gued that this strange metal behaviour might be caused by superconduct-ing fluctuations [37], just as might be the case in the pseudogap regime. The most interesting explanation for strange metal behaviour is that elec-tron flow in near optimally doped cuprates experiences hydrodynamic ef-fects. Based on the small mean free path length, one would not expect hydrodynamic behaviour in the first place. This would therefore classify as so-called ’unconventional hydrodynamics’, on which Ch. 4 will elabo-rate in detail.

3.5 Quantum Critical Point 27

3.5

Quantum Critical Point

For the pseudogap and the strange metal, both poorly understood phases, it has been suggested that these point to the existence of a quantum critical point (QCP) [23][33][35][38]. A quantum critical point is a phase transition at zero temperature and therefore powered by quantum fluctuations. It is assumed to be located somewhere around optimal doping. Since this is deeply hidden by the superconductive dome, requiring huge critical fields to lift superconductivity, experimentation on this part of the phase dia-gram is extremely difficult. Many different types of QCPs have been the-oretically suggested [39]. A definite and conclusive experimentally based answer to this debate has not yet been given, however. Nevertheless, the theoretical importance of the possible existence of a QCP is a motivation for the further study on the poorly understood strange metal phase, as it would be surprising if all the separate phases in the phase diagram were not related. Since the strange metal phase is the connecting phase in the phase diagram, understanding the strange metal region possibly holds the key to understanding the physics of cuprates and even unconventional su-perconductivity as a whole. This motivates the study for a strange metal explanation, which is done in the next chapter.

Chapter

4

Unconventional Hydrodynamics

Multiple theories that try to explain strange metal phenomena are based on mathematics from string theory, in the form of the so-called holo-graphic duality. The fact that the nature of quantum gravity might be based on the same paradigm as the observed phenomena for the cuprate strange metal phase [40] is a deeply fundamental idea, which has struck interest in many both theoretical and experimental physicists. The holo-graphic theories for the explanation of strange metal behaviour predict hydrodynamic behaviour in systems where one doesn’t expect to observe it based on the mean free path length (see Ch. 2 and 3). Because of this, we will now briefly discuss the holographic principle as well as what hy-drodynamics entails and how one might observe it experimentally.

4.1

Holography

Holographic duality is the principle that the information of a certain volume of space is deeply connected to the characteristics of the area surrounding this volume. One can deduce what is inside a certain enclosed volume by only observing the boundaries, which are of a lesser dimension than the entire volume, justifying the name of holography. As an example, one might think of a black hole. Nothing can escape from it and contrary to a many intuition entropy scales with the area of the horizon. As once said by one of the early ’holographists’ Leonard Susskind, cited by [1]:

30 Unconventional Hydrodynamics

The duality that is the most important in all fields of string theory and also has the most relevance with regard to this problem is the AdS-CFT correspondence. Originally coined by Maldacena in 1997 [41], this mathematical correspondence links the gravity theory in a weakly curved (d+1)-dimensional Anti-de Sitter (AdS) spacetime to the d-dimensional strongly coupled quantum Conformal Field Theory (CFT) on the bound-aries of the anti-de Sitter spacetime [42]. The AdS spacetime represents a negatively curved spacetime, featuring a negative cosmological constant. This spacetime is infinitely large, while it still has a boundary from which it takes finite time to travel to the spacetime interior [40]. On the other side of the correspondence, the CFT represents a category of field theo-ries used for describing a broad spectrum of physical phenomena, among which certain condensed matter phenomena. The crucial feature of the AdS-CFT duality is that it is a strong-weak duality. Fields that are coupled strongly in the quantum field theory of the CFT are only weakly coupled in the gravitational AdS theory. Since the strange metal regime clearly needs a theory based on strong interactions, the AdS-CFT correspondence is ideally suited.

