Oxide-Graphene Interfaces for Graphene Spintronics.

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ABSTRACT

STUART, SEAN CLAYTON. Oxide-Graphene Interfaces for Graphene Spintronics. (Under the

direction of Dan Dougherty and Marc Ulrich.)

Graphene’s high carrier mobility and low spin-orbit scattering allow for efficient spin

transport, which has been demonstrated by several publications over useful length scales.

Spintronic devices require an oxide tunneling barrier to allow for efficient spin injection

from a magnetic contact and can employ magnetic oxide gates for spin manipulation. This

thesis concerns the production and characterization of oxide films for graphene based

spintronics.

Pulsed laser deposition (PLD) was used to grow thin, uniform MgO films on graphene of

suitable quality for tunneling barriers. This was an important result, improving on previous

deposition techniques significantly.

Progress toward more sophisticated spintronic devices requires controllable

manip-ulation of spin polarized charge carriers. We have identified Cr₂O₃as a material whose

magnetoelectric properties would enable voltage controlled switching of the exchange

interaction. Magnetoelectric Cr2O3films were produced by PLD. These films were

character-ized by x-ray diffraction, photoelectron spectroscopy and atomic force microscopy (AFM).

The magnetoelectric properties of Cr2O3were characterized by a novel combination of

electrostatic (EFM) and magnetic force microscopy (MFM). Magnetoelectric annealing was

used to produce varying sized magnetoelectric domains imaged by MFM. A local electric

field was applied with a conducting AFM tip, and the local switching of the polarization

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Oxide-Graphene Interfaces for Graphene Spintronics

by

Sean Clayton Stuart

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

Physics

Raleigh, North Carolina

2015

APPROVED BY:

J.E. Rowe J.P Maria

Dan Dougherty

Co-chair of Advisory Committee

Marc Ulrich

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DEDICATION

To my family, for their support, encouragement and motivation.

To my grandfather who suggested I become a Doctor. To my mother who explained

that this did not necessarily mean a medical doctor, and spent years teaching me math and

research skills. To my father for the wealth of technical and troubleshooting knowledge. To

my siblings for keeping me grounded.

To my in-laws for their support, childcare and countless meals which helped me

gradu-ate and allowed me to finish this thesis.

To my wife for the love, support, and active encouragement provided throughout my

graduate and undergraduate education. She pushed me to be the best version of myself.

Finally to my son who gives me incredible joy and who provided much needed distractions

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BIOGRAPHY

The author developed an interest in physics while attending Bellport Sr. High School in

Bellport NY. There he attended an authentic research class which allowed him to read about

quantum computing and related physics topics. Following this interest he attended the

University of Maryland at Baltimore County for a bachelors in physics. While attending

UMBC and for a year afterwards he had the opportunity to work at the Naval Research

Lab where he developed skills in machining, solidworks modeling and vacuum equipment

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ACKNOWLEDGEMENTS

I would like to thank my advisers for their years of support and enormously helpful

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LIST OF TABLES

Table 2.1 List of important X-Ray lines produced by magnesium94_{. . . .} ₅₂

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LIST OF FIGURES

Figure 1.1 3D band structure of graphene (left) and 2D band structure around the K point (right) based on the results of the preceding calculations 2

Figure 1.2 2D hexagonal structure of graphene. Carbon atoms shown in green, with dashed lines outlining a primitive unit cell containing 2 carbon atoms. Thea1anda2vectors define the unit cell of graphene. . . 4

Figure 1.3 AFM of graphene grown on copper foil. The right image shows topog-raphy which is dominated by the copper foil morphology. The left image shows the phase contrast indicating regions with graphene in red and without graphene in blue. The graphene is slightly (< 1nm) higher in the topography image however the phase contrast is necessary to locate the boundaries. . . 12

Figure 1.4 AFM image of a commercial graphene/Si sample. The roughess shown is attributed to PMMA residue on the graphene. . . 13

Figure 1.5 AFM image of graphene grown on silicon carbide (SiC) by evaporation in ultra high vacuum. Left shows a large scale image. The major features are SiC steps. The image at right is a zoom in showing the single layer and brighter bilayer graphene. . . 14

Figure 1.6 Diagram of the Elliott-Yafet (EY) and Dyakonov-Perel (DP) effect. The left side shows the EY effect where the spin ocassionally flips along the conduction path. The green line shows the linear dependence of spin relaxation on number of scattering events characteristic of EY effect. The right side shows the DY effect where the spin gradually precesses along the conduction path. The Red line shows the inverse dependence of spin relaxation on number of scattering events charac-teristic of DP effect. The spin relaxation in materials with both effect is still under debate as illustrated by the question mark. Diagram from Boross et. al33_{. . . .} ₁₆

Figure 1.7 Spectral gap via spin-orbit interactions (2λl) induced by a∆shift of a

carbon atom out of the plane. The red and blue lines show the results with and without including d-orbitals. Figure from Gmitra et al41_{. . .} ₁₉

Figure 1.8 Mott model of charge transport across an interface between a fer-romagnet and a semiconductor. The spin transport is modeled by separating the current into a spin up (I_↑) and a spin down(I_↓) channel. The effective resistance (R_↑,R_↓) in the magnet is spin dependent. The large difference between (R↑- R↓) and RS C makes the spin injection

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Figure 1.9 A. SEM image of a non local spin valve on graphene. B. Side View diagram showing the device geometry. C and D. the voltage of spin up vs spin down electrons with the film magnetizations shown by black arrows. The figure is from Tombros et. al.44 _{. . . .} ₂₄

Figure 1.10 Datta-Das device, an electron wave analog of the electro-optic mod-ulator52 _{. . . .} ₂₇

Figure 1.11 Graphene Spin-FET geometry from Semenov et. al.55_{. A} Ferromag-netic Dielectric (FMD) is used to alter the polarized electron spins, injected at the source S, in the graphene channel. The effect is tuned through the gate voltage Vg and detected at the source D. The arrows

in the graphene channel illustrate the spin precession induced by exchange interactions with the FMD. . . 28

Figure 1.12 Diagram of the time reversal and spatial inversion symmetries of different magnetic and multiferroic orders. When both time reversal and spacial inversion symmetries are present multiferroic proper-ties (including magnetoelectric properproper-ties) are present. Figure from Eerenstein et. al.56 _{. . . .} ₃₁

Figure 1.13 Chromium (III) oxide structure. The chromium atoms are shown in green and oxygen shown in grey. The chromium atoms are arranged in two sub-lattices, one with blue and one with red arrows corre-sponding to up and down magnetic polarization. The terminating chromium atoms are are aligned with the same magnetization. . . 32

