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PURDUE UNIVERSITYGRADUATE SCHOOL
Thesis Acceptance
This is to certify that the thesis prepared
By
Entitled
Complies with University regulations and meets the standards of the Graduate School for originality
and quality
For the degree of
Final examining committee members
, Chair
Approved by Major Professor(s):
Approved by Head of Graduate Program:
Date of Graduate Program Head's Approval:
Ahmet Ali Yanik
Spin Dependent Electron Transport in Nanostructures
Doctor of Philosophy
Ronald Reifenberger Gerhard Klimeck
Supriyo Datta
Hisao Nakanishi
Yuli Lyanda-Geller
7-25-2007
Ronald Reifenberger
Supriyo Datta
Nicholas J. Giardano
Graduate School ETD Form 9(01/07)
SPIN DEPENDENT ELECTRON TRANSPORT IN NANOSTRUCTURES
A Dissertation
Submitted to the Faculty
of
Purdue University
by
Ahmet Ali Yanik
In Partial Fulfillment of the
Requirements for the Degree
of
Doctor of Philosophy
August 2007
Purdue University
West Lafayette, Indiana
ii
To my mother (annem) Vesile and my father (babam) Yasar.
iii
ACKNOWLEDGMENTS
First and foremost, I would like to thank my advisor Prof. Supriyo Datta for
introducing me to the exciting world of nanoelectronics and giving me the opportunity to
work with him. He has witnessed and shaped my development as a scientist in-training
and I am much grateful to him for all he has done for me. It is through his guidance, his
enthusiasm for science and talent for picking and assigning the right project that this
thesis has been possible.
During the last three years, I have been fortunate enough to have a brilliant
second mentor, from whom I have learnt a great deal. Among many other things, I would
like to thank Gerhard Klimeck for being always ready to help, for being ready to go in
details with me and get his hands dirty. Without his help completing this thesis work
would be impossible.
I would also like to thank Professor Yuli Lyanda-Geller and Professor Hisao
Nakanishi for serving on my committee. Even though I haven’t interacted academically
with, I especially would like to thank Professor Ronald Reifenberger for his friendly
guidance after my prelim defense and giving me the courage to finish this work.
I would like to thank all of my friends in NSF Network for Computational
Nanotechnology , Physics Department and School of Electrical Engineering for making
Purdue an exciting place to study and a fun place to live. I would like to thank my friends
Sayeef Salahuddin, Prabhakar Sivastava, Kirk Bevan, Titash Rakshit, Albert Liang,
Magnus Paulsson, Avik Gosh and many others whose names I forgot to list. I would like
to particularly thank Kirk Bevan, Eliza Ekins for reviewing my dissertation and helping
me with corrections.
I would like to give my special thanks to my girlfriend Eliza Ekins who has
touched my life in many different ways with her friendship, support and love. She has
iv
been by my side through the most difficult steps that it has taken to finish this
dissertation.
I am thankful to my brother Fatih Yanik for his never ending support, helpful
advice and encouragement throughout my Ph.D. More than a brother, he was a friend,
mentor and a colleague. I also would like to thank him for listening me when I need it
most.
The last but not least, I am deeply grateful to my mother and father for their love,
care and all the sacrifice they made for my education. Their passion for education ignited
my interest in science and academics. I am most thankful to them for giving me the
opportunity to pursue my goals and successfully finish this dissertation.
v
TABLE OF CONTENTS
Page
LIST OF FIGURES .......................................................................................................... vii LIST OF SYMBOLS ......................................................................................................... ix USEFUL CONVERSION FACTORS................................................................................ x ABSTRACT....................................................................................................................... xi CHAPTER 1. INTRODUCTION ....................................................................................... 1
1.1. Input Parameters to the NEGF model ...................................................................... 1 1.1.1. Channel Region.................................................................................................. 2 1.1.2. Contacts.............................................................................................................. 3
1.2. Dephasing Processes ................................................................................................ 6 1.3. The NEGF Formalism .............................................................................................. 7
1.3.1. Coherent Regime Solution Procedure.............................................................. 10 1.3.2. Incoherent Regime Solution Procedure ...........................................................12
1.4. Outline .................................................................................................................... 14 CHAPTER 2. MTJ DEVICE: COHERENT REGIME .................................................... 16
2.1. Preliminaries........................................................................................................... 16 2.2. Method.................................................................................................................... 21
2.2.1. Choice of Basis ................................................................................................ 21 2.2.2. Source Drain Self-Energy Matrices ................................................................. 24 2.2.3. Current and JMR Ratio .................................................................................... 26
2.3. Results .................................................................................................................... 27 2.4. Summary................................................................................................................. 30
CHAPTER 3. MTJ Device: Incoherent Transport............................................................ 31 3.1. Preliminaries........................................................................................................... 31 3.2. Spin Exchange Interaction and Scattering Tensors ................................................ 33 3.3. Solution of NEGF Equations.................................................................................. 35
3.3.1. Calculation Scheme for the Green’s Function and Scattering Self-Energies... 36 3.3.2. Current Equations: ........................................................................................... 39
3.4. Results .................................................................................................................... 40 3.5. Summary................................................................................................................. 46
LIST OF REFERENCES.................................................................................................. 48 APPENDICES
Appendix A. Spin Exchange Scattering Tensors........................................................... 53 Appendix B. Direct Calculation Scheme For In-scattering and Correlation Matrices.. 58 Appendix C. Source Code ............................................................................................. 60
vi
Page
VITA................................................................................................................................. 67
vii
LIST OF FIGURES
Figure Page Figure 1.1 A schematic illustration of inputs needed for NEGF calculations is shown.