The Ads-CFT correspondence can be seen as a form of duality in the same manner that holds for the particle-wave duality. The true nature of the physical phenomena is not contained in one theoretical formulation only, but it is rather a matter of perception. In the same way the Fourier Transform links the particle to the wave, this AdS-CFT correspondence mathematically links classical gravity to the quantum CFT [40].

To account for strange metal behaviour, a black hole is studied in the deep interior of the AdS spacetime. The origin for the use of a black hole lies in the discovery by Hawking and Bekenstein that black holes are trinsically thermodynamic objects [43][44]. The black hole in the deep in-terior encodes for a finite temperature in the boundary of the CFT, which is equal to the Hawking temperature arising from Hawking radiation. Plus, the entropy on the boundary coincides with Bekenstein-Hawking entropy set by the area of the black hole horizon [40]. One should keep in mind that these black holes are by no means astrophysical objects, but rather math-ematical descriptions, as is the case for the non-physical AdS spacetime. In this sense, Maldacena’s version of the AdS-CFT correspondence linked the dynamics of the black hole horizon with a certain quantum liquid (the N=4 Yang-Mills liquid) [41].

The (supposedly) non-physical black hole that is used for the explana-tion of strange metal phenomena is a so-called Reissner-Nordstr ¨om (RN) black hole. This is a charged black hole placed in the AdS bulk, char-acterised by only its mass, charge and angular momentum. Black holes 30

4.1 Holography 31

that are characterised by these three observable parameters exclusively are called no-hair black holes. Interestingly, also pseudogap behaviour has an AdS-CFT explanation, but then for a black hole dressed with a cer-tain form of ’hair’, meaning other parameters are needed to characterise it [40]. In the case of the RN black hole in the bulk, however, this leads to a finite density in the boundary, looking quite similar to the landscape observed in strange metal cuprates.

FIG. 1: An illustration of the holographic approach to understanding non-Fermi liquids. The physics of a strongly-coupled many-body theory is exactly dual to that of a gravitational theory in a curved spacetime with one extra dimension. A finite density of charge in the many-body problem corresponds in the gravity theory to a black hole with an extra emergent conformal symmetry near its horizon: these correspond to the symmetries of two-dimensional Anti de-Sitter space. This space (when projected onto a two-dimensional plane) forms the basis of the M.C. Escher print Circle Limit IV, which is thus used here to represent the horizon of the black hole. Understanding fermionic response in this system corresponds to scattering bulk fermions off the black hole and shows excitations characteristic of Fermi or non-Fermi liquids.

surface are not Fermi-liquid-like quasiparticles [8, 9], and suggests that the system may be

governed by a quantum critical point at low energies [10–17]. Similar anomalous behavior

has also been observed in heavy fermion systems near a quantum phase transition [18–20],

where quantum critical behavior is more clearly established. The anomalous behavior of

a strange metal can be characterized by a phenomenological model called the “marginal

Fermi liquid” (MFL) [8], which assumes that the spin and charge excitation spectra of

the system are momentum-independent (or have weak momentum dependence) and have a

specific scale-invariant form.

1

In this paper we report on a class of non-Fermi liquids discovered using the AdS/CFT

1

A microscopic model which produces this critical fluctuation spectrum has been proposed recently in [21].

Figure 4.1: A schematic illustration of the curved 2-dimensional AdS spacetime. A 2-dimensional projection of this spacetime is depicted in the form of M.C. Es-cher’s Cirkellimiet IV (’Hemel en Hel’), also found on the title page. Although there are infinitely many figures, all of the same size (in the spacetime metric), the cir-cumference is still finite, which is the analogy to the AdS spacetime. In the strange metal case, Escher’s piece of art is thus used to display the horizon of the black hole. Fermionic response is understood by the scattering of bulk fermions on the black hole itself. The inset then shows excitations apparent in the CFT boundary that are characteristic for both Fermi liquids and non-Fermi strange metals. Taken from [42].

