Graphene heterostructures for spin and charge transport Zomer, Paul Joseph

Graphene heterostructures for spin and charge transport Zomer, Paul Joseph

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for spin and charge transport

Paul Zomer

ISBN: 978-94-034-1712-7

ISBN: 978-94-034-1711-0 (electronic version)

The work described in this thesis was performed in the research group Physics of Nanodevices of the Zernike Institute for Advanced Materials at the University of Groningen, the Netherlands. This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Nether- lands Organisation for Scientific Research (NWO).

Cover design: Artwork by Suze Zomer

Printed by: Ipskamp Printing

for spin and charge transport

PhD thesis

to obtain the degree of PhD at the University of Groningen

on the authority of the Rector Magnificus, prof. E. Sterken,

and in accordance with the decision by the College of Deans.

This thesis will be defended in public on Monday 24 June 2019 at 12.45 hours

by

Paul Joseph Zomer born on 20 May 1984

in Groningen

Assessment committee:

Prof. A. A. Bol

Prof. M. M. G. Kamperman

Prof. B. J. Kooi

1 Introduction 1

1.1 Graphene . . . . 2

1.1.1 Quality improvement . . . . 3

1.2 Spin transport . . . . 4

1.2.1 Graphene based spin transistors . . . . 6

1.3 Outline . . . . 6

References . . . . 9

2 Theory 13 2.1 Graphene properties . . . 14

2.1.1 Crystal lattice . . . 14

2.1.2 Electronic band structure of graphene . . . 15

2.1.3 Electronic band structure of bilayer graphene . . . 16

2.2 Graphene electronics: field effect transistors . . . 18

2.2.1 Bilayer graphene . . . 21

2.3 Quantum Hall effect . . . 23

2.3.1 2DEG . . . 23

2.3.2 Graphene . . . 24

2.4 Graphene spintronics . . . 26

2.4.1 Spin injection and detection . . . 27

2.4.2 Non-local spin valve devices . . . 29

2.4.3 Hanle spin precession . . . 31

2.4.4 Spin relaxation . . . 34

2.4.5 Conductivity mismatch and the R-parameter . . . 34

References . . . 37

v

3 Experimental techniques 41

3.1 Graphene sources . . . 42

3.2 Device fabrication . . . 43

3.2.1 Exfoliation . . . 43

3.2.2 Localizing exfoliated flakes . . . 45

3.3 Graphene and h-BN Transfer . . . 47

3.3.1 Transfer with Elvacite . . . 47

3.3.2 Pick up transfer . . . 49

3.3.3 Other transfer methods . . . 51

3.4 Device fabrication . . . 52

3.4.1 Substrate markers . . . 52

3.4.2 Contact exposure and development . . . 54

3.4.3 Metal deposition . . . 55

3.5 Etching . . . 56

3.5.1 Graphene etching . . . 56

3.5.2 Heterostructure etching . . . 57

3.6 Measurement setups . . . 58

References . . . 61

4 A transfer technique for high mobility graphene devices on commercially available hexagonal boron nitride 65 4.1 Introduction . . . 66

4.2 Device fabrication . . . 66

4.3 Measurements . . . 68

4.4 Conclusion . . . 71

4.5 Acknowledgments . . . 71

References . . . 72

5 Fast pick up technique for high quality heterostructures of bilayer graph- ene and hexagonal boron nitride 73 5.1 Introduction . . . 74

5.2 Device fabrication . . . 75

5.3 Measurements . . . 78

5.4 Conclusion . . . 80

5.5 Acknowledgments . . . 80

References . . . 81

6 Quantum Hall transport as a probe of capacitance profile at graphene edges 83 6.1 Introduction . . . 84

6.2 Model . . . 84

6.3 Application to experimental data . . . 86

vi

6.4 Acknowledgments . . . 89

References . . . 91

7 Long Distance Spin Transport in High Mobility Graphene on h-BN 93 7.1 Introduction . . . 94

7.2 Device fabrication . . . 95

7.3 Measurements . . . 95

7.4 Conclusion . . . 101

7.5 Acknowledgments . . . 102

References . . . 103

8 Robust spin transport in highly irradiated few-layer graphene 105 8.1 Introduction . . . 106

8.2 Fabrication . . . 107

8.3 Measurements . . . 108

8.3.1 Charge transport measurements . . . 108

8.3.2 Spin transport measurements . . . 109

8.4 Discussion . . . 111

8.5 Conclusion . . . 115

8.6 Acknowledgments . . . 115

References . . . 116

9 Conclusion 117 9.1 Conclusion . . . 118

9.2 Outlook . . . 119

References . . . 121

Summary 123

Samenvatting 127

Acknowledgements 131

List of publications 135

vii

1

Introduction

Abstract

Graphene, a single atom thick layer of graphite, is a material that has always been around

us, but only recently became accessible to researchers worldwide. This could happen due

to a very simple process dubbed the scotch tape method, where the graphite is pulled apart

by tape and deposited on a substrate. The initially produced graphene devices showed

good electronic properties, but this significantly improved after the substrate was com-

pletely removed or substituted by hexagonal boron nitride. One area where graphene

immediately showed its potential is in spintronics, where the electron spin is used to

convey information similar to charge in electronics. Micrometer spin transport length

scales could be reached at room temperature in graphene and by placing graphene on

hexagonal boron nitride, even larger distances can be bridged. This thesis contains two

approaches to fabricate high quality heterostructures of graphene and hexagonal boron

nitride. These heterostructures are then used for studying the capacitance profile near the

graphene edge as well as spin transport in high mobility graphene. Finally the robustness

of spin transport in few layer graphene devices is demonstrated using devices that have

been intentionally damaged by proton irradiation.