Magnetization direction of the drain is defined relative to the source( )θ∆ ................ 2
Figure 1.2 Open boundary conditions due to the contacts are treated as self-energy matrices [ΣC] to the device Hamiltonian [H]. .............................................................. 4
Figure 1.3 Shifting and broadening of a discrete energy level after contacting with a contact are illustrated. .................................................................................................. 5
Figure 2.1 (a) A schematic illustration of the density of states of ferromagnetic Ni. (b) The exchange split band structure of bulk Ni in [110] direction is shown in the Brillouin zone for the (left) majority-spin electrons and (right) minority-spin electrons. The electronic structure of the Ni is characterized by the dispersive s-like bands and the more localized d-like bands while the tunneling current is mainly dependent on s-like itinerant electrons (heavily solid curve) [31,34]. ....................... 17
Figure 2.2 (a) Schematics of the MRAM devices. (b) Resistance of the tunneling spin-valve device (MTJ) is dependent on the relative magnetization state of the ferromagnetic contacts. The ferromagnetic layer with lower coercivity is called the soft layer. Its magnetization state is controlled by external magnetic fields [14]...... 18
Figure 2.3 Magnetoresistance of a spin-valve device is defined as the resistance difference ( R∆ ) between the parallel and anti-parallel magnetization states............ 19
Figure 2.4 Energy band diagram for the model MTJ devices considered in this work. Right contact assumed to be the soft ferromagnet whose magnetization direction can be manipulated with an external magnetic field......................................................... 22
Figure 2.5 Source (left) contact self-energy matrix can be partitioned into two independent spin-channels. Self energy matrix of this contact can be written in its magnetization spin-basis as a diagonal matrix........................................................... 24
Figure 2.6 Drain (right) contact self-energy matrix can be partitioned into two independent spin-channels. Self energy matrix of this contact can be written in its magnetization spin-basis as a diagonal matrix........................................................... 26
Figure 2.7 Thickness dependence of theJMR ratios for different barrier heights are shown in comparison with experiments [35-44]. The TMR ratios given in experimental measurements are converted to JMR ratios when it’s necessary. Slonczewski’s asymptotic limit result is also shown (flat lines)................................ 28
viii
Figure Page Figure 2.8 An energy resolved analysis of ( )zJMR E (left-axis) and normalized ( )zEω
distributions (right-axis) are presented for a device with a tunneling barrier height of 1.6eV. ......................................................................................................................... 29
Figure 3.1 (a) A schematic illustration of the vertical MTJ device with δ-doped magnetic impurities is given. (b) Reduction in MR values are observed with increasing impurity content [5].................................................................................................... 32
Figure 3.2 Energy band diagram for the model MTJ devices considered in this work. Right contact assumed to be the soft ferromagnet whose magnetization direction can be manipulated with an external magnetic field......................................................... 36
Figure 3.3 Flowchart for the iterative calculation of Green’s function G and scattering self-energy ΣS. ............................................................................................................ 38
Figure 3.4 For MTJ devices with impurity layers, variation of JMR ratios for varying barrier thicknesses and interaction strengths (0.6-0.3-0 eV) are shown. Normalized JMR values are proved to be thickness independent as displayed in the inset. ......... 40
Figure 3.5 For MTJ devices with impurity layers a detailed energy resolved analysis is shown. Nomalized ( )zEω distributions are unaffected by exchange interactions
reflecting the inelastic nature of the scattering processes. ......................................... 41 Figure 3.6 JMR values for different barrier heights are shown. ....................................... 42 Figure 3.7 Using a barrier height dependent constant ( )barrc U , one can shown that JMR
values for different barrier heights scale to a universal curve.................................... 43 Figure 3.8 Experimental data taken at 77K/300K is compared with theoretical analysis in
the presence of palladium magnetic impurities with increasing impurity concentrations. ........................................................................................................... 44
Figure 3.9 Experimental data taken at 77K/00K is compared with theoretical analysis in the presence of nickel magnetic impurities with increasing impurity concentrations..................................................................................................................................... 45
Figure 3.10 Experimental data taken at 77K/300K is compared with theoretical analysis in the presence of cobalt magnetic impurities with increasing impurity concentrations. ........................................................................................................... 46
ix
LIST OF SYMBOLS
Constant Units
h Planck’s Constant 6.626 x 10-34 J s
ℏ / 2h π 1.055 x 10-34 J s
, q e Charge of electron 1.602 x 10-19 C
Bµ Bohr Magneton: magnetic moment
of electron
24 29.27400949 10 A m−×
(SI)
219.27400949 10 emu−×
(CGS)
oµ Permeability of free space 7 24 10 A/mπ −×
S�
Pauli Spin Matrix
0 1
1 02xS
=
ℏ
0
02y
iS
i
− =
ℏ
1 0
0 12zS
= −
ℏ
J s
x
USEFUL CONVERSION FACTORS
Conversion between J and eV 1 eV = qJ
Conversion between SI and CGS unit of
magnetic moment of electron
2 31 Am 10 emu=
Conversion between Tesla (T)
and Oersted (Oe)
1 A/m = 34 /10π Oe
1 A/m = 1 T/ oµ
1 T = 10000 Oe
Conversion between SI and CGS unit of magnetization -3 31 A/m = 10 emu/cm
Since 3
emu1 4 Oe
cmπ=
Conversion between SI and CSG unit of work 10-7 J = 1 erg
xi
ABSTRACT
Yanik, Ahmet A. Ph.D., Purdue University, August, 2007. Spin Dependent Electron Transport in Nanostructures. Major Professors: Supriyo Datta and Ronald Reifenberger.