Furthermore, the 2-dimensional AdS spacetime describes a boundary metal characterised by quantum criticality, which is exactly what exper-iments seem to suggest (although still much discussion is present sur-rounding this topic, see Ch. 3). This 2-dimensional version of the AdS-CFT correspondence is depicted in fig. 4.1.

32 Unconventional Hydrodynamics

4.2

Towards Unconventional Hydrodynamics

The prediction of strange metal behaviour in cuprates follows from the holographic dual of resistivity, namely the black hole entropy. In the same way that black hole entropy scales with Hawking temperature linearly, so does resistivity scale linearly with temperature. Apart from the fact that holographic theories explain the T-linear resistivity, they also predict hydrodynamic behaviour, originating in so-called Planckian dissipation and unparticle physics as argued by Zaanen [40]. Following his argumen-tation, he states that due to the strong interactions in the strange metal regime, one should think of this regime as being many-body entangled. As already discussed in Ch. 3, the quasiparticle description doesn’t fit the strange metal phase. By stating that all charge carriers are entangled with each other, Zaanen argues that ’everything knows about everything’. It is precisely this prediction in holographic view that simplifies the mathe-matics since in field theories, entanglement often becomes very hard very fast, both analytically and from a computer-related perspective.

The consequence of this reasoning is also that the holographic dual of the strange metal predicts a viscosity to entropy ratio for the electron fluid in the strange metal that is extremely small compared to normal fluids. This so-called minimal viscosity is many orders of magnitude smaller than the values that Quantum Chromodynamics predicts [45]. This prediction therefore forms the perfect experimental test case of the holographic the-ory. A breakthrough in favour of the holographic explanation came in 2005, when the newly created quark-gluon plasma appeared to behave as a hydrodynamical fluid with a viscosity close to this minimal value that holography predicts [46]. This observation raises hope for the AdS-CFT correspondence as a possible tool in explaining the strange metal be-haviour in cuprates.

Next to all of this, one also expects turbulent electron flow at the nanoscale due to this minimal viscosity prediction [40]. This is another phenomenon unthinkable seen from the framework of conventional elec-tron transport as explained in Ch. 2.

The clear contrast that thus arises is that conventional theory of elec-tron transport forbids hydrodynamic behaviour in the dirty cuprates, whereas holography does predict hydrodynamics. Even in extremely clean systems, the viscosity of the hydrodynamic electron flow is orders of magnitude higher than the minimal value following from holography and unparticle physics. Clearly, the predicted unconventional holographic hydrodynamic behaviour for the strange metal cuprate forms the perfect experiment to uncover more details about the strange metal phase.

4.3 Hydrodynamic Framework 33

4.3

Hydrodynamic Framework

Before the discussion of several experiments on probing hydrodynamics, we first briefly establish a framework on what hydrodynamics concep-tually entails. We should thereby keep in mind that what follows is not limited to unconventional hydrodynamics. Neither is most of it limited to hydrodynamic behaviour in metals only. The compelling idea of hydro-dynamic electron flow is that the well-established field of the physics of hydrodynamic fluids can now be applied to the field of electromagnetic transport.

Already one aspect of hydrodynamics that makes it distinct from the quasiparticle electron flow, is that due to the long-wavelength excitations of the medium [1] following from the unparticle agglomerate [40], one can only describe the behaviour of macroscopic conserved observables of mass, momentum, charge and energy. These individual particle character-istics are lost due to the strong interactions. The language to describe the behaviour of these bulk conserved quantities is hydrodynamics. Specif-ically for holographic hydrodynamics, this might ring a bell, since also a hairless black hole is characterised only by its charge, mass and angu-lar momentum, making hydrodynamic behaviour intuitively understand-able.