1

1.1 Graphene

Carbon atom chicken wire. This would be a very basic description of what graphene looks like; a single atom thick layer of hexagonally arranged carbon atoms as shown in figure 1.1. Only over the past decade this deceivingly simple material has become world famous[1, 2]. Yet interestingly, it has been present around us for a much, much longer time. Graphene is the elemental building block of graphite, which can be seen as a stack of graphene sheets bound together by van der Waals force. To give an indication, 1 mm thick graphite would contain a stack of almost 3 million graphene layers, given that the interlayer spacing in graphite is 3.5 ˚ A. The weakly bound layered structure of graphite is in fact at the core of many of its application.

Layers can easily slide over each other, which makes a good lubricant, for example.

An even more common example are pencils. When writing or drawing with a pencil one is actually, on a microscopic scale, depositing thin layers of graphite and maybe even graphene on paper. Fittingly, the word graphene originates from the Greek word graphein which means to write. And so it seems that we all have been creating graphene since we were making our first pencil drawings, but the story is not that simple.

Figure 1.1:

Graphical impression of what graphene looks like, the black carbon atoms are arranged in a hexagonal lattice. The atoms are spaced by ∼0.14 nm.

Until 2004 graphene in fact mostly remained the domain of theoreticians. Mo- tivated by building an understanding of graphite, graphene was described theoret- ically for the first time in 1947 by P.R Wallace[3]. Further efforts where fueled by the interest in carbon nanotubes (CNT)[4, 5], the 1 dimensional allotrope of graphite.

Single layers of graphene were however deemed to be too unstable to exist thermo-

dynamically by themselves[6]. This belief was proven wrong when a method was

developed and demonstrated in 2004 by Andrei Geim and Konstantin Novoselov

that is as simple as effective for the mechanical exfoliation of graphene[1]. They

demonstrated that by pulling a piece of graphite apart using simply adhesive tape

1

and pressing the tape on a substrate one could obtain monolayers of graphite, a procedure which has become known as the ”scotch tape method”. The discovery of graphene quickly turned it into the subject of intense study[7], demonstrating graph- ene’s remarkable properties such as its electrical properties[1, 8], strength[9], thermal conductivity[10] or impermeability[11]. Andrei Geim and Konstantin Novoselov have been awarded the Nobel prize in physics of 2010 for their discovery.

However, the scotch tape method also has it’s downside. This method has a relatively low yield of small flakes (typically in the 10 µm order) that are distributed randomly on the substrate. As a consequence the process of producing an actual graphene device requires one to first identify a useable flake. This may work in a laboratory environment, but is unacceptable for larger scale use. Therefore, after graphene was demonstrated to exist in a stable form, more methods for graphene production have been developed. These include chemical vapor deposition[12–17], epitaxial growth on silicon carbide[18–20] and liquid exfoliation[21, 22]. While these are important developments for graphene to find its way into real life applications or to realize large area graphene, none of these methods have exceeded the scotch tape method in terms of quality so far. But that does not mean that there is no room for quality improvement where exfoliated graphene is concerned.

The fact that graphene is merely a single atom thick layer makes it also extremely sensitive to its environment. This can be considered an interesting characteristic to exploit for sensor application[23–25]. However, this additional functionality is un- desired in other applications of graphene. Graphene’s excellent mobility, which can be used as a measure of quality, is for example considerably limited by the typically used silicon dioxide substrate. Furthermore, unwanted adsorbates on the graphene can unintentionally dope the graphene. Therefore, new fabrication methods were required to truly explore the intrinsic properties of graphene.

1.1.1 Quality improvement

The first and most straightforward method to obtain something closer to pristine graphene is to remove the substrate. Several methods were developed to do so by removing the substrate underneath the graphene[26–29]. However, the processing steps required to fabricate a graphene device also introduce a considerable amount of contaminants that need to be removed. For suspended devices this was done by current annealing, meaning that a large current is sent through the graphene in order to heat it up and hence clean it of adsorbates. Current annealing is effective since it can be done in situ, however it is also difficult to control and in most cases the graphene will break[30].

A safer method was developed with the use of hexagonal boron nitride (h-BN)[31–

33], which is also referred to as white graphene as might be understood from fig-

ure 1.2. The indicated similarity comes from its crystal structure, which is hexagonal

1

a) b)

Figure 1.2:

a) Graphite flakes (size ∼1 to 2 mm). b) Hexagonal boron nitride flakes (size

∼1 mm). Copyright HQ Graphene.

with layers held together by van der Waals force. The in plane lattice constants of h-BN are actually very similar to graphene with a mismatch of only 1.7%[34]. Be- sides that h-BN makes a good dielectric, with a relative permittivity  ≈ 3.9 similar to SiO

and breakdown voltage of ∼0.7 V/nm[31]. By including a sufficiently thick (∼15 nm) h-BN layer between the SiO

and graphene the influence of the original substrate can be eliminated and indeed the initial reports showed a considerable in- crease in graphene mobility. While this new substrate yielded promising results, the mobilities were still lagging to what could be achieved for suspended devices.

The next developments focused on reducing the possible contamination of the graphene and h-BN flakes to the absolute minimum during the fabrication of a het- erostructure. Since building stacks of graphene and h-BN is done by using polymers which will always leave some contaminants behind, this required another approach.

Instead of building the stack by depositing one layer at a time, the inverse was done by picking up the layers one by one, stacking them in the process[33, 35]. No more polymer is included in the stack this way. The reported devices realized this way have shown the highest possible quality, on par with suspended devices.