Figure 1.14 Spin polarized UV photoelectron spectroscopy of a Cr2O3film on sapphire. The red and blue data show alternate spin polarization of collected electrons. The green data are the difference spectrum, revealing the existence of a spin polarized state near the Fermi level. Spectrum reproduced from He et. al.60 _{. . . .} ₃₃

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Figure 2.1 Schematic of the inter-atomic forces involved in atomic force mi-croscopy. The tip - sample interaction is dominated by the atomic forces between a terminating atom (or group of atoms) of a tip and the atoms at the surface of a sample. The tip is moved along the contour B by tracking these forces and adusting the tip (or sample) position. The atom A is drawn to show how atomic forces can be probed in an STM measurment when a atom transduces a atomic force to a tunneling current signal. Illustration from Binnig et. al64_{. .} ₃₇

Figure 2.2 An AFM tip that is large (radius R₂) compared to the surface features (radius R₁) will not resolve the topography correctly. In this case the tip geometry is a dominant feature in the measured topography (dashed line). Diagram from Westra et. al73_{. . . .} ₃₉

Figure 2.3 Schematic of atomic force microscopy with optical deflection mea-surment. Left shows the tip deflected up where the laser reflection is measured by the blue detector. Right shows a less deflected tip where the reflection moves to the red detector. . . 39

Figure 2.4 The choice driving frequency relative to the harmoic frequency of a AFM tip determines wether the measured forces are attractive or repluslive. Figure from Seo et. al67_{. . . .} ₄₃

Figure 2.5 An example of magnetic force microscopy. The red and blue features are magnetic bits encoded into a hard drive platter. . . 44

Figure 2.6 Magnetic Force variant of Atomic Force Microscopy. The red is a magnetic film on the tip and sample. The magnetic force between the tip and sample are probed by offsetting the tip from the surface, decreasing the effects of shorter range inter-atomic forces. The phase shift of the tip oscillation is related to the magnetic force. . . 45

Figure 2.7 The Asylum AFM showing the scan head and the sample stage. Pic-ture from Asylum Research86 _{. . . .} ₄₈

Figure 2.8 The Asylum AFM showing the optical path and z- piezoelectric inside the scan head. Picture from Asylum Research86_{. . . .} ₄₉

Figure 2.9 Hemispherical electron analyzer geometry. Electrons with selected energy (green) will pass through the electric field of the Hemispheres and be detected. Electrons with energy too high (red) or too low (blue) will be discarded. . . 53

Figure 2.10 Photo of the Riber XPS system. . . 55

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Figure 2.12 Time evolution of the plasma produced in a pulsed laser deposition system. The top-left image shows the location of the target, laser and the heater (where the sample is placed). The plasma expands over 15

µs (top panels) untill it hits the heater and rebounds to fill the target -heater area. Image from Lowndes et. al.98 _{. . . .} ₅₈

Figure 2.13 Plot of velocity of ablated particles as a function of laser fluence. Plot from Tasaka et. al100_{. . . .} ₅₉

Figure 2.14 Plot of velocity of ablated particles as a function of chamber pressure and distance from target. Plots from Riabinina et. al101_{. . . .} ₆₀

Figure 2.15 Image of pulsed laser deposition system used in this research. . . 62

Figure 2.16 Diagram of pump system used in PLD chamber. The bottom port of the chamber is connected to a turbo pump which is separated by a gate valve. Two smaller metal valves allow for the indtroduction of nitrogen for venting and for use of a secondary pump used for the initial chamber evacuation. A O₂leak valve allows for controlled introduction of oxygen. . . 63

Figure 3.1 AFM images showing the initial stages of MgO growth on HOPG. All three depositions grown at 10−5_{Torr of O}

2at room temperature. a) 10 shots, 50 Hz, at 380 mJ laser energy. Small_∼5Ahigh islands are uniformly spread across the surface rms∼0.07 nm b) 50 shots, 50 Hz, 400 mJ 50% covered surface with the MgO islands<1 nm in height. rms 0.4 nm c) 75 shots, 50 Hz 400 mJ. A completely coverage surface, the pinholes seen in b) are filled in by the additional MgO rms∼0.1 nm. . . 72

Figure 3.2 RMS roughness of MgO on HOPG as a function of oxygen pressure during deposition. All films were deposited with 100 shots at 400 mJ and 50 Hz at room temperature. . . 74

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Figure 3.4 XRD spectrum and AFM image from_∼100 nm MgO films deposited on HOPG at elevated substrate temperatures. A) XRD of films grown at room temperature, 150◦_{C, and 200}◦_{C showing emergence of MgO} (200) peak at elevated temperatures. The unlabeled peaks are present in the XRD spectrum of the HOPG substrate. B) AFM image of MgO features. C) AFM line profile of 200nm high MgO features observed at elevated temperatures. . . 76

Figure 3.5 XPS spectra of a MgO film grown on HOPG with 100 shots at 400 mJ and 50 Hz at room temperature and 10−5_{Torr of oxygen before and} after a 300◦_{C anneal for 30 min in UHV. The shaded region marks} the hydroxide peak and the dashed line the O-Mg peak. The larger hydroxide peak at 30◦_{photoelectron emission angle as compared to} normal emission indicates that the hydroxide is predominantly on the surface. Final state effects are anticipated as the cause for the shift in the hydroxide peak after annealing. . . 77

Figure 3.6 XPS (Mg 1s) spectra of a MgO Film grown on HOPG with 100 shots at 400 mJ and 50 Hz at room temperature and 10−5Torr of oxygen before and after a 300◦_{C anneal for 30 min in UHV. . . .} ₇₈

Figure 4.1 AFM images illustrating the growth of Cr₂O₃on HOPG at 100◦_{C with a} roughness of 0.8 nm RMS. Topography is shown at left and associated line profile is on the right . . . 86

Figure 4.2 AFM images illustrating the topography of Cr₂O₃on a HOPG sample at 400◦C. The large scale features are due to the HOPG topography. . 87

Figure 4.3 AFM images illustrating the growth of Cr₂O₃on a SiC - Graphene sam-ple at 400◦_{C. A. The topography after deposition. B. The topography} of the initial SiC- graphene sample . . . 88

Figure 4.4 AFM images illustrating the growth of Cr₂O₃on HOPG at 600◦_{C with} increasing film thicknesses. A, and associated line profile B, show the initial growth is seen to nucleate in 10-20 nm high islands. In C and D a nominally 100 nm thick film is shown, with a roughness of 0.7 nm RMS. A nominally 300 nm film is shown in E and F, with a mainly smooth surface is interrupted by deep gaps. . . 89

Figure 4.5 AFM images illustrating the growth of Cr₂O₃on HOPG at 600◦_{C with} a post anneal at 1000◦_{C. Topography A, and associated line profile B.} ₉₀