Spin-electronic devices, exploiting the spin degree of freedom of the current
carrying particles, are currently a topic of great interest. In parallel with experimental
developments, theoretical studies in this field have been mainly focused on the coherent
transport regime characteristics of these devices. However, spin dephasing processes are
still a fundamental concern [1-6].
The Landauer transmission formalism has been the widely used method in the
coherent transport regime [7]. Recently this formalism has been adapted to incorporate
spin scattering processes by introducing random disorder directly into the conducting
medium and subsequently solving the disordered transport problem over a large ensemble
of disorder distributions [8-10]. Although proposed to be a way of incorporating spin
scattering processes, what this approach basically offers is an averaged way of adding
random coherent scatterings (similar to the scatterings from boundaries) into the transport
problem. Certainly such a treatment of spin-dephasing processes misses the incoherent
and inelastic nature of the scattering processes. As a result, a rigorous way of treating the
spin scattering processes is still needed [10-12].
The objective of this thesis is to present a quantum transport model based on non-
equilibrium Green’s function (NEGF) formalism providing a unified approach to
incorporate spin scattering processes using generalized interaction Hamiltonians. Here,
the NEGF formalism is presented for both coherent and incoherent transport regimes
without going into derivational details. Subsequently, spin scattering operators are
derived for the specific case of electron-impurity exchange interactions and the model is
xii
applied to clarify the experimental measurements [5]. Device characteristics of magnetic
tunnel junctions (MTJs) with embedded magnetic impurity layers are studied as a
function of tunnel junction thicknesses and barrier heights for varying impurity
concentrations in comparison with experimental data. For MTJs with embedded magnetic
impurity layers, this model is able to capture and explain three distinctive experimental
features reported in the literature regarding the dependence of the junction magneto-
resistances (JMRs) on (1) barrier thickness, (2) barrier heights and (3) the concentrations
of magnetic impurities [5,6,29,46]. Although in this dissertation our treatment was
restricted to the electron-impurity spin exchange interactions, the NEGF model presented
here allows one to incorporate other spin exchange scattering processes involving nuclear
hyperfine, Bir-Aranov-Pikus (electron-hole) and electron-magnon interactions. This
model is general and can be used to analyze and design a variety of spintronic devices
beyond the large cross-section multilayer devices explored in this work.
1
CHAPTER 1. INTRODUCTION
This thesis presents a rigorous formalism to study spin quantum dependent
transport in nanostructures. Within the last decade, spin-electronic devices have
stimulated considerable interest due to their potential in information storage, sensor
technology and magneto-optics applications [13-18]. In particular, interest in
magnetoresistive devices is remarkable, as they constitute the central part of the available
spin-electronic commercial products [19]. The diversity of the physical phenomena
governing these devices also makes them interesting from the fundamental physics point
of view. Rigorous physics based models incorporating different physical mechanisms are
needed.
In this chapter, NEGF formalism [21-23] is introduced as a quantum transport
model able to tackle various challenges presented by these structures. Sections 1.1.1 and
1.1.2 summarize the inputs needed for NEGF calculations for non-degenerate spin
systems in the coherent transport regime. The section 1.2 explains how to integrate non-
coherent processes into the transport equations. A general solution scheme of the NEGF
equations without going into the details is presented in section 1.3.1 and section 1.3.2 for
the coherent and incoherent regimes, respectively. This is followed by an outline of this
dissertation.
1.1. Input Parameters to the NEGF model
The open system (Figure 1.1), consisting of the device region (channel) and
contacts stretching out to infinity, can be truncated into smaller subsets by treating
contacts as self energies to the device region. This is a common treatment in many-body
physics to incorporate non-coherent interactions within the channel region. Nevertheless,
2
a single-electron approximation is used in this dissertation. lnputs needed for the NEGF
calculations are divided into channel region [ ]H U+ , the source/drain contact self-
energies ,L R Σ and the scattering self-energy [ ]SΣ . This is illustrated in Figure 1.1 with
some of the nomenclature.
1.1.1. Channel Region
Channel properties are defined by the device Hamiltonian matrix [ ]H including the
applied bias potential. The effective potential matrix [ ]U is reserved for charging effects
due to the change of the number of electrons in the channel region. In the following
chapters, these charging effects are neglected due to the low bias and tunneling regime
operation of the devices considered here. A detailed explanation of the NEGF formalism
with charging effects can be found elsewhere [20].
Figure 1.1 A schematic illustration of inputs needed for NEGF calculations is shown.
Magnetization direction of the drain is defined relative to the source( )θ∆ .