1.2 Spin transport

In electronics the charge of an electron is used to convey information. In a transistor

for example, a voltage applied to a gate is used to electrostatically deplete a semicon-

ducting channel of charge carriers, switching the transistor from an on to an off state

1

where no charge current can flow. In spintronics on the other hand, the electron spin is used to carry information. The electron spin is an intrinsic magnetic moment with two eigenstates, spin up and spin down. The imbalance between electrons with spin up or down is the main source for the occurrence of ferromagnetism, which plays an important role in spintronic devices. Graphene made its entry in the field of spin- tronics in 2007 and since then secured its position well, for reasons that will follow.

First, we will shortly address the relevance of spintronics.

A key element for spintronics is the spin transistor, the equivalent of a regular transistor for electronics. A simple spin transistor can be created by applying a volt- age between two ferromagnets separated by a non-magnetic material. By individu- ally controlling the magnetization of the ferromagnets with respect to each other one can either align them parallel or anti-parallel. Respectively this results in a low or high resistance, the equivalence of an on and an off state. The significant change in resistance due to a switch from parallel to anti-parallel magnetization of the ferro- magnets in a spin transistor is known as giant magnetoresistance (GMR). This effect was discovered by Albert Fert and Peter Gr ¨unberg in 1988, a discovery that was awarded with the Nobel prize in physics of 2007. GMR mainly finds its application in magnetic field sensors. Well known examples that are of great importance today are the hard disk drive (HDD) and magnetic random access memory (MRAM) or more recently spin transfer torque random access memory (STT-RAM). In a HDD the data is essentially stored by magnetizing a thin ferromagnetic layer on a disk with a write head. The data can be subsequently read with a read head which is sen- sitive to the relative magnetization between the head and the disk due to GMR. In MRAM two perpendicularly aligned grids of wires are separated by memory cells.

The memory cells, consisting of a pinned and a free magnetic layer spaced by a tun- nel barrier, can be magnetized by sending simultaneous current pulses through the crossing wires. This allows for creating parallel or anti-parallel aligned cells, which can be seen in terms of computer language as a ”1” and ”0” respectively. What sepa- rates STT-RAM from MRAM is that spin transfer torque is used to assist in switching the free magnetic layer. A spin polarized current injected into the free layer will ap- ply a torque to the layer when their magnetizations are misaligned. With the help of this torque a lower writing current is needed to switch the free layer.

In the devices just described, the separating medium between the (ferro-) mag- nets acts just as a spacer. However, in the charge-based transistor mentioned at the start of this section the medium between the source and drain electrodes is acted upon by an external gate. When something similar could be used for the spin tran- sistor, this would add extra functionality. Spins can then be acted upon while they move between the source and drain electrodes. And indeed, a similar approach can be taken for the spin transistor by using ferromagnetic source and drain electrodes.

The medium that connects the two needs to meet certain specific requirements how-

ever and that is where graphene comes in.

1

1.2.1 Graphene based spin transistors

An important requirement is that spins traversing the medium in between the fer- romagnets do not lose their orientation, i.e. information, on their way. Spins will unavoidably relax to the equilibrium state of the medium, but some factors will influence the rate at which relaxation takes place. One factor is the spin-orbit in- teraction, the interaction between the magnetic moment and orbital motion of an electron. The magnitude of this effect scales with the atomic weight as Z

. An- other source for spin relaxation is hyperfine interaction between the electron spin and nuclear spin. Graphene appears as a favorable choice where both mechanisms are concerned[36, 37]. Firstly, carbon is a relatively light atom and secondly, it has no nuclear spin

. A key measure to determine the relaxation rate is the relaxation time τ

, which represents the time it takes to reduce the spin imbalance by a fac- tor e

. When the space between the two ferromagnets is concerned, spin diffusion is another important factor. The spin diffusion constant D

provides a measure for the speed at which spin travels through the medium. Combining D

and τ

, the spin relaxation length can be obtained: λ = √

τ

D

. Initial experiments on the spin transport in graphene confirmed that it indeed carries great potential[38–40], with spin relaxation lengths of ∼2 µm at room temperature[41]. The improvement of the electronic quality of graphene is also relevant for spintronic devices. By changing from a SiO

substrate to h-BN, spin signals could be observed after actually being transported over 20 µm[42]. Further improvements in device fabrication such as encapsulation with h-BN[43, 44] or following a bottom-up procedure with h-BN on top [45] allowed for increasing spin relaxation lengths, up to 30 µm. The improve- ment in spin relaxation length is achieved both through higher diffusion constants and longer relaxation times.

1.3 Outline

• In Chapter 2 the theoretical background of graphene will be discussed. This starts with the crystallographic and electronic properties and concludes with graphene spintronics.

• Chapter 3 mainly discusses the fabrication steps involved when preparing graph- ene h-BN heterostructure devices as well as the basic measurement setup used for device characterization.

• Chapter 4 shows a method for fabricating a h-BN graphene stack by using a sac- rificial polymer layer. Importantly it also demonstrates for the first time that

1In fact, C-12 has no nuclear spin, which comprises 99% of the naturally occurring carbon.

1

with commercially available h-BN one can achieve high quality heterostruc- tures.

• In Chapter 5, a fabrication method for heterostructures comprised of more than two layers is developed. Individual flakes are picked up from a substrate while creating the stack instead of being deposited one by one. This is an important step in order to achieve clean interfaces.

• Chapter 6 shows how by using a high electronic quality graphene device, one can probe the charge density profiles at graphene’s edges using the quantum Hall effect.

• In Chapter 7 spin transport is measured on graphene h-BN heterostructures, allowing for the detection of non-local spin signals over much longer distances than before.