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Figure 4.7 XRD of a film of Cr₂O₃grown on HOPG at 400◦_{C. The peak shown} at 39.75◦is consistent with Cr2O3(0001) crystalinity. The other XRD peak is from HOPG via a tungsten X-ray line . . . 93

Figure 4.8 XPS of a film of Cr₂O₃grown on HOPG at 400◦_{C. The peak shape is} characteristic of Cr2O3 . . . 94

Figure 4.9 XPS performed on a partial film of Cr₂O₃films on HOPG deposited at various temperatures. The top spectrum is a large area scan showing a single C 1s peak considtent with sp2_{bonds. Detailed view (bottom)} show a small attenuation in theπ∗_{shakeup peak after deposition. . .} ₉₅

Figure 5.1 Structural and chemical properties of Cr2O3films grown on HOPG. Figure 5.1A shows the topography of a chromium-oxide films grown on hopg with major features representative of the HOPG surface, and particulates common to the PLD process. Figure 5.1B shows a Cr2O3 <0006>peak with a FWHM of 0.32◦for a film grown at 400◦C. Figure

5.1C shows Cr 2p XPS with a structure characteristic of Cr₂O₃. . . 102

Figure 5.2 Magnetoelectric domains produced by magnetoelectric annealing in

∼1T applied magnetic field and variable electric fields. Figure 5.2A shows small domains (_∼2µm) produced by a 1.75 kV/cm electric field. Figure 5.2B shows larger domains (_∼5µm) produced by 1.875 kV/cm electric field. Figure 5.2C shows a much larger (∼ 15µm) domain produced by a 2kV/cm field. . . 103

Figure 5.3 Results of applying a local electric field varying from 2.5V to -2.5V with a conducting AFM tip in contact with the surface. Figure 5.3A, B, and C shows the topography, EFM in red and MFM in blue. Figure 5.3D shows line profiles from the MFM and EFM in Figures 5.3B and C. 104

Figure 6.1 Chromium - graphene lattice mismatch. The graphene is shown in blue and the chromium atoms are in green. . . 114

Figure 6.2 A simulation of a Hanle measurment of spin transport in graphene. The governing equation (Equation 6.3) is used with the values from Han et. al30_{. in Equation 6.5. A scale factor is applied to obtain} numerical agreement with previously published values of∆RN L30. . 117

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Figure 6.4 Calculation of a graphene - Cr₂O₃Spin-FET device performance. The non-local resistance is plotted as a function of applied voltage. A change in resistance of severalΩis easily measurable for a reason-able electric field. The device modeled assumes a 100nm Cr₂O₃ thick-ness, a 5µm spin transport length and tranport properties matching

published values30_{. . . 119}

Figure 6.5 Part (a) The Hanle measurment device geometry where spin current is injected at E2and a change in voltage is measured between E3and E₄. By adding a resistor R_{s e n s e} the V_{s e n s e} is translated into a current (Io u t). The measured Vs e n s e is shown in part (b) as a function of the Rs e n s e and the applied magnetic field. The measured Io u t is shown in part (c), showing a very similar signal to V_{s e n s e}. Figure from Wen et. al.140_{. . . 121}

Figure A.1 Vacuum diagram of the Riber XPS . . . 137

Figure A.2 Peak resolution at different focusing settings . . . 144

Figure A.3 Pins on filament connection to X-Ray Gun . . . 145

Figure A.4 X Ray interlock flowchart . . . 147

Figure B.1 AFM, EFM and MFM of a pattern prepared on 40_±10nm Cr₂O₃-HOPG sample. Vertical bars were patterned magnetoelectric writing using biases of 2, 1, -1 and -2 V. Image A is an AFM image showing the film topography in the area used for all images shown. B and C are EFM images taken with a non-magnetic tip charged to+0.5V and -0.5V respectively. D,E and F are MFM images taken with a magnetic tip charged to 0,+0.5 and -0.5V respectively. The lack of change in the relative contrast for MFM images D,E and F shows that contrast is primarily magnetic as opposed to electrostatic. . . 149

Figure B.2 AFM, EFM and MFM of two Cr₂O₃films deposited on a graphene -SiC substrate. A NSCU logo is used to define a magnetic pattern in the film shown at right. Image A shows the topography in gold, image B shows the EFM in red and image C shows the MFM in Blue. A linear ramp in voltage is used to form a pattern in a second sample at left. Image D shows the topography, image E shows the EFM and image F shows the MFM. . . 150

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Figure B.4 XPS-based thickness calibration for Cr₂O₃films grown on HOPG by PLD. The calibration curve features 4 films grown under 400◦C in 100 mTorr of O2. Each of the points represents the ratio of the carbon 1s peak area to the chromium 2p3₂peak area . The decay was fitted toAe−tx +y

0. The calculated decay constant (t) of 4900±1300 was determined to be the number of shots corresponding to the mean free path of the electrons within the film (corrected for the 15◦_take-off angle of phototelectron leaving the film). The constanty0accounts for trace carbon contamination of the samples and the constant A accounts for the sensitivity and density factors. The mean free path of the electrons in the Cr2O3films was taken to be 2.0 nm in the NIST database95_{. The thickness of the film corresponding to 4900}_±₁₃₀₀ shots is 2.0_±0.5 nm. . . 152

Figure C.1 A simulation of a Hanle measurment of spin transport in graphene. The governing equation (Equation 6.3) is used with the values from Han et. al30_{. shown in Equation 6.5. A scale factor is applied to the} calculate results correspond to the measured values of RNL previously published30_{. . . 154}

Figure C.2 Calculation of a graphene - Cr₂O₃Spin-FET device performance. The non-local resistance is is plotted as a function of applied voltage. A change in resistance of severalΩis easily measurable for a reasonable electric field. The device modeled assumes a 100nm Cr₂O₃thickness, a 5µm spin transport length and tranport properties matching pub-lished values30_{. . . 157}

Figure D.1 The effect of capactive forces on the average phase in EFM images. The phase shift parabolically dependent on the tip bias as expected. 161

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CHAPTER

1 INTRODUCTION

1.1 Graphene

The isolation of Graphene by Gaim and Novoselov1_{in 2004 spawned a remarkable amount}

of research into the electrical and physical properties of two dimensional (2D) materials2,3.

Graphene has a 2D Electron Gas (2DEG) because it is a truly 2D material and, under ideal

conditions, a semimetal. The Graphene band structure can be see in Figure 1.1. Near

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1.1. GRAPHENE CHAPTER 1. INTRODUCTION

the dispersiond E/d K is linear near the fermi level the effective mass of the electrons is

zero. This property of graphene makes electrons behave as massless dirac fermions4_{. This}

is a rare example of quantum relativistic effects observable in a solid state experimental

system4_.