Spin Array
Lµ
[ ]LΣz
[ ]SΣ
Channel
[ ]H U+
SourceDVI I
Drain
Rµ
[ ]RΣ z z
RM
∆θ
LM
qωℏ
Spin Array
Lµ
[ ]LΣz
[ ]SΣ
Channel
[ ]H U+
SourceDVI I
Drain
Rµ
[ ]RΣ z z
RM
∆θ
LM
qωℏ
3
1.1.2. Contacts
In an isolated system, eigenvalues of the device Hamiltonian define the discrete
energy levels and the localized eigenfunctions. Once contacted, the continuous
wavefunctions of the open system overlap with the localized device eigenfunctions
giving rise to the broadening and the shifting of the otherwise discrete energy levels. In
NEGF formalism, self-energy matrices are employed to incorporate the effects of the
open contacts, enabling one to work in a small subset of the complete system. In this
fashion, self-energy matrices ,L R Σ could be viewed as modifications to the channel
Hamiltonian [ ]H causing a finite state lifetime for the electrons in the channel region.
However, self energies are more than simple modifications. Unlike Hamiltonian matrices,
self-energy matrices are energy dependent and non-hermitian.
Self-energy matrices are obtained by using the definition of the retarded Green’s
function of the complete system. This is illustrated in the model system consisting of an
isolated device and a single open contact in Fig. 2.2. In an orthonormal basis set, Greens
function of the complete system is defined as:
( ) ( ) 1
0C CG E lim E i I H
ηη
+
−
→= + − (1.1)
where [ ]CH is the Hamiltonian of the “complete system” and [ ]I is an identity matrix
of same size. After a simple re-indexing, system Hamiltonian can be written in separate
blocks as in:
†L
C
HH
H
ττ
=
(1.2)
where suffix L refer to the left contact. Substituting Eq. 2.2 into Eq. 2.1:
[ ] ( )( )
1†L
C
E i I HG
E i I H
η ττ η
− + − −
= − + − (1.3)
and using the following identity:
1A B a b
C D c d
−
=
(1.4)
4
with 11d D CA B
−− = − , it can be shown that the only contributing part of the complete
Green’s function matrix to the channel region of the device is:
( ) ( ) [ ]1 1
0L LG E lim E i I H EI H
ηη
+
− −
→= + − − Σ ≈ − − Σ (1.5)
where;
( ) ( )L LE g Eτ τ +Σ = (1.6a)
( ) ( ) -1-L Lg E E i I Hη= + (1.6b)
and ( )Lg E is the Green’s function of the contact.
Figure 1.2 Open boundary conditions due to the contacts are treated as self-energy matrices [ΣC] to the device Hamiltonian [H].
The [ ]τ coupling matrix is non-zero only for a small number of ( ),m n indexed points
coupling the contact to the channel. The ( ),Lg m n elements of the contact Green’s
function are relevant to the contact self energy ( )L EΣ . As a result, contact self energy
matrices have the same size of the isolated device Hamiltonian [ ]H can be calculated
through recursive techniques instead of the inverting the total contact Hamiltonian
extending to infinity. This will be detailed in the following chapters for ferromagnetic
contacts with specific examples.
Physically speaking, self-energy matrices stand for the open boundary related
effects such as shifting and broadening of the otherwise discrete energy levels in an
[ ]LH
Channel
[ ]H
Left Contact
Channel + Contact
[ ]LH + Σ[ ]τ[ ]LH
Channel
[ ]H
Left Contact
Channel + Contact
[ ]LH + Σ[ ]τ
5
isolated channel. This could be shown by separating the self-energy matrices into the
real and the imaginary parts:
† †
2 2L L L L
LH H Σ + Σ Σ − Σ+ Σ = + +
(1.7)
Shifting ( )† 2L LΣ + Σ is the real part of the self-energy matrices added to the isolated
device Hamiltonian [ ]H as a correction:
†
2L L
LH H Σ + Σ= +
ɶ (1.8)
Figure 1.3 Shifting and broadening of a discrete energy level after contacting with a contact are illustrated.
Broadening is the anti-hermitian part of the contact self-energy matrices,
2L LH H i+ Σ = − Γɶ (1.9)
where:
( ) ( ) ( )†-L L LE i E E Γ = Σ Σ (1.10)
Broadening matrix ( )L EΓ is proportional with the strength of the coupling and can
also be interpreted as the inverse residence time of the escaping electrons from the
device. Accordingly, multiple contacts can be treated using self-energy matrices. For a
system consisting of source and drain contacts as shown in Figure 1.1, Green’s function
is given by;
( ) ( ) 10D D L RG E E i I H
−+ = + − − Σ − Σ (1.11)
nε
( )nδ E-ε
nε
ε∆
( )RR
δ E-ε∑
nε
( )nδ E-ε
nε
ε∆
( )RR
δ E-ε∑ ( )RR
δ E-ε∑
6
with broadening matrices:
( ) ( ) ( )( )†, , ,-L R L R L RE i E EΓ = Σ Σ (1.12)
In/out-scattering matrices ,,
in outL R Σ are defined as a quantum mechanical
description of the rate at which electrons are scattered in/out of a state. This is similar to
the semi-classical treatment; but generalized to incorporate phase-correlations among
different quantum states. The broadening matrix multiplied by the corresponding Fermi
function ( )0 ,L Rf E µ− (analogous to the occupancy of the contact state) is the in-
scatttering matrix:
( ) ( ) ( ), 0 , ,inL R L R L RE f E EµΣ = − Γ (1.13)
while the out-scattering matrix is given by the broadening multiplied by ( )0 ,1 L Rf E µ− −
(analogous to the vacancy of the contact state):
( ) ( )( ) ( ), 0 , ,1-outL R L R L RE f E EµΣ = − Γ (1.14)
Here, the scattering processes from/to contacts to/from the channel are assumed to be
elastic and the Fermi functions in the contacts are given by:
( ) ( )0 ,
,
1
1 expL R
L R B
f EE k T
µµ
− = + −
(1.15)
where ( ), 2L R f BE eVµ = ± is the chemical energy for the left/right contact at a bias BV .