• Chapter 8 takes another approach to demonstrate the potential of graphene for spintronic applications. Few layer graphene devices are damaged using proton irradiation and nevertheless show remarkably robust spin transport.

• Chapter 9 concludes this thesis and gives a brief outlook of how the techniques

developed here are applied in other research.

1

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1

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2

Theory

Abstract

Here the theoretical aspects required for the understanding of chapters 4 to 8 are explained

further. First the characteristic honeycomb crystal lattice and linear dispersion relation,

that dictates the charge transport properties of graphene, are looked into. This is followed

by how the quantities that are measured in graphene field effect transistors are extracted

from the devices. The next section discusses the quantum Hall effect, which can be used

to check the quality of a device. The final part is about spin transport in graphene, start-

ing with the basic non-local spin valve measurements and ending with Hanle precession

measurements that are used to extract the spin transport parameters.

2

2.1 Graphene properties

Graphene is made special by its lattice. This section will give a background to the origin of graphene’s extraordinary electronic properties.

2.1.1 Crystal lattice

Carbon, the building block of graphene, has two stable isotopes. Of the naturally occurring carbon about 99% of the atoms are C-12 with nuclear spin I = 0 and 1%

are C-13 with I = 1/2[1]. The atomic ground state of a carbon atom has its electrons arranged in the 1s

2s

2p

configuration[2, 3]. In the presence of other carbon atoms it is energetically favorable to excite one electron from the 2s orbital to the third 2p orbital. For single layer graphene (SLG) this consequently leads to sp2 hybridization through the superposition of the |2si with the |2p

i and |2p

i states to form covalent σ -bonds. The three resulting orbitals lie in a plane with respective angles of 120

, yielding a hexagonal lattice of carbon atoms. The 2p

orbitals remain unhybridized and perpendicular to this plane. They form π-bonds, where the π-electrons are delo- calized over the entire graphene structure.

A B

x y

a₁ a₂

φ a ≈ 1.42 Å

Figure 2.1:

Schematic of the graphene lattice. The two carbon atoms spanning the basis of the hexagonal lattice are labeled A and B, the unit cell spanned by a

and a

is outlined.

The hexagonal Bravais lattice of graphene as shown in figure 2.1 is spanned by

the unit vectors − → a

and − → a

, where |a

| = |a

| and their angle φ = 120

. The unit

vectors shown here are defined as:

2

−

→ a

= √

3a, 0, 0 

, (2.1)

−

→ a

= − √ 3 2 a, 3

2 a, 0

!

, (2.2)

where a is the nearest neighbor spacing between the carbon atoms in graphene. It is found that a ≈ 1.42 ˚ A, which is the average between the spacing of the double bonds and the single σ-bonds. The corresponding value of |a

| and |a

| is p(3)a ≈ 2.46 ˚ A.

There are two atoms, labeled A and B, in the basis of the graphene lattice which each form a hexagonal sublattice. The outline in Fig.2.1 marks the unit cell, the two sublattices are distinguished by the dark- and light gray atoms. The corresponding first Brillouin zone is hexagonal in shape and is rotated by 90

with respect to the graphene lattice. The reciprocal lattice vectors are:

−

→ b

=

 2π

√ 3a , 2π

3a , 0



, (2.3)

−

→ b

=

 0, 4π

3a , 0



. (2.4)

Important to the electronic properties of graphene are the six corners of the Bril- louin zone, which are alternatingly labeled the K and K’ points

. At these points, also known as the Dirac points, the valence band touches the conduction band as shown in figure 2.2. The valence band of undoped graphene is filled up to the Dirac points.

Graphene is therefore defined as a zero-gap semiconductor or a semimetal[4].

2.1.2 Electronic band structure of graphene

The band structure of a pristine graphene layer was already calculated in the late fifties in order to build a better understanding of graphite[5, 6]. For low energy excitations

(|E| < 1 eV) with respect to the Fermi energy, the dispersion relation around a Dirac point is found to be linear by approximation and conical in shape[3, 7]:

E

( − →

k ) = ±v

~| − →

k |, (2.5)

where − →

k is the wave vector relative to the Dirac point, ~ is the reduced Planck constant and v

is the Fermi energy. The latter has been calculated to be v

≈ 10

m/s[8]. This linear behavior has been imaged experimentally using Angle- resolved photoemission spectroscopy (ARPES)[9].

1K and K’ are inequivalent since they cannot be connected using the primitive reciprocal lattice vectors.

2The measurements described in this thesis are within this limit.

2 _k

E

k

a) b)

k E

E

K K’

Figure 2.2:

a) Graphene dispersion relation showing the conduction band (red) and valence band (blue). The Brillouin zone is indicated with a dashed line, the K and K’ points are where the bands meet. b) Two-dimensional representation of the dispersion relation close to the Fermi energy E

.

The hexagonal lattice of graphene thus has a remarkable consequence: charge carriers traverse the graphene at a velocity that is independent of their momentum or energy. This is similar to massless relativistic particles like photons, with the speed of light c being replaced by the Fermi velocity which in turn is a fraction of c of about 1/300. Furthermore, the presence of the inequivalent K and K’ points leads to a two- fold valley degeneracy g

= 2 in addition to the spin degeneracy g

= 2 for spin up and spin down. The density of states for low excitations that follows from the dispersion relation 2.5 is:

ν(E) = g

g

|E|

2π~

v

, (2.6)

In section 2.2 the density of states will be required in the calculation of the charge diffusion constant in graphene based transistors (equation 2.13).