Now for hydrodynamic behaviour, it is the viscosity that plays the role of a sort of internal resistance of the fluid. For hydrodynamics applied to electron transport, this often results in quite complex underlying theories, since both the Maxwell equations that govern electrodynamics and the Navier Stokes equations that govern hydrodynamics have to be satisfied. The most basic equation that the hydrodynamic electron velocity profile v(b), directed along the a-direction, has to satisfy (if the sample width is large) is [47]): ∂v ∂t =η ∂2v ∂b2 + e mE (4.1)

Here η is the shear viscosity of the electron flow, and E is the electric field along a. This equation is strongly dependent on the boundary conditions of the current path. If v = 0 at the boundary, a parabolic current pro-file will appear, thus evidence for hydrodynamics. This experimentally testable flow profile will be further elaborated on in section 4.4.

By introducing a magnetic field in the system’s c-direction, one breaks time reversal symmetry. Just as in the description of conventional elec-trodynamics in Ch. 2, this brings up the need of introducing a viscosity tensor, of which the off-diagonal part displays the non-dissipative term,

34 Unconventional Hydrodynamics

also called the Hall viscosity. Thus, if we follow the same system descrip-tion as in Ch. 2, with the 2-dimensional a, b-plane as being the plane in which electron flow takes place, equation 4.1 will take the following form [3]: ∂~v ∂t =ηplane∇ 2_~v+ ηab∇2~v× ˆc+ e m  ~_{E+ ~v× ~B} _(4.2) In accordance with Ch. 2, ηplane is the diagonal dissipative viscosity of

the a, b-plane, while ηab represents the Hall viscosity. ˆc is the unity basis

vector in the direction orthogonal to the current plane, along the c-axis. Now by assuming a stationary regime of a current in the a-direction again, this results in two separate equations [3]:

ηplane d2va db2 + e mEa =0 (4.3) −ηab d2va db2 + e mEb = eB mva (4.4)

As was the case for equation 4.1, where no magnetic field was present, these equations are very sensitive to the boundary conditions. For the sake of argument, and in accordance with the argumentation of Scaf-fidi et al. [3], we choose the conventional no-slip boundary condition va(b = ±W/2) = 0, with W the width of the current path. This then

links the resistivity of Ch. 2 to the hydrodynamic regime of viscosity in the following way:

ρplane =ηplane 12 W2 m e2_n (4.5) ρab =ρbulkab  1−ηab 12 W2 m Be  (4.6) Here n is the charge carrier density, and ρbulk_ab = −B/en is the conventional Hall resistivity, just named ρabin Ch. 2. This then gives an experimentally

observable characteristic of hydrodynamic flow. In conclusion, we can say that the concept of holographic hydrodynamics is a crucial component of a possible explanation for the cuprate strange metal phase, possibly leading to a deeper insight in unconventional superconductors as a whole. We have briefly established a framework of hydrodynamic behaviour.

4.4 Probing Hydrodynamics 35

4.4

Probing Hydrodynamics

The theory and experimentation on the phenomenon of electron hydrody-namics in metals commenced with the pioneering work of Gurzhi [48][49]. We now review experiments that have been conducted since then and pos-sibly can be used on the cuprate Bi2Sr2CuO6+x. It must be pointed out that

the hydrodynamic behaviour suggested in strange metal cuprates is not unconventional in itself. It is the underlying theory that makes strange metal hydrodynamics unconventional. This then opens up the world of conventional hydrodynamic experimentation in ultraclean Fermi liquids, to be applied to the strange metal cuprates. We suggest three types of ex-periments possibly applicable to strange metal cuprates: hydrodynamic boundary effects, viscous backflow and the sign-reversing Hall effect.