Figure 1.1.3D band structure of graphene (left) and 2D band structure around the K point (right) based on the results of the preceding calculations

The high mobility in graphene makes it appealing for a range of electronic applications.

The highest measured electron mobility in graphene is 200,000 cm2_V−1_s−1_{in a suspended}

graphene sample5_{. However, for most methods of graphene production the effects of}

substrates, defects, and impurities limit the mobility significantly6_{. Several methods of}

pro-ducing graphene have been developed to obtain high quality graphene including solution

processing7_{, and CVD for on various materials}8,9_{. There is usually a trade-off between high}

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1.2. GRAPHENE BAND STRUCTURE CHAPTER 1. INTRODUCTION

exfoliation, including stacks of different 2-D materials10, however this method has a very

low yield and is not suitable for large scale production.

In addition to graphene’s high mobility, the nature of the band gap in graphene is of

interest for applications replacing a traditional semiconductor11_{. Graphene has no intrinsic}

band gap, since the occupied and unoccupied band structure meet at or near the fermi

energy. This fermi energy changes depending on the level of electrostatic or chemical

doping. This is an issue for devices such as a transistor, where a band gap determines the

"on-off" threshold voltage. There has been significant work towards inducing a band gap12_,

however this necessarily reduces mobility. Instead of modifying graphene to replace a

traditional semiconductor, applications that do not require a band gap are immediately

accessible with graphene and are able to use its full potential. For example, graphene’s

high mobility combined with its low spin-orbit interaction make it an ideal material for

spintronics.

1.2 Graphene Band Structure

To understand the electrical properties of graphene, it is necessary to calculate it’s band

structure13,14_{. The crystal structure of graphene is a hexagonal lattice with a carbon-carbon}

distance of 1.42Å. This structure is shown in Figure 1.2, the dashed lines represent the

diamond shaped primitive lattice which contains 2 carbon atoms. Carbon has a 1s2₂_s2₂_p2

electron structure. In graphene each carbon atom forms threes p2 _{bonds, leaving one}

conduction electron per atom. In this section we calculate and plot the band structure

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1

2 a

₁

a

₂

3

4

5

6

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band structure shown in Figure 1.1 is calculated using a linear combination of atomic

orbitals (LCAO) approximation, which begins with the physical structure of the material.

To calculate the band structure a Bloch function is used:

ψk~=

X

~

R∈G

eik~·R~φ(x~−R~) (1.1)

WhereG denotes the set of all lattice vectors,R~is a specific lattice vector,k~is the

momen-tum, andφis an atomic wave function andφ(x~)is a linear combination of thepz orbitals

of both carbon atoms in the primitive cell. We can label these asφ₁andφ₂so thatφ(x~)is

defined as:

φ(x~) =b1φ1(x~) +b2φ2(x~) (1.2)

where b1and b2are the respective amplitudes.

The Hamiltonian for an electron in graphene (at positionx~) is then given by:

H = p~

2

2m +

X

~

R∈G

Va t(x~−x~1−R~) +Va t(x~−x~2−R~)

(1.3)

where thex~1andx~2are the positions of the two carbon atoms in the unit cell. TheR~term

describes the set of vectors that translate between equivalent lattice sites.

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1.2. GRAPHENE BAND STRUCTURE CHAPTER 1. INTRODUCTION φ1 H φ₁

=ε1

φ1 φ₁ + φ1 X ~

R6=0

Va t(x~−x~1−R~) +Va t(x~−x~2−R~)

+Va t(x~−x~2)

! φ₁

(1.4)

Whereε₁is the energy ofφ₁,Va t is the potential between the location atx~and a specific

carbon atom. Theφ₁atomic wavefunction is located at first atom in the unit cell, so the sum

does not includeR~=0 to prevent double counting that term. Equation 1.4 can be simplified

using gauge invariance to setε1,2=0 and defining the sum (in the large parentheses) as

U1,2corresponding toφ1,2.

To calculate the band structure we need to solve the Schrodinger equation:

Hψ_k~

=E(k~)ψ~_k

(1.5)

We can do this by writing theψwave function in the basisφas shown:

ψ_k~

=X i φi ψ_k~

(1.6)

Using this basis the Schrodinger equation (Equation 1.5 changes to:

Hψ_K~

=Hφi

ψ_k~

= φi U_i ψ_k~

(1.7)

The first term to calculate isφ1

ψ_k~

:

φ1

ψ~_k

= φ1

eik~·~0+ei~k·a~1+_eik~·a~2_{. . .}+_eik~·a~n _b 1

φ₁

+b2

φ₂

(1.8)

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nearest neighbor interactions. This allows us to set theeik~·a~n>2 _{terms in Equation 1.8 to zero.}

We will consider the interaction between the wavefunctionφ₁and each ofφ₂,φ2(x+a1),

andφ2(x+a2). These interactions take into account the nearest neighbors from the number

1 lattice site shown in Figure 1.2. The termsφ∗

1andb1φ1are evaluated only atR =0 to

avoid including second order interactions. The results of this approximation are:

φ1

ψ

=b1+b2

Z

φ∗ 1φ2

1+e−ik~·a~1+_e−i~k·a~2=_b

1+b2γα(k~) (1.9)

φ2

ψ

=b2+b1

Z

φ∗ 2φ1

1+eik~·a~1+_eik~·a~2=_b

2+b1γα∗(k~) (1.10)

Here the integrals are equivalent due to symmetry and renamedγ. These integrals

calcu-late the overlap between nearest neighborpx orbitals. The sum of exponentials in both

equations are namedα(k~). The last difference between these equations is the sign of the

exponential. This can be explained by the difference in sign of the basis vectors needed to

go from the first atom in the unit cell. From Figure 1.2 one can see that the first atom in the

unit cell, labeled 1, has atoms 3 and 4 for nearest neighbors. Atoms 3 and 4 are equivalent

to atom 2, with the subtraction of vector a1and a2.

The right hand side of the Schrodinger equation (Equation 1.7) is:

φ1 U₁ ψ = φ1 U₁

1+e−ik~·a~1+_e−ik~·a~2 _b 1

φ₁

+b2

φ₂

(1.11)

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1.2. GRAPHENE BAND STRUCTURE CHAPTER 1. INTRODUCTION φ1 U₁ ψ =

(b1

Z

φ∗ 1U1φ1

+b2

Z

φ∗ 1U1φ2

1+e−ik~·a~1+_e−ik~·a~2_. _(1.12)

We can remove theRφ₁∗U1φ1term, because there is (trivially) no potential difference betweenφ1andφ1. For simplicity we can defineη=

R

φ∗

1U1φ2. And due to symmetry:

η=R

φ∗

2U2φ1=

R

φ∗

1U1φ2, so we can easily write the results of

φ2 U₂ ψ

as well as:

φ1 U₁ ψ

=b2ηα(k~), (1.13)

and φ2 U₂ ψ

=b1ηα(k~). (1.14)

Using these equations along with equations 1.9 and 1.10 and applying them to Schrodinger

equation stated in Equation 1.7 results in:

E(k~) b1+b2γα(k~)

=b2ηα(k~), (1.15)

and

E(k~) b2+b1γα(k~)

=b1ηα(k~) (1.16)

.