2.1.3 Electronic band structure of bilayer graphene

As the name indicates, bilayer graphene (BLG) can be seen as a stack of two graphene sheets. Looking perpendicular to the graphene plane, the energetically favorable way of stacking the two layers is by shifting them in the (x, y) plane by (

√3 2

a,

a) without any rotations[10, 11], which is called AB- or Bernal stacking[12]. This is indicated in figure 2.3.

Similar to SLG, the first Brillouin zone of BLG is hexagonal in shape and rotated

2

a₁ a₂

φ

x y

Figure 2.3:

Schematic of a AB-stacked bilayer graphene lattice, the top layer is black and the bottom layer gray. The unit cell spanned by a

and a

is also indicated.

by 90

with respect to lattice of the individual flakes in the BLG layer. Also here the conduction and valence bands touch at the corners of the Brillouin zone, K and K’, and both sublattices contribute. The dispersion relation for BLG may be obtained using a tight binding approach[13, 14] and is no longer linear, but becomes parabolic.

At low energies and without an applied external electric field, this relation can be simplified to:

E

( − →

k ) = ~

v

γ

| − →

k |

, (2.7)

where γ

= 0.37 eV is the interlayer coupling between two overlapping atoms. A schematic representation of the dispersion relation is given in Fig.2.4a). The density of states becomes:

ν

(E) = g

g

4π~

v

(2E + γ

), (2.8) where the energy dependent term is the same as for graphene.

Interestingly and unlike SLG, BLG allows for the opening of a band gap by apply-

ing an electric field perpendicular to the BLG plane[14–17]. By altering the respective

potential of the two layers, the inversion symmetry is broken. For a small potential

V between the layers the band gap is given by:

2

a) b)

k E

E

k E

E

Δ

Figure 2.4:

a) Bilayer graphene parabolic dispersion relation. b) Opening of a gap ∆ after creating a potential difference between the individual layers of bilayer graphene.

∆(V ) = 2V − 4V

γ

. (2.9)

The opening of a gap ∆ is also depicted in Fig.2.4b).

The possibility to locally open a band gap in BLG by local gating has led to in- teresting new possibilities. The most straightforward option is to build a transistor with a high on-off ratio, something that is very difficult to achieve with SLG. Fur- thermore, by gating specific areas in the BLG one can realize constrictions that allow for quantized conductance or even create quantum dots[18–20].

2.2 Graphene electronics: field effect transistors

The main approach used throughout this thesis for the characterization of graphene, is electronic (or spintronic) measurement by use of a field effect transistor (FET) ge- ometry. Several useful parameters can be extracted from a field effect device, such as the mobility µ, mean free path l

or charge diffusion constant D

. A basic FET consists of a source and a drain electrode that are interconnected by means of a, preferably semiconducting, channel. Here graphene will serve as the channel. The final component of the FET is a gate that is separated from the channel by a insulator.

The substrate of choice for graphene FET devices is a heavily doped silicon (n++)

wafer. The top side of the wafer is thermally oxidized so the silicon dioxide (SiO

)

will act as the insulating dielectric between the silicon gate and the graphene chan-

nel. The choice in thickness of the SiO

may vary and, as will be addressed in chapter

10 µm

2

h-BN SiO₂

graphene

s d

graphene

SiO₂ Si n++

a) b)

V_bg A

V_sd

Figure 2.5:

a) Schematic of a graphene based transistor. A potential difference V

drives a current through the graphene. A gate voltage V

is applied to the Si backgate, which is separated from the graphene by a layer of SiO

. b) SEM top view image of an actual graphene based transistor with 7 contacts. The graphene flake is placed on top of a h-BN crystal.

3, will affect the optical contrast of the graphene flakes it supports. For graphene FET devices based on hexagonal boron nitride (h-BN) there will be an additional dielec- tric layer in series with the SiO

.

The operation of a graphene FET is done by application of a source-drain volt- age V

to drive a current I

through the graphene layer. With these two values the resistance R = V

/I

can be calculated. A voltage V

applied to the silicon gate then allows for tuning of the Fermi level around the Dirac point. As seen from equation 2.6, this will affect the density of states. Effectively one can tune the carrier type between electrons and holes, making the channel n- or p-type respectively by application of a positive or negative gate voltage.

A typical measurement consists of monitoring the changes in R while tuning V

. For graphene this will yield a characteristic plot where a resistance peak is observed that marks the Dirac point. Ideally the resistance maximum is found at an offset V bg = 0 V, but in many cases extrinsic elements, such as the substrate or adsor- bates may dope the graphene and hence the position of this peak is referred to as the charge neutrality point V

. The graphene flake and the silicon can be approxi- mated as a parallel plate capacitor to obtain the induced charge carrier density n:

n = α(V

− V

). (2.10)

The factor α is given by:

α = 



de , (2.11)

where 

is the relative permittivity of the dielectric layer with thickness d between

the channel and the gate, 

is the vacuum permittivity and e is the elementary

2

charge. For SiO



= 3.9 and for h-BN 

= 3 − 4 [21]. For devices where the insulat- ing layer is composed of both SiO

and h-BN, an effective α

= α

α

/(α

+ α

) can be used.

With n know, the mobility can be calculated using Drude’s formula:

σ = neµ, (2.12)

where σ is the conductivity of the graphene layer. The conductivity is obtained from the resistance by taking the graphene width w and length l between the source and drain into account: σ = l/(Rw). An example of the mobility calculated using equa- tion 2.12 is shown in figure 2.6b.

a) b)

-3x100 ¹²-2x10¹²-1x10¹² 0 1x10¹²2x10¹²3x10¹² 1

2 3 4 5

R

(k

)

n (cm

)

RT K 77 K

-3x100 ¹²-2x10¹²-1x10¹² 0 1x10¹²2x10¹²3x10¹² 20000

40000 60000 80000 100000

RT 77K



(cm

V

s

)

n (cm

)

Figure 2.6:

a) Measurement of the resistance for fully h-BN encapsulated a graphene flake (see figure 3.7). b) Mobility extracted using equation 2.12.