PHYSICS

Fig. 1. (A) Current streamlines (black) and potential color map for viscous flow through a constriction. Velocity magnitude is proportional to the den-sity of streamlines. Current forms a narrow stream, avoiding the bound-aries where dissipation occurs and allowing the resistance, Eq. 1, to drop below the ballistic limit value. (B) Current distribution in the constriction for different carrier collision mean-free-path values. The distribution evolves from a constant in the ballistic regime to a semicircle in the viscous regime, Eq. 10, illustrating the interaction-induced streaming effect. Parameters used are L = 3w and b = 105v. The distributions are normalized to unit

total current. A Fourier space filter was used to smooth out the Gibbs phe-nomenon.

scatterers spaced by a distance L, the typical time of

momen-tum transfer is τ ∼ L

2

/ν ≈ L

2

/v

T

`, whereas, for an ideal gas, this

time is τ

0

= L/v

T

, where v

T

is thermal velocity and ` is the mean

free path. For `  L, the viscous time τ is much longer than the

ballistic time τ

0

.

The peculiar correlations originating from fast particle

colli-sions in proximity to scatterers can be elucidated by a spatial

argument: Particle collisions near a scatterer reduce the

aver-age velocity component normal to the scatterer surface, v

⊥

,

which slows down the momentum loss rate per particle, mv v

⊥

/L.

Momentum exchange makes particles flow collectively, on

aver-age staying away from scatterers and thus lowering the resistance.

The viscosity-induced drop in resistance can be used as a

vehi-cle to overcome the quantum ballistic limit for electron

conduc-tion. Indeed, we can compare the values R

vis

and R

ball

by putting

them in a Drude-like form R = m/ne

2

τ

, with m as the carrier

mass and τ as a suitable momentum relaxation time. Eq. 1 can

be modeled in this way using the time of momentum diffusion

across the constriction τ = w

2

/ν, whereas R

ball

can be put in a

similar form with τ

0

= w /v

F

as the flight time across the

constric-tion. Estimating ν =

1₄

v

F

l

ee

, we see that Eq. 1 predicts resistance

below the ballistic limit values so long as τ & τ

0

, i.e., in the

hydro-dynamic regime w & l

ee

.

Understanding the behavior at the ballistic-to-viscous

cross-over is a nontrivial task. Here, to tackle the crosscross-over, we use a

kinetic equation with a simplified ee collision operator chosen in

such a way that the relaxation rates for all nonconserved

harmon-ics of momentum distribution are the same. This model provides

a closed-form solution for transport through VPC for any ratio of

the length scales w and l

ee

, predicting a simple additive relation

G

VPC

= G

ball

+ G

vis

.

[3]

This dependence, derived from a microscopic model,

interpo-lates between the ballistic and viscous limits, w  l

ee

and w  l

ee

,

in which the terms G

ball

and G

vis

, respectively, dominate.

The Hydrodynamic Regime

We start with a simple derivation of the VPC resistance in Eq. 1

using the model of a low-Reynolds electron flow that obeys the

Stokes equation (29).

(η∇

2

− (ne)

2

ρ)

v(r) = ne∇φ(r).

[4]

Here φ(r ) is the electric potential, η is the viscosity, and the

sec-ond term describes ohmic resistivity due to impurity or phonon

scattering. Our analysis relies on a symmetry argument and

invokes an auxiliary electrostatic problem. We model the

con-striction in Fig. 1A as a slit −

w₂

< x <

w₂

, y = 0. The y → −y

sym-metry ensures that the current component j

y

is an even function

of y whereas both the component j

x

and the potential φ are odd

in y. As a result, the quantities j

x

and φ vanish within the slit

at y = 0. This observation allows us to write the potential in the

plane as a superposition of contributions due to different current

elements in the slit,

φ(x , y ) =

Z

w

2 −w₂

dx

0

R(x − x

0

, y )j (x

0

),

[5]

where the influence function R(x , y) =

β(y_(x22_+y−x2₎22)

describes

potential in a half-plane due to a point-like current source at the

edge, obtained from Eq. 4 with no-slip boundary conditions and

ρ = 0

(30). Here β =

_π(en)2η 2

, and, without loss of generality, we

focus on the y > 0 half-plane.