1.2. Dephasing Processes
In NEGF formalism, dissipative/phase-breaking processes present in the channel
region are treated with a boundary condition reflecting the nature of the scattering
processes (scattering self-energy):
( ) [ ]L R SG E EI H= − − Σ − Σ − Σ (1.16)
where the subscript ‘s ’ refers to the scattering. But unlike regular contacts, the physical
effect of the scattering processes in the device region is strongly dependent on state of the
7
initial wavefunctions and the vacancy of the final state itself. Accordingly, there is no
straightforward way of calculating the self-energy matrix [ ]SΣ for the scattering
processes at the onset of the NEGF calculation scheme. The scattering contacts [ ]SΣ can
only be incorporated into the transport calculations through the self-consistent solution of
the NEGF equations as outlined in the sections 1.3.2. Similarly, there is no simple
relationship between the broadening matrix [ ]SΓ and the in/out-scattering matrices
,in outS Σ due to the absence of a ( )sf E Fermi function for the scattering processes.
Nevertheless the broadening matrix is still the sum of in-scattering and out-scattering
matrices as such for regular contacts (following Eq 1.13 and 1.14):
( ) ( ) ( )Sin outS SE E E Γ = Σ + Σ (1.17)
Once the broadening matrix, the imaginary part of the self-energy matrix, is determined
the real part of the self-energy can be obtained via Hilbert transform as for any causal
function (Eq. 1.18):
( ) ( ) ( )S
Re Im
' '1
2 ' 2S SdE E E
E iE EπΓ Γ
Σ = −−∫
������� �����
(1.18)
This scheme requires integrations over the all energy grid.
1.3. The NEGF Formalism
At zero bias, the contact Fermi levelsLµ and Rµ are equal, and the device is in
equilibrium with the contacts. Applying a positive bias voltage (B L RV µ µ= − ) will lower
the energy levels in the right contact with respect to the left. Contacts seeking to bring the
channel into equilibrium with their Fermi energies will create a non-equilibrium electron
distribution (current) in the channel region. The NEGF formalism can be used to obtain
the electron density n , the charging potential [ ]U and the current. In this part, we will
discuss the NEGF formalism in detail.
8
Given the input parameters listed in the Subsections 1.1 and 1.2, NEGF formalism
describes how to calculate the spectral function (whose diagonal elements are the local
density of states):
( ) ( ) ( )†A E i G E G E = − (1.19)
In semi-classical picture, one can describe the electron distributions by specifying
a distribution function ( )f k�
which tells us the number of electrons occupying a particular
state k�
. In quantum description of electron transport, the concept of ( )f k�
distribution
functions is extended into a correlation matrix ( ), '; , 'nG k k t t
� � in order to include the
additional information regarding the phase correlations. In general, a two-time correlation
function ( ), '; , 'nG k k t t (analogous to the electron density) is defined:
( ) ( ) ( )†, '; , ' 'nk kG k k t t a t a t= (1.20)
where ka and †ka are the creation and annihilation operators. Note that the density matrix
can still be obtained from the correlation function by setting 't t= :
( ) ( )'
, '; , '; , 'n
t tk k t G k k t tρ
= = (1.21)
Subsequently, the steady-state solutions can be obtained by eliminating one of the time
variables as the correlation function only depends on the time difference ( 't t− ). The
diagonal elements of the correlation functions is the number of electron occupying a
particular state. Using Fourier transformation relationship between energy and time
difference coordinate ( 't t− ), one can show that:
( ) ( )', '
1( ) , '; , ' , ;
2n n
k k t tf k G k k t t G k k E dE
π= = = ≡ ∫� � � �
� � � � � (1.22)
where ( 't tτ = − ). The validity of this is relationship is not restricted to the k-space
representation. For example, in the real space representation electron density can be
defined as:
( ) ( )1, ;
2nn r G r r E dE
π= ∫ (1.23)
9
Similar to the electron correlation function, one can also define a hole correlation
function ( ), '; , 'pG k k t t (analogous to the hole density):
( ) ( ) ( )†, '; , ' 'pk kG k k t t a t a t= (1.24)
describing the hole distribution and phase correlations among holes. In difference with
the conventional definition, this function refers to the holes in the conduction band, not
the ones in the valence band. Accordingly, the spectral function is defined as:
( ) ( ) ( )n pA E G E G E= + (1.25)
In the NEGF formalism, the electron/hole correlation function is related to the
Green’s function of the device through,
( ) ( ) ( ) ( ), , �n p in outG E G E E G E = Σ (1.26)
where:
( ) ( ) ( ) ( ), , , ,in out in out in out in outL R SE E E EΣ = Σ + Σ + Σ (1.27)
representing the in/out-scattering from a state due to the contacts and the scattering
processes (Eqs. 1.13 and 1.14).