In order to calculate D

, the Einstein relation can be used:

σ = e

νD

, (2.13)

where ν is obtained through equation 2.6. The value of E can be extracted from a measurement by considering the relation between the n and E:

n(E) = g

g

E

4π~

v

. (2.14)

Later the spin diffusion constant D

will be defined. This will allow for a comparison between charge and spin diffusion. Finally the mean free path can be calculated using:

l

= σh

2e

k

, (2.15)

2

where k

= √

πn is the Fermi wave vector[22].

When taking short and long range scattering into account, a density independent mobility µ

can be obtained. At low charge carrier densities scattering is dominated by charged impurities which shifts to short range scattering at high densities. These two regimes are respectively reflected by a residual conductivity σ

at the charge neutrality point and a contribution in resistivity due to short range scattering ρ

. The conductivity is then[23]:

1

σ = 1

neµ

+ σ

+ ρ

. (2.16)

By fitting the σ(n) with σ

known, one can obtain both µ

and ρ

. While the mobility obtained from graphene FET devices will immediately provide informa- tion about the relative electronic quality of devices, knowledge of ρ

will give more insight in the cause of the quality difference in terms of scattering. For graphene devices based on h-BN it is found that the contribution of charged impurities is sub- stantially reduced with respect to SiO

based devices[23].

Additional information about the charge carrier density inhomogeneity in the graphene flake can be obtained from full-width half-maximum (FWHM) of the re- sistivity peak at the charge neutrality point. Strong inhomogeneity, resulting in the formation of electron-hole puddles, will lead to reduced maximum resistivity (in- creased σ

) and larger FWHM of the peak. The FWHM is used to give an upper bound for the fluctuations in charge carrier density.

2.2.1 Bilayer graphene

As mentioned in section 2.1.3, it is possible to open a gap in bilayer graphene when subject to a large displacement field. Using a BLG flake that is sandwiched between two h-BN flakes, it is straightforward to include local gates to the transistor in ad- dition to the global Si backgate[20]. The fabrication method discussed in chapter 5 provides an excellent tool for easy fabrication of such devices.

An example of a dual gated BLG device is shown in figure 2.7 a) and b). From bottom up the device consists of a Si/SiO

substrate, few layer graphene (FLG), h- BN (7 nm thick), BLG and h-BN (17 nm thick). The bottom and top gate each produce displacement fields, D

, which are calculated as follows:

D

= ±

V

− V

d

. (2.17)

The average displacement field is:

D = (D ¯

+ D

)/2. (2.18)

2

10 µm

s

V

tg

bg V

d

d

d

ɛ

ɛ

SiO

Si n++

h-BN

BLG FLG

a) b)

c)

top h-BN

bottom h-BN

s tg d

-30 -20 -10 0 10 20 30

0.00 0.05 0.10 0.15 0.20 0.25

dI/dV(mS)

V_bias(mV) D = 0 V/nm D = -0.1 V/nm D = -0.2 V/nm D = -0.3 V/nm

-1.8 -1.2 -0.6 0.0 0.6 1.2 1.8 -0.6

-0.4 -0.2 0.0 0.2 0.4 0.6

Vbg(V)

V_tg(V)

0 2 4 6 8 10 12 14 G(e²/h)

d)

Figure 2.7:

a) Optical microscope image of h-BN encapsulated BLG. The encapsulated region is indicated with white dashed lines. b) Schematic cross section of a BLG flake encapsulated in h-BN, including source, drain, top-gate and backgate contacts (respectively s, d, tg and bg).

The backgate consists of a FLG flake. c) Conductance measurement at 4.2 K as function of both back- and topgate. The diagonal blue line is where the conductance goes to zero. d) Conductance traces for various displacements fields ¯ D as function of bias voltage.

The displacement fields in equation 2.17 give rise to two separate effects[24].

Firstly a doping is induced in the BLG that is determined by the difference D

− D

. Secondly, ¯ D breaks inversion symmetry, resulting in the gap[14–17].

Both effects are clearly visible in figure 2.7 c). The conductance depends on both

the top- and backgate. Since the backgate covers a larger area of the BLG channel,

a conductance minimum that does not depend on the topgate remains visible as a

horizontal line. A gap opens where the gates are of opposite polarity. The slope of

the diagonal conductance minimum can be used to derive the relative thickness of

the h-BN top- and bottom flakes, indicating that the top h-BN layer thickness should

be ∼2.4 times the bottom layer thickness.

2

2.3 Quantum Hall effect

2.3.1 2DEG

When electrons flow through a sample in the presence of en electric field − → E and a magnetic field − →

B , they will be subject to the Lorentz force[2]:

−

→ F = −e( − →

E + − → v × − →

B ) (2.19)

For a charge current − →

j flowing through a finite sample this will result in a charge accumulation along the edges parallel to the current path as shown in Fig.2.8. In response, an electric field (Hall field) develops in the direction − →

B × − →

j that will cancel the Lorentz force in the steady state. This is known as the Hall effect, which was discovered in 1879[25, 26]. By measuring the potential difference due to the Hall field, one can determine the Hall coefficient R

, defined as R

= E

/j

B

= −1/ne . This allows for extraction of the charge carrier density n.

j

B

B

+ +

+

- -

- v

a) b)

z

y x

Figure 2.8:

Schematic of a two dimensional sample in a magnetic field. a) When a current j

flows between the left and right sides of the sample in a perpendicular magnetic field B

, it experiences a Lorentz force in the y direction. b) The drift velocity v

of the electrons is consequently altered, leading to an accumulation of charge along the edge of the sample and the formation of a electric field (Hall field) that will cancel the Lorentz force.