Crucially, rather than providing a solution to our problem, the

potential-current relation Eq. 5 merely helps to pose it. Indeed,

a generic current distribution would yield a potential that is not

constant inside the slit. We must therefore determine the

func-tions j (x ) and φ(x , y) self-consistently, in a way that ensures that

the resulting φ(x , y) vanishes on the line y = 0 inside the slit.

Namely, Eq. 5 must be treated as an integral equation for an

unknown function j (x ). Denoting potential values at the

half-plane y ≥ 0 edge as φ

+0

(x ) = φ(x , y )

_y=+0

, we can write the

rela-tion Eq. 5 as

φ

+0

(x ) = −

β

2

Z

∞ −∞

dx

0



j (x

0

)

(x − x

0

_{+ i 0)}

2

+

j (x

0

)

(x − x

0

_{− i 0)}

2



,

[6]

where j (x ) is the current y component, which is finite inside and

zero outside the interval [−

w₂

,

w₂

].

A solution of this integral equation such that φ

+0

(x )

vanishes

for all −

w₂

< x <

w₂

can be obtained from a 3D electrostatic

prob-lem for an ideal-metal strip of width w placed in a uniform

exter-nal electric field E

0

= λˆ

x

. The strip is taken to be infinite, zero

thickness, and positioned in the Y = 0 plane such that

−

w

2

< X <

w

2

,

Y = 0,

−∞ < Z < ∞

[7]

(for clarity, we denote 3D coordinates by capital letters).

Poten-tial Φ

3D

(X , Y )

is a harmonic function, constant on the strip and

Guo et al. PNAS | March 21, 2017 | vol. 114 | no. 12 | 3069

Figure 4.2: Flow profile of a current directed in the y-direction. The orthog-onal x-direction is normalised by the width w of the current path. By adjust-ing this width, one changes the ratio with the momentum conservadjust-ing mean free path length lee, thereby changing the electron flow behaviour between the viscous

regime and the ballistic regime. Taken from [50].

36 Unconventional Hydrodynamics

4.4.1

Boundary Effects

Already in equation 4.1, we saw that the boundary conditions of the cur-rent path are of great influence in the resulting velocity profile of the hy-drodynamic electron current. Guo et al. [50] have modelled the velocity profile in order to distinguish between the conventional hydrodynamic regime and the ballistic regime. This gives the velocity profiles as given in fig. 4.2.

Showing conventional hydrodynamic behaviour using boundary ef-fects all comes down to the measurements of two facts. In the first place, one should know the ratio of the momentum-conserving mean free path lMC (or lee) to the width of the current path W. This then gives a measure

of which regime one would expect based on conventional theory (Ch. 2). These measurements can be controlled either by varying the width of the current path by using different sizes of constrictions (as in fig. 4.2), or by varying the temperature, thereby altering the lMC(as in fig. 4.3). Secondly,

the velocity current profile has to be measured. Sulpizio et al. [51] have developed a measure for doing this in graphene. By varying the temper-ature, they could clearly see the ballistic and hydrodynamic regime in fig. 4.3

11

Figure 2: Imaging ballistic and Poiseuille electron flow profiles. a. Graphene channel with overlay

indicating region over which flow profiles are imaged. 1D profiles are taken along the dashed line, 2D profiles are imaged across the region enclosed by the black square (scale bar 2.5μm). b. Potential of flowing electrons, 𝜙, as function of the 𝑦 coordinate (dashed line, panel a) imaged at 𝐵 = 0 (blue, 𝑇 = 7.5K). Dashed yellow

Figure 4.3: The distinction of the velocity profiles between the more ballistic regime (left) and a clear conventional hydrodynamic regime (right) as observed in graphene. The current flows in the x-direction. Both the x- and the y-axis are normalised by the current path width W. The flow profile is measured by the elec-tric field in the y-direction Ey, normalised by some classical E-field Ecl. Increasing

the temperature decreases lMC due to thermal motion, moving the electron flow

behaviour towards the hydrodynamic regime. Figure taken from [51].