Alternatively, the in/out-scattering matrix is related to the electron/hole
correlation function through [20]:
( ) ( ) ( ) ( ), ;,
, '; , '; , '; i j i j k l k l
k l
in n nS r r E D r r G r r E dσ σ σ σ σ σ σ σ
σ σω ω ω Σ = − ∑∫ ℏ ℏ ℏ (1.28a)
( ) ( ) ( ) ( ), ;,
, '; , '; , '; i j i j k l k l
k l
out p pS r r E D r r G r r E dσ σ σ σ σ σ σ σ
σ σω ω ω Σ = + ∑∫ ℏ ℏ ℏ (1.28b)
where the spin indices ( ,k lσ σ ) and ( ,i jσ σ ) refer to the (2x2) block diagonal elements of
the on-site electron/hole correlation functions and in/out-scattering matrices, respectively.
Here, the n pD D are fourth-order scattering tensors, describing the spatial
correlation and the energy spectrum of the underlying microscopic spin-dephasing
mechanisms. These tensors are determined by the detailed description of the scattering
mechanism using perturbative approaches and can get increasingly complicated
expressions depending on the degree of the approximation. For exchange interaction spin
scattering processes, a detailed derivation of these spin scattering tensors is given in
10
appendix A. Spin scattering processes as well as spin-conserving ones are discussed in
this dissertation.
One can calculate the current at any channel in terms of the matrices listed above.
For the numerical implementation presented in chapters 2 and 3, we do not compute the
charging potential [ ]U self-consistently. Charging in the tunnel barriers is neglected and
assumed not to influence the electrostatic potential. This allows one to focus on the
dephasing due to the spin-flip interactions.
Once the retarded Green’s function is calculated through the self consistent
solution of the above equations, current at any terminal “i” can be calculated via
( ) ( ) ( ) ( )( )in ni i i
qI dE tr E A E tr E G E
h
∞
−∞
= Σ − Γ ∫ (1.29)
Further details of the self consistent calculation scheme are discussed in chapter 3. One
useful quantity to keep track of the consistency of the solution is the current density:
( ) ( )( ){ }, 1 1, nj j j j j
qJ dE Re tr H E i G E
h
∞
+ +−∞
= ∫ (1.30)
Here the index ‘j’ refers the grid point (in a real space representation) where the current
density is calculated. Calculating current density has “no numerical advantage”.
However it enables one to follow the convergence of the self-consistent solution scheme,
since the conservation of charge requires current density to be conserved through out the
device.
In the following subsections 1.3.1 and 1.3.2, the similarities and the differences
for coherent and the incoherent transport regimes are illustrated. The solution procedures
are also briefly discussed.
1.3.1. Coherent Regime Solution Procedure
Physics of the system at hand is fully contained in the input matrices defined
above (subsections 1.1.1 and 1.1.2). Given these matrices, the NEGF formalism provides
11
a straightforward procedure calculating the transport properties of the system in the
absence of the dephasing processes (, 0n pD = ):
0SΣ = (1.31a)
, 0in outSΣ = (1.31b)
Accordingly, after substituting Eqs. 1.13-1.14 and 1.25-1.27 in Eq. 1.29:
� ( ) ( )L L L L R L L L R R
ini nA G
qI tr f G G tr G f f G
h+ +
Σ
= Γ Γ + Γ − Γ Γ + Γ
ɶ������� ���������
(1.32)
current relation will simplify to the commonly used Landauer transmission formula in
the ballistic transport limit:
( ) ( )L R
qI T E f E f dE
h = − ∫ (1.33)
where the transmission function ( )T E is defined as:
( ) ( ) ( ) ( ) ( )L D R DT E tr E G E E G E+ = Γ Γ
(1.34)
Solution scheme for the coherent transport regime is relatively easy and it is
outlined step by step in the following.
1. For a given contact and device, first we need to choose an appropriate basis set
adequate to describe the device system. For the model systems (MTJ devices)
considered here real space basis is specified.
2. The next step is to write down a suitable Hamiltonian for the device in the chosen
basis set. In the following chapters, effective mass approximation is used to
obtain the device Hamiltonian within the real space basis set.
3. Similarly self-energy matrices are obtained for the left and right contacts. Details
of this calculation for a MTJ device are given in chapter 2.
4. Charging effects in principle can be solved in a self-consistent manner.
Nevertheless the charging potential [ ]U is neglected due to the pure tunneling
nature of quantum transport in these devices and the lack of any device areas
where charge could accumulate.
12
5. Using the matrices , ,, , , ,L R L RH U G Σ Γ and the Fermi levels Lµ and Rµ at a bias
( B L RV µ µ= − ), one can calculate the current using NEGF equations summarized
above.