As B

is increased, the corresponding cyclotron radius r

= ~k

/eB is reduced.

In a semi-classical picture, the electrons in a two-dimensional electron gas (2DEG) will traverse the sample by skipping along the edges when r

becomes sufficiently small. This can be observed within specific device geometries[27]. Electrons that move in an orbital motion in the central region of the 2DEG and do not hit the edges no longer contribute to the transport.

In an energy picture the allowed energy states become discrete and are separated

2

by ~ω

. The following quantized energy levels are allowed for a 2DEG:

E

= (n + 1

2 )~ω

, (2.20)

where ω

= eB/m

is the cyclotron frequency and m

is the effective mass, or in the case of graphene the cyclotron mass. Equation 2.20 describes the integer quantum Hall effect (IQHE) and the index n = 0, 1, 2, ... refers to the different energy states known as Landau levels (LL). The Nobel prize in physics was awarded to Klauss von Klitzing in 1985 for the discovery of the IQHE in 1980[28].

a) b)

z

y x

B

Figure 2.9:

a) Skipping orbit trajectories of electrons in a 2DEG subject to a perpendicular magnetic field B

b) Landau levels in an energy diagram, following equation 2.21. Current flows at the edges, where the Landau levels intersect with the chemical potential.

2.3.2 Graphene

Graphene displays an unusual case of the QHE, the half-integer quantum Hall effect[3, 7, 8, 29, 30]. The cyclotron frequency for the massless Dirac fermions in graphene be- comes ω

= √

2v

/l

where the magnetic length is l

= p

~/(eB)[7]. Important to note is the square root scaling between ω

and B, which was linear for the 2DEG case.

Consequently the quantum Hall effect in graphene can be observed even at room temperature[31]. The Landau level energy spectrum becomes the following[8]:

E

= sign(N )v

p 2e~B(N + 1/2 ± 1/2). (2.21) The ± sign refers to the sublattice pseudospin in graphene and importantly, the low- est Landau level E

appears at E = 0 eV and can only be achieved when the pseu- dospin term is subtracted.

In a graphical picture one can image the allowed energy level as planes in the

(x,y) space, with E on the z-axis, as shown in Fig.2.9b). Towards the edges of the

2

-8x10¹¹ -4x10¹¹ 0 4x10¹¹ 8x10¹¹

0.00 0.25 0.50 0.75 1.00

B (T)

n (cm

)

1E1 1E2 1E3 1E4

R

(  )

0.0 0.5 1.0 1.5 2.0 2.5

0 500 1000 1500

R

(  )

V

(V)

-10 -6 -2

G

(e

/h)

a)

b)

Figure 2.10:

a) Onset of the quantum Hall effect measured at 4.2 K for a h-BN encapsulated graphene Hall bar (see figure 3.7). Peaks in R

start taking shape around 0.2 T b) Measure- ment of R

and G

at 12 T and ∼17 mK. Landau levels show steps of 1 e

/h. Data from P. J.

Zomer et al., unpublished.

2DEG the energy levels bend up towards infinity. Wherever the chemical potential

µ intersects with the energy levels conducting channels will form. By tuning the

applied gate voltage one can thus control the number of channels that are populated.

2

The number of occupied Landau levels for a certain charge carrier density n, called the filling factor ν, is:

ν = n

eB/h . (2.22)

The quantized Hall conductance for graphene is:

G

= g

g

e

h (N + 1

2 ). (2.23)

When transitioning from one transverse conductance plateau to another, the longi- tudinal resistance R

will peak. The development of such peaks with increasing magnetic field is shown in figure 2.10 a).

For high quality devices in large magnetic fields the spin and valley degeneracy can be further broken, resulting in the appearance of quantum Hall conductance plateaus at intermediate filling factors[32]. Such degeneracy breaking is shown in figure 2.10 b). Conductance plateaus form at steps of e

/h instead of 4e

/h .

2.4 Graphene spintronics

At the basis of spintronics[33, 34] lies the quantum mechanical spin, an intrinsic form of angular momentum ~/2 of the electrons[35]. In equilibrium the number of electrons in graphene having spin up and spin down are equal. One can however influence the spin density by injection of a spin polarized current (j

6= j

)

j = j

+ j

, (2.24)

j

= j

− j

, (2.25)

which will generate a spin imbalance in the graphene. Here equation 2.24 represents the charge current density and equation 2.25 represents the spin current density. Fur- thermore j

and j

are the respective contributions to the current having spin up or down. Spin polarized electrons will diffuse through the graphene and eventually relax back to the equilibrium state. The spin polarization of the current is:

P = j

j . (2.26)

During the diffusion process the spins can be manipulated, for example by a per-

pendicular magnetic field that induces spin precession. The spin projection can be

probed at some distance from the injection point using a detection contact. As long

as the spin imbalance does not decay to equilibrium before reaching the detection

point, the injection, manipulation and detection of spin can be used to extract the

spin relaxation time τ

and spin diffusion constant D

. All these aspects will be ad-

dressed in detail in the following section.