This can be written in matrix form:

 

E(k~) α(k~)(E(k~)γ−η)

α∗₍_k~₎_E₍_k~₎_γ _E₍_k~₎

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Taking the determinant yields:

E2(k~)−α(k~)

2

(E(k~)γ−η)2=0. (1.18)

Expanding:

E2(k~)−α(k~)

2

(E2(k~)γ2+η2−2E(k~)γη) =0. (1.19)

In the tight binding approximationγis small, so we set the terms containingγto zero

and obtain:

E2(k~)−α(k~)

2

η2₌

0. (1.20)

The following equations show the steps taken to simplify this result.

E(k~) =±α(k~)

η (1.21)

E(k~) =±ηÇ 1+e−ik~·a~1+e−ik~·a~2 1+eik~·a~1+eik~·a~2 (1.22)

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1.3. GRAPHENE PRODUCTION CHAPTER 1. INTRODUCTION

E(k~) =±ηp3+eik~·a~1+e−ik~·a~1+e−ik~·a~2+eik~·a~2+e−ik~·(a~1−a~2)+eik~·(a~1−a~2) (1.24)

From here we use the identity stated in Equation 1.25 to re-write the equation in terms of

cosine functions, as shown:

2 cos(x) =ei x+e−i x (1.25)

E(k~) =±η

Ç

3+2 cos k~·a~1

+2 cos k~·a~2

+2 cos k~·(a~1−a~2)

(1.26)

Equation 1.26 is a good approximation of the band structure of graphene. A plot of this

equation is shown in Figure 1.1. This band structure shows the linear dispersion which is

characteristic of graphene.

1.3 Graphene Production

The dominant methods of graphene production are; Chemical Vapor Deposition (CVD),

de-composition of silicon carbide16_{and exfoliation. Exfoliation is accomplished by repeatedly}

cleaving a Highly Oriented Pyrolitic Graphite (HOPG) crystal using adhesive tape. This was

the first method of obtaining few layer graphene flakes, and the method is useful for several

2-D materials such as hexagonal boron nitride17_{and molybdenum selenide}10_{. The method}

is versatile and inexpensive, producing high quality graphene, however it is particularly

(29)

located via optical microscopy, and the size, location and number of layers transferred is

random.

1.3.1 Graphene Production by Chemical Vapor Deposition

The need for wafer scale graphene motivated the development of other methods of graphene

growth. Graphene is most commonly produced commercially by CVD. This process was

first used to grow graphene on a Ni foil18_{, however copper foil allows more control and}

better quality single layer graphene growth19_{. The CVD process uses a hydrocarbon gas}

(such as methane) which is "cracked" by interaction with a hot metal substrate, removing

the hydrogen and leaving behind carbon. The challenges in this method are producing

uni-form graphene with a large grain size. The substrate purity, uniuni-formity, and the particulars

of temperature, pressure, time, and methane flow all have an effect on the growth.

I have produced graphene by this method, and characterized it by AFM and XPS. The

topography of the copper dominates the standard AFM image, making it difficult to resolve

the graphene on the surface, as shown on the left of Figure 1.3. However phase contrast

reveals the graphene by resolving the chemical difference between the graphene and copper

substrate.

For most applications, the CVD graphene needs to be transferred from the metal foil to

a more suitable substrate. A substrate is chosen to electrically isolate and thereby use the

graphene’s electrical properties. The usual choice is a silicon wafer with a significantly thick

thermal oxide layer. To transfer the graphene, a polymer (PMMA or similar) is used to hold

the graphene while the copper is etched away. These polymer and wet chemistry processes

(30)

Figure 1.3.AFM of graphene grown on copper foil. The right image shows topography which is dominated by the copper foil morphology. The left image shows the phase contrast indicating regions with graphene in red and without graphene in blue. The graphene is slightly (<1nm) higher in the topography image however the phase contrast is necessary to locate the bound-aries.

graphene. The topography is also affected, as shown in the topography of commercially

obtained graphene on a Si/SiO₂wafer (Figure 1.4).

1.3.2 Graphene Production From Silicon Carbide

Graphene films produced by CVD typically have too much disorder and have a grain size

too small to obtain the mobility desired for many applications. Epitaxial growth of graphene

on silicon carbide (SiC) produces higher quality graphene. Since it is an epitaxial growth,

the graphene has no rotational disorder, and since it is a wafer scale process, it has the

potential to produce large, uniform areas of graphene. SiC is also a useful substrate since it

has relatively low conductivity eliminating the need for transfer. An example of graphene

grown on SiC that I have produced at NCSU is shown in Figure 1.5. I produced graphene

(31)

Figure 1.4.AFM image of a commercial graphene/Si sample. The roughess shown is attributed to PMMA residue on the graphene.

kept below 1x10−9_{Torr. This process was developed on our system by Dr. Andreas Sandin}20_.

At 1300◦_C _{the silicon evaporates leaving behind a carbon rich surface, this process is slower}

and more controllable at the silicon rich termination of the SiC wafer as compared to the

carbon terminated surface. The initial carbon layer is still partially bound to the silicon and

is called a buffer layer. When more silicon leaves via evaporation, the buffer layer decouples

from the silicon and becomes graphene. A new “buffer layer” is formed at the interface

between the graphene and the SiC bulk.

This growth method was first used at Georgia Institute of Technology the used to create

graphene films used to demonstrate ballistic transport in graphene, among other unique

electrical properties21,22_{. It produces high quality graphene that has been used to create}

(32)

1.4. SPIN RELAXATION IN GRAPHENE CHAPTER 1. INTRODUCTION

Graphene produced in the manner is partially isolated from the silicon carbide but

there are some substrate effects in the graphene’s transport properties. The lattice

mis-match causes ripples in the graphene, affecting it’s electron transport. For use in the most

demanding applications, the buffer layer needs to be passivated. Hydrogen24,25_{and other}

light elements26_{have been used to bond with the buffer layer and reduce it’s effect on the}

graphene. In addition, bilayer graphene has also been used to separate the top graphene

channel from the buffer layer. Two layer graphene on SiC has been shown to exhibit

ex-tremely good spin transport properties27_.