1.3.2. Incoherent Regime Solution Procedure
In Landauer-Buttiker formalism [24], electron transport in the active device
region is essentially assumed to be coherent; while the incoherent processes within the
contacts are assumed to be maintaining the local equilibrium state. This viewpoint is
advantageous due its conceptual simplicity and relative ease in incorporating incoherent
phase-breaking processes by the conceptual voltage probes which extract the electrons
from the device and re-inject them after phase randomization. However, phase breaking
and dissipative processes involve subtle issues beyond the extent of these
phenomenological models. As a result, it is desirable to express the electron transport
with microscopic theories which enabe one to properly relate the incoming electron
wave-functions with the scattering potential.
The concept of self-energy has been extensively used in many-body physics to
describe non-coherent electron-electron and electron-phonon interactions. We could do
the same in principle and use a self-energy function SΣ to describe the effect of non-
coherent interactions of the localized magnetic impurities and device with its
surroundings.
In general, these scattering tensors can be obtained starting from a spin scattering
Hamiltonian of the form:
( ) ( )int -j
j jRH r J r R Sσ= ⋅∑
��� � � (1.35)
where jr R��
are the spatial coordinates and jSσ��
are the spin operators for the channel
electron / (j-th) magnetic-impurity. In this dissertation, we assume a delta interaction
model (see Appendix A) and show that n pD D can be written as:
13
( ) ( ) ( ), , ,, '; , '; , '; n p n p n p
sf nsfD r r D r r D r rω ω ω = + ℏ ℏ ℏ (1.36)
The first part describing the process of spin-flip transitions (subscript sf) due to
the spin-exchange scatterings in the channel region is given by:
( ) ( ) ( )
( )
n,p 2
sf
,
,
D , '; '
,
,
0 0 0
q
qI
k l
i j
u d q
d u
r r r r J N
F
F
ωω δ ω
σ σ
σ σ
δ ω ω
δ
↑↑ ↓↓ ↑↓ ↓↑
↑↑
↓↓
↑↓
↓↑
= −
→
↓
∑ℏ
∓
( ) 0 0 0
0 0 0 0
0 0 0 0
qω ω
±
(1.37)
while the second part corresponding to the spin-conserving exchange scatterings
(subscript nsf for "no spin-flip") in the channel region is defined as:
( ) ( ) ( ) ( )n,p 2
nsf
1D , '; '
4
,
,
1 0 0 0 0 1 0 0
0 0 1 0
0
q
I q
k l
i j
r r r r J Nω
ω δ δ ω ω
σ σ
σ σ↑↑ ↓↓ ↑↓ ↓↑
↑↑
↓↓
↑↓
↓↑
= −
→
↓
−
∑ℏ
0 0 1
−
(1.38)
Here ( )I qN ω is the number of the magnetic impurities with qωℏ energy difference
between spin states and u dF F represents the fractions of the spin-up/spin-down
impurities for an uncorrelated ensemble ( 1u dF F+ = ). Spin-flip transitions of electrons
due to the exchange scattering processes can be elastic/inelastic depending on the
degeneracy ( 0 0ω ω= ≠ℏ ℏ ) of the impurity spin states. A similar expression for both of
the contributing parts has been previously shown by Appelbaum [2,3] using a tunneling
Hamiltonian treatment.
14
Solution scheme for the incoherent transport regime is summarized below without
going into detailed description of the self-consistent schemes:
1. Selection of basis set (same with coherent regime calculations).
2. Device Hamiltonian (same with coherent regime calculations).
3. Determination of contact self-energies (same with coherent regime
calculations). The matrices listed under subsections 1.1.1 and 1.1.2 are
specified and fixed at the outset of any calculations.
4. Charging effects are neglected.
5. Self-consistent solution of , ,, , ,n p in outS S SG Σ Γ Σ (details are discussed in
chapter 3). While the[ ]U , in outS S Σ Σ matrices under the subsection 1.2
depend on the correlation and spectral functions requiring an iterative self-
consisted solution of the NEGF Equations [Eqs. 1.16-1.19 and 1.25-1.28].
Details of the self-consistent scheme are discussed in the following chapters
whenever it is necessary.
6. Current is calculated using Eq. 1.29.
1.4. Outline
This preliminary report is organized as follows. In chapter 2, we show how to use
NEGF formalism summarized above in analysis of spin dependent electron transport in
MTJs. Details of the matrices defined above are obtained by using a real space basis set
for an impurity free MTJ device. Definitions of device characteristics are also specified in
this part. Formalism is applied and compared with experimental measurements for
impurity free devices (ballistic regime) and the device parameters are benchmarked. In
chapter 3, the spin-dependent electron transport formalism will be extended to
incorporate exchange interaction spin-scattering processes. Derivation of scattering
matrices particular to electron-impurity spin exchange mechanism will be supplied in
Appendix A while the application of the formalism is illustrated in chapter 3 for MTJ
15
devices with magnetic impurity layers. Theoretical estimates and experimental
measurements are also compared in this part.
16
CHAPTER 2. MTJ DEVICE: COHERENT REGIME
In this part, a theoretical analysis of MTJ devices in the absence of magnetic
impurity layers is presented and compared with experimental data for varying tunneling
barrier heights and thicknesses. Calculations are done considering the spin channels
independent in the absence of any spin relaxation mechanism and spin-orbit coupling
[25-32].
2.1. Preliminaries
In 3d-transition ferromagnets, the narrow d-bands are exchange split between spin
up and down electrons leading to an unequal density of states (DOS) at Fermi level [31].