2

N

(E) N

(E) E

N

(E) N

(E) E

N

(E) N

(E) E

FM

NM a)

E

µ

µ

x 0

-1 1

b)

c)

e)

f)

µ

µ

Figure 2.11:

Schematic representation of spin injection from a ferromagnet FM into a non- magnetic material NM, in this case graphene. a), b) and c) show the density of states for spin up and down. d) shows a schematic of the device. The tunnel barrier is represented by the dashed region. e) shown the corresponding exponential decay of the spin imbalance in the NM.

2.4.1 Spin injection and detection

The creation of a spin imbalance in a non-magnetic material like graphene can be done with the use of ferromagnetic electrodes. Unlike in the case of non-magnetic materials, the density of states for spin up and spin down at the Fermi energy is unequal for a ferromagnet. When a current is injected into a non-magnetic material through a ferromagnet this can therefore generate a spin accumulation directly un- derneath the contact[36]. A spin chemical potential µ

is associated with the spin imbalance:

µ

= µ

− µ

2 , (2.27)

2

where µ

and µ

are the chemical potentials for spin up and spin down respectively.

Moving away from the injection point, µ

will decay exponentially towards equi- librium. The parameters that are characteristic to this decay are the spin relaxation time τ

, spin diffusion constant D

and spin relaxation length λ, which are related as follows:

λ = p

τ

D

. (2.28)

The spin relaxation time and diffusion constant can be determined with the Bloch equation[34]:

d− → µ

dt = − ω →

× − → µ

+ D

∇

− → µ

−

−

→ µ

τ

. (2.29)

Under the steady state condition

= 0 and considering transport in one dimension with µ

(x → ∞) = 0, this leads to the solution:

µ

= µ

e

, (2.30)

representing the exponential decay of the spin chemical potential to equilibrium from its injection point:

µ

= P eIR

. (2.31)

Here P is given by equation 2.26, which is a measure for injection efficiency. R

is introduced as material specific spin resistance, thus for graphene in this case. It is given by:

R

= R

λ

W , (2.32)

the square resistance R

times the spin relaxation length λ over the graphene width W . Combining equations 2.30, 2.31 and 2.32 yields the spin accumulation in graph- ene at a location x from the contact:

µ

(x) = 1

2 P eIR

λ

W e

, (2.33)

where the factor

accounts for spin diffusing in both the negative and positive x- direction.

Having established a spin accumulation in the graphene, a method of detection

is required as well. Here a second ferromagnet can be used which is assumed to

have the same efficiency P as the injector. While having its magnetization aligned

with that of the injector (parallel state), it will probe the majority spin chemical po-

tential. Alternatively, when aligned anti-parallel it will probe the minority popula-

tion. Hence by switching between parallel and anti-parallel configurations one can

2

effectively observe a switching behavior between a high (parallel) level and a low (anti-parallel) level.

Considering a detector contact placed a distance L from the injector, with its ref- erence at µ

(∞) = 0 , then it will probe a spin potential V

= P µ

(L)/e . With equa- tion 2.33 this results in a spin resistance:

R

= ± 1

2 P

R

λ

W e

. (2.34)

The ± sign is used to distinguish between parallel (+) and anti-parallel (-) configura- tions. In the following section a method for the measurement of spin accumulation will be addressed in more detail.

2.4.2 Non-local spin valve devices

The actual generation and detection of a spin accumulation in graphene comes with some challenges that have not yet been discussed. One issue relates to the contacts and relative resistivity between these and the graphene, as will be addressed in sec- tion 2.4.5. Another relates to the used measurement configuration. When measuring in a transistor scheme similar to that in figure 2.5 a), the charge current is measured simultaneously with the spin current

. This has some disadvantages, for example the small spin signal is easily lost in the noise or spurious charge related effects may show, such as the magneto Coulomb effect[37] or anomalous Hall effects[38].

An effective way to avoid such charge related issues, is to exploit the spin diffu- sion process. In a device with four contacts as schematically shown in figure 2.12 a), the left sided contact pair (contacts 1 and 2) can be used to drive a charge current through the graphene flake. Charge current will be limited to the region between these contacts. Spin, on the other hand, can diffuse away from this region and reach the right sided contact pair (3 and 4), which will be used for spin detection. This device geometry is known as a lateral non-local spin valve[39–41].

Ideally only the inner contact pair is used for the injection and detection of spin, respectively contacts 2 and 3 in figure 2.12 a). The non-local resistance would in this case be given by equation 2.34. For fabrication specific reasons however, it is more convenient for all four contacts to be ferromagnetic. Consequently, the outer contacts can also contribute to a measurement, be it by extracting spin or probing a specific spin population. The impact this may have on a spin valve measurement becomes apparent from the schematic in figure 2.12 b), which displays the spin chemical po- tential for different magnetization schemes of the contacts. The corresponding mea- surement of the non-local resistance is shown on the right in figure 2.12 c). While a device with only one injector and one detector would have only two levels in the

3In the previous section the charge current was only used to create a spin accumulation, but ignored in the measurement.

2

0.0 0.5

0

-0.5 0.0 0.5

0

-0.5 0.0 0.5

0

-0.5 0.0 0.5

0

-0.5 0.0 0.5

0.00 0.25 0.50 0.75 1.00 0

µ

/ µ

x/λ B (a.u.)

R

(a.u .)

I _nl V _nl

a)

b) c)

1 2 3 4

B x

y

Figure 2.12:

a) A non-local measurement configuration. b) Chemical potential for spin up (blue) and down (red) for different contact magnetization configurations, shown above the plots. The magnetic field is continuously increased, opposite to the contact magnetization.

The contribution of individual contacts is shown with a gray dotted line, the probed chemical

potential is shown by open dots. c) Spin valve measurement corresponding to the configura-

tion shown left to the respective plots.