Figure 1.5.AFM image of graphene grown on silicon carbide (SiC) by evaporation in ultra high vacuum. Left shows a large scale image. The major features are SiC steps. The image at right is a zoom in showing the single layer and brighter bilayer graphene.

1.4 Spin Relaxation in Graphene

In general spin relaxation occurs when an electron precesses in a non uniform magnetic

(33)

magnetic field from a magnetic material. But a non-uniform field from impurities,

spin-orbit interactions, or symmetry considerations will cause the spin polarization of charge

carriers to decay.

In many materials the spin-orbit (SO) interactions dominate the spin relaxation process.

The spin orbit interaction can be understood by looking at the electric field from the nucleus

as experienced by an electron. The electron moving in an electric field experiences an

effective magnetic field:

B=

₁

c

~

E ×v~ (1.27)

This magnetic field interacts with the electron spin as expected. The primary parameter in

this interaction is the atomic number (Z) of the material. The electric field is proportional

to eZ, and therefore the SO interaction is proportional to the atomic number. There are

extrinsic effects that can cause SO interactions that are not directly related to the electric

field from the nucleus. Two examples are Dresselhaus SO coupling, which comes from

bulk inversion asymmetry28_{and Rashba SO interactions which appears due to inversion}

asymmetry in 2D structures29. Both of these mechanisms may influence spin transport in

graphene devices.

Since graphene, a very low Z material, has a negligible intrinsic SO interaction it should

have a very high spin lifetime. However the measured spin lifetimes in graphene on the

order of 2ns are much shorter than the_∼1µs theoretically predicted30_{. This observed}

spin lifetime is comparable to many other semiconductors, however the spin diffusion

length and spin signals are larger in graphene due to the higher mobility30_{. To understand}

(34)

contributing to the spin relaxation in graphene.

The most often discussed graphene spin relaxation mechanisms are the Elliott-Yafet(EY)

Mechanism31_{, the Dyakonov-Perel (DP) Mechanism}32_{which are described in Figure 1.6}

and in the following sections.

Figure 1.6.Diagram of the Elliott-Yafet (EY) and Dyakonov-Perel (DP) effect. The left side shows the EY effect where the spin ocassionally flips along the conduction path. The green line shows the linear dependence of spin relaxation on number of scattering events characteristic of EY ef-fect. The right side shows the DY effect where the spin gradually precesses along the conduction path. The Red line shows the inverse dependence of spin relaxation on number of scattering events characteristic of DP effect. The spin relaxation in materials with both effect is still under debate as illustrated by the question mark. Diagram from Boross et. al33_.

1.4.1 Elliott-Yafet Spin Relaxation

The Elliott-Yafet (EY) mechanism describes the change in spin polarization from

interac-tions with impurities, lattice defects, or phonons. These sources of electric fields create

an effective magnetic field through a spin-orbit interaction as described previously. The

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EY effect in graphene34. The Elliott relation shown in Equation 1.28 describes the linear

dependence of spin relaxation time (τ_s) and the momentum scattering time (τ_p). Theα

term is related to the spin flip probability in a momentum relaxation event34_.

τs =

τp

α (1.28)

In graphene this relationship is dependent on the carrier concentration because of the

“absence of a energy gap between the conduction and valenceπbands”34_{. The relevant}

Elliott equation (Equation 1.29) in graphene can be calculated with reasonable

approxima-tions34.

τs '

ε2

F

∆2

S O

τp (1.29)

Hereε_F is the Fermi energy and∆_{s o}is the spin-orbit splitting energy. Since experimental

evidence35_{shows a linear dependence between the spin relaxation time and the diffusion}

coefficient, matching the dependence predicted by Ochoa et. al34_{, this is a good model in}

some systems. Additionally in CVD based spin valves, the EY mechanism (As described by

Avsar et al.36_{) can describe the spin relaxation dynamics}34_{. This is attributed to the large}

number of grain boundaries in CVD graphene34.

1.4.2 Dyakonov Perel Spin Relaxation

The Dyakonov-Perel (DP) mechanism comes into play in non-centrosymmetric

semicon-ductors and describes spin relaxation occurring between scattering events37–39_{. The DP}

mechanism is proportional to the inverse of elastic scattering time τ_{s p i n}_∝_τ 1 e l a s t i c

(36)

occurs between scattering events. This is the inverse of the relationship observed in the EY

mechanism τ_{s p i n}_∝τ_{e l a s t i c}. The additional Hamiltonian term describing the DP effect is:

ħ

hΩ~(p~)S~ (1.30)

WhereΩ~(p~)is the effective magnetic field experience by an electron with spinS~. ThisΩ~(p~)

is a function of the momentum as shown:

Ωx ∼px px2−p

2

z

Ωy ∼py pz2−p

2

x

Ωz∼pz

p_x2−p_y2 (1.31)

When the electron’s momentum changes by collisions or other scattering processes in

the material, the effective magnetic field changes (since it is a function of the momentum).

This causes the spin to precess in between collisions, meaning that the DP spin relaxation

actually increases with decreasing impurities. Though single layer graphene is a

centrosym-metric 2D material, the substrate and impurity interactions or simply bilayer graphene

break that symmetry, allowing DP spin relaxation to occur.

In graphene a number of experiments show that spin relaxation that does not scale

proportionally to the density of charged impurities. Examples include measurements of

spin transport in by-layer graphene27,40_{, and measurements where the effects of the density}

of charged impurities was examined40. The DP mechanism is also used to explain the effects

of ripples in the graphene on spin transport. Conceptually, the ripples force changes in the

(37)

the spin-orbit coupling induced by various effects have been conducted by Gmitra et al41,

including the effects of ripples as shown in Figure 1.7.

Figure 1.7.Spectral gap via spin-orbit interactions (2λl) induced by a∆shift of a carbon atom

out of the plane. The red and blue lines show the results with and without including d-orbitals. Figure from Gmitra et al41_.

1.4.3 Spin Relaxation Experiments

A informative paper by Han et al.42gives some information about when DP or EY interactions

are the dominant effects. By comparing the results of single and bi-layer graphene at low

temperature they show that the spin lifetime has a different dependence in each case,

meaning that EY interactions are dominant in single layer graphene and DP interactions

are dominant in bilayer graphene. However there is still uncertainty about the details of spin

dynamics of graphene30_{. The effects of substrates, defects, and other non-uniformities on}

(38)

1.5. GRAPHENE SPINTRONICS CHAPTER 1. INTRODUCTION

overall spin and electrical transport properties of graphene make it an excellent choice for

spintronics. It is useful to compare graphene to silicon, where the spin transport has been

demonstrated43_{. Applebaum et al. showed spin signals in the pA range and spin lifetimes}

of 1 ns. These figures were only observed at low temperatures (85◦_{K) where graphene has a}

2ns lifetime at room temperature.

1.5 Graphene Spintronics

Graphene is an ideal choice for spin transport due to it’s low spin orbit interaction and high

mobility. Since transporting spin polarized carriers does not require a band gap, the highest

quality and highest mobility graphene can be used. The first graphene spin transport

measurements demonstrated spin diffusion lengths in the micron range44_{. This would easily}

accommodate a device based on the manipulation of electron spin in a graphene channel.

More recent measurements have shown that the use of high quality oxide tunneling barriers

and epitaxial graphene improves the measured spin transport dramatically27_{. Dublak et.}

al27_{. measured spin diffusion lengths in the 100}_µ_{m range and spin signals in the M}_Ω_range,

orders of magnitude above previously observed values30_{. The high quality oxide layers were}

used for tunneling junctions for spin injection.

1.5.1 Tunnel Junctions

The first step in any spin dependent device is to have a spin polarized source of electrons.

In a ferromagnetic metal the fermi level is dominated by a single spin polarization, so

(39)

Figure 1.8.Mott model of charge transport across an interface between a ferromagnet and a semiconductor. The spin transport is modeled by separating the current into a spin up (I↑) and a spin down(I↓) channel. The effective resistance (R↑,R↓) in the magnet is spin dependent. The large difference between (R↑- R↓) and RS C makes the spin injection into semiconductors very

inefficient.

interface between the ferromagnetic metal and the spin conducting semiconductor, the

electron transport can be modeled with two transport channels. One path carries spin

"up" electrons and the other carries spin down electrons as shown in Figure 1.8. The

difference in Fermi level for spin up and down electrons in the magnet makes the chemical

potential different for carriers with different spin. Therefore each path has a different

effective resistance. However a semiconductor’s resistance is very large compared to the

difference in resistance between the two spin paths. Since the total resistance of the spin up

and down paths (including both the ferromagnet and semiconductor) are nearly identical,

the spin polarization of carriers injected into the semiconductor is negligible. This problem

is called the conductivity mismatch.

To avoid conductivity mismatch a tunnel junction can be used. Using Quantum

(40)

via tunneling is modeled using Fermi’s Golden Rule:

λi f =

2π ħ h ψi V ψ_f

ρf(Ef)Ef'Ei (1.32)

Hereλ_{i f} is the transition probability per unit time,ψ_iandψ_f are the initial and final states

of the tunneling electron,ρ_f(EF)is the density of final states,Ei is the energy of the initial

state and finally V is the potential due to the oxide barrier.

Equation 1.32 states that the transition probability per unit time (λi f) is proportional to

the density of the initial and final states. The transmission probability is dependent on the

initial states due to theψi

V

ψ_f

term. Since the initial statesψiis spin dependent in a

ferromagnet, the transmitted electrons preserve that spin asymmetry since the ψ_iV ψ_f

term couples the initial and final states. Notice that this process is independent of

con-ductivity in either the metal or semiconductor thereby solving the concon-ductivity mismatch

problem.

1.5.2 Producing Tunnel Barriers on Graphene

The potential barrier necessary for quantum tunneling should be as spatially uniform as

possible. The oxide or other non-conducting material should be chemically and physically

uniform. Since we want significant conduction through tunneling barrier, it must be no

more than a few nanometers thick. MgO and Al2O3 oxide barriers are typical for spin

injection into graphene45. Early attempts at producing oxide films on graphene failed due

to the low sticking coefficient of graphene46_{. To overcome the low reactivity of graphene,}

oxide films have been produced by using fluorine to increase the adhesion of oxide species47_.

(41)

to graphene46. The quality of the oxide interface is particularly important to a device as

it directly effects the quality of spin injection and the quality of the graphene after oxide

deposition. We have found that pulsed laser deposition can produce high quality MgO

films quickly and relatively easily, as described in Chapter 3. This process avoids a chemical

modification of the surface, avoiding possible damage to the graphene or detrimental

effects on mobility.

1.5.3 Non Local Spin Valves

Spin diffusion lengths and related quantities can be measured by a non local spin valve48_.

It is comparable to a four contact resistance measurement where two contacts are used to

inject current and another two contacts are used to measure a potential drop due to the

resistance of the material. The four contacts are used to eliminate contact resistance from

the measurement. Injection of spin-polarized electrons is accomplished using a

ferromag-netic metal and a tunnel junction. The decay in spin polarization as the electron translate

through the material causes a potential difference which can be measured using another

magnetic contact and tunneling barrier setup. In addition to injection and measuring spin

polarized current, this interface will also probe the spin dependent density of states.

The first reported non local spin valve used to measure spin transport in graphene

is shown in Figure 1.9. This measurement, conducted by Tombros et. al., measured a

spin transport length of 2µm44_{. Subsequent measurements have claimed a 100}_µ_{m spin}

transport length27_{. The difference in voltage measured at the contacts 1 and 2 corresponds}

to the decay in spin polarization as the electrons are transported in the graphene channel.

(42)

(43)

calculated. The equation modeling the 2-D spin transport44is:

Rn o n−l o c a l =

P2

s λs f

2Wσe

−L

λs f _(1.33)

WhereRn o n−l o c a l is the measured resistance,Ps is the spin polarization of the contacts, L

and W are the length and width of the graphene strip between the central electrodes, and

λs f is the spin relaxation length and theσis the conductivity.

1.5.4 Hanle measurment

In a Hanle measurment, a change in the non local resistance is measured in a spin valve

geometry as a result of an applied magnetic field. The measurement is similar to the non

local spin valve, in that spin injection and detection is accomplished with a magnetic

material and a tunneling barrier. However, the addition of a varying magnetic field causes

the electron spins to precess. As shown in Equation 1.34, the length of the channel (L) and

the spin lifetime (τ_s) and the electron’s precession frequencyωare the dominant factors.

RN L∝ ±

Z ∞

0

1

p

4πD te

−L2 4D t

cos(ωt)e

−t

τs

dt (1.34)

The spin precession frequency is a function of the magnetic field as expected:

ω=gµbB⊥ ħ

h (1.35)

Hanle measurements are an excellent way to quantify spin transport unambiguously49,50

(44)

1.5.5 Datta Das Device and the Rashba Effect

A device that modifies the spin polarization of electron waves was proposed by S. Datta

and B. Das in 198952_{. To explain the device, they drew an analogy to the modification of}

light polarization using a electro-optic material. An electro-optic material has a direction

dependent dielectric constantεx xorεy y so that a 45◦polarized wave of light is shifted from

equal combination of x and y polarized light:

  1 0  +   0 1  =   1 1  →  

ei kxL

ei kyL



 (1.36)

Wherekxandky are momentum of light polarized in the x and y direction and L is the path

length.

The polarization shift is modifiable (in an electro-optic material) by applying an electric

field, which changes the ratio ofkx toky and thereby the polarization of the transmitted

photons. When placed in between a polarizer and analyzer the transmitted photons can be

modulated with the voltage applied to the eletro-optic material.

The analogous device where electron spin polarization is being manipulated instead

of light polarization is shown in Figure 1.10. This device uses the spin orbit interaction

of electrons through a small bandgap semiconductor (such as InGaAs) where the 2DEG

experiences a zero field splitting due to the Rashba effect. The Rashba term in the effective

mass Hamiltonian is:

(45)

Figure 1.10.Datta-Das device, an electron wave analog of the electro-optic modulator52

Whereη is the Rashba coupling,σ_x_,_y is the x or y Pauli matrix vector, and Kx,y is the

momentum in the x or y direction.

The Rashba effect occurs in 2-D materials. An electric field (effective or applied) breaks

the symmetry of spin up and down conductors moving in two dimensions, and thereby

breaks the degeneracy of the two states. This interaction is very similar to that of the

standard spin-orbit interactions (B= 1_cE~×v~) where in this case the E field comes from the symmetry breaking instead of a nucleus. By applying a electric field across a 2DEG in a

material (or interface) with the proper symmetry, the electron spins can be manipulated.

In most cases this effect is not seen in graphene since it does not show this symmetry,

although there have been attempts at pairing graphene with a suitable material to induce

this effect, such as gold53_{and nickel}54_.

1.5.6 Graphene Spin Field Effect Transistor

Instead of using the Rashba effect to manipulate spin, an exchange interaction can be used.

(46)

between spin polarized electrons in a magnetic material and electrons in graphene. This

is necessary when the spin-orbit coupling is negligible, such as in graphene. By using the

effective magnetic field’s interaction with electrons in graphene, the high mobility and low

intrinsic spin orbit interaction will allow for an interesting and useful device. This setup

what proposed by Semenov et. al in 200755_{with the device structure shown in Figure 1.11}

Figure 1.11.Graphene Spin-FET geometry from Semenov et. al.55. A Ferromagnetic Dielectric (FMD) is used to alter the polarized electron spins, injected at the source S, in the graphene channel. The effect is tuned through the gate voltage Vg and detected at the source D. The arrows

in the graphene channel illustrate the spin precession induced by exchange interactions with the FMD.

This device relies on the magnetic exchange interaction with the “Ferromagnetic

Di-electric” (FMD) layer. The exchange interaction depends on the overlap of the graphene

wavefunction with that of the (uncompensated) spins of the magnetic ions in the FMD. The

magnetization in the dielectric is namedS~_λ,j designating the spin moment at siteλof unit

cell j. Again following the notation and results of Semenov et. al.55_{the exchange interaction}

(47)

1.6. MAGNETOELECTRIC CHROMIA CHAPTER 1. INTRODUCTION

He x=

1

N

X

j=1

X

λ

J(R~_λ,j)S~λ,j·s~ (1.38)

The interaction between the electron spin,s~, and the magnetization in the material (via

the exchange interactions) causes the spin precession on which the device relies.

1.6 Magnetoelectric Chromia

To realize the device described by Semenov et. al55_{. requires the selection of a ferromagnetic}

dielectric (FMD) that will induce a spin precession in the graphene channel. A

magnetoelec-tric (ME) is an excellent choice for the FMD as it would allow for an elecmagnetoelec-trically switchable

exchange interaction thus providing control over the spin-FET. Chromium Oxide is a very

good choice for this application since it exhibits a significant room temperature ME effect.

1.6.1 Magnetoelectrics

Magnetoelectric materials have a coupling between the the electric and magnetic ordering

in the material. This can be identified by the relevant components of the Landau free energy

expansions:

F(E~,H~) =F0−PisEi−MisHi−

1

2ε0εi jEiEj − 1

2µ0µi jHiHj −αi jEiHj (1.39)

whereF0is the initial free energy,Pis andM s

i are the spontaneous polarization and

mag-netization,E andH are the electric and magnetic field respectively, andεandµare the

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1.6. MAGNETOELECTRIC CHROMIA CHAPTER 1. INTRODUCTION

are not uniform in all directions. Theαis the linear magnetoelectric coefficient coupling

the electric and magnetic fields.

The magnetization of a ME material is composed of the static magnetizationMs i, the

response to magnetic fieldHiand the contribution fromαi jEjHj. Theαi j term couples the

magnetization to the electric fieldEi. The “ij” subscripts in theαterm are included because

the value of alpha changes depending on the symmetry of the material, i.e. αx x 6=αz z.

Taking the derivative _dd_HF

i gives this magnetization:

Mi E~,H~

=−dF

dHi

=M_iS+µoµi jHi+αijEi (1.40)

For ME materials, the electric polarization is dependent on the external magnetic field.

By taking the derivative of the Landau free energy expansion shown in Equation 1.39, _dd_EF i

gives the polarizationPi(E~,H~)which is proportional toαi,jHj.

Only materials with the proper symmetry have anα₆=0. The material must have Polarity

(P)_·Time (T) symmetry but not P or T symmetry individually. This is shown in Figure 1.12.

Conceptually, time reversal inverts the magnetization, and spacial inversion inverts the

electric polarization. In a material with ME symmetry, the original order is restored only

when both time and space inversion are applied.

Intuitively this effect can be understood by considering Cr₂O₃, a magnetoelectric,

anti-ferromagnetic and piezoelectric material. The electric field induces a strain in the material

that pushes some chromium atoms out of their sub-lattice by a piezoelectric effect. These

Cr atoms have their spin flipped by an exchange interaction with chromium atoms in the

other sublattice that have an opposing magnetization. The two sub-lattices have opposite

Oxide-Graphene Interfaces for Graphene Spintronics.

ABSTRACT

DEDICATION

BIOGRAPHY

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

CHAPTER

1

INTRODUCTION

1.1

Graphene

1.2

Graphene Band Structure

1

2

a

a

3

4

5

6

1.3

Graphene Production

1.3.1

Graphene Production by Chemical Vapor Deposition

1.3.2

Graphene Production From Silicon Carbide

1.4

Spin Relaxation in Graphene

1.4.1

Elliott-Yafet Spin Relaxation

1.4.2

Dyakonov Perel Spin Relaxation

1.4.3

Spin Relaxation Experiments

1.5

Graphene Spintronics

1.5.1

Tunnel Junctions

1.5.2

Producing Tunnel Barriers on Graphene

1.5.3

Non Local Spin Valves

1.5.4

Hanle measurment

1.5.5

Datta Das Device and the Rashba Effect

1.5.6

Graphene Spin Field Effect Transistor

1.6

Magnetoelectric Chromia

1.6.1

Magnetoelectrics