This exchange splitting ( excE∆ ) is the origin of the magnetization (M ) of the material
and it can be associated with
excE mJ Mα∆ ≈ ≈ (2.1)
whereJ is the exchange integral as calculated by Brooks [33], M is the magnetization,
and the constant α is a product of the exchange integral with the cell volume. The
exchange interaction shifts the spin-up and spin-down states with respect to each other,
leading to a preferential occupation of the spin-up band and an overall reduction of the
total energy.
In the absence of spin-orbit coupling and spin-scattering interactions, spin states
of the electrons are conserved and the exchange split spin channels yield independent but
parallel currents. In the tunneling transport regime, conductances of the spin channels
depend on the spin channel DOSs in the source and drain contacts [26]. This leads to an
imbalanced electric current carried by the tunneling spin channels [26]. This
17
phenomenon, also called spin-dependent tunneling (SPT), was first observed in 1970 by
Tedrow and Meservey [25] in their tunneling current experiments between a
superconducting aluminum layer and a ferromagnetic nickel film. However, experimental
measurements have revealed inconsistency with the predicted DOS for bulk 3d
ferromagnetic metals. This observation was later explained by Stearns by distinguishing
the localized and itinerant electrons of ferromagnetic metals [31]. She proposed that the
tunneling conductance is not only dependent on the number of electrons at the Fermi
energy but also on the tunneling probability of the various electronic states.
Figure 2.1 (a) A schematic illustration of the density of states of ferromagnetic Ni. (b) The exchange split band structure of bulk Ni in [110] direction is shown in the Brillouin
zone for the (left) majority-spin electrons and (right) minority-spin electrons. The electronic structure of the Ni is characterized by the dispersive s-like bands and the more localized d-like bands while the tunneling current is mainly dependent on s-like itinerant
electrons (heavily solid curve) [31,34].
The localized d-band electrons with larger effective masses have little contribution to the
tunneling current due to the faster decaying rates in the tunnel barriers. On the other
hand, the s-like electrons are highly mobile and have slower decaying rates [31].
Accordingly, the spin dependent tunneling current is expected to show only the DOS
features of the itinerant s-band like electrons. Following Stearn’s argument, at the thick
barrier limit the polarization (FMP ) of the ferromagnets can be approximated to [31];
18
( ) ( )( ) ( )
maj min maj minF F F F
FM maj min maj minF F F F
E E k kP
E E k k
ρ ρρ ρ
− −= =+ +
(2.2)
where ( ) ( )maj minF FE Eρ ρ is the effective DOS for the majority/minority electrons and
maj mink k is the Fermi wavevector for the s-like bands (shown in Figure 2.1 with heavily
solid curve).
Julliere replaced the superconducting film with another ferromagnet using
exchange-split states of ferromagnets, thereby making a magnetic tunnel junction (MTJ)
device consisting of a single tunneling barrier sandwiched between two ferromagnets
[26]. Ferromagnets with different magnetic coercivities are utilized to independently
manipulate the contact magnetization by external magnetic fields (Figure 2.2).
Figure 2.2 (a) Schematics of the MRAM devices. (b) Resistance of the tunneling spin-valve device (MTJ) is dependent on the relative magnetization state of the ferromagnetic
contacts. The ferromagnetic layer with lower coercivity is called the soft layer. Its magnetization state is controlled by external magnetic fields [14].
The tunneling magnetoresistance ratio (TMR G G= ∆ ) for this system is defined as the
difference in conductance between parallel and antiparallel magnetizations of
ferromagnetic contacts, normalized by the anti-parallel magnetization state conductance:
P AP P AP
AP P
G G R RTMR
G R
− −= = (2.3)
19
Another commonly used device parameter is the junction magnetoresistance ratio
JMR R R= ∆ formulated in terms of the change of resistance;
P AP P AP
AP P
R R G GJMR
R G
− −= = (2.4)
Figure 2.3 Magnetoresistance of a spin-valve device is defined as the resistance difference ( R∆ ) between the parallel and anti-parallel magnetization states.
Assuming spin conservation in the tunneling barrier and equal electron tunneling
probabilities, Julliere formulated the first and the simplest expressions for the tunneling
magnetoresistance ratios observed in the FM/I/FM tunneling systems. Although
simplistic, Julliere’s formula has proven to be qualitatively consistent with the
preliminary experiments. For parallel configuration of the ferromagnetic contacts,
majority/minority electrons in the source electrode tunnel through the barrier into the
majority/minority carrier bands in the drain electrode leading to a conductance relation:
maj maj min minP L R L RG G G ρ ρ ρ ρ↑ ↓= + ∝ + (2.5)
In anti-parallel alignment of the ferromagnetic contacts, conductance is given by:
maj min min majAP L R L RG G G ρ ρ ρ ρ↑ ↓= + ∝ + (2.6)
Parallel Contacts Anti-parallel Contacts
Soft Layer
Hard Layer
Tunneling Oxide
F
F
Soft Layer
Hard Layer
Tunneling Oxide
F
F
∆
FE
minoritycE
majoritycE
FEminoritycE
majoritycE
∆∆
FE
minoritycE
majoritycE
FEminoritycE
majoritycE
∆∆
FE
minoritycE
majoritycE
FEminoritycE
majoritycE
∆
20
due to the tunneling of the majority/minority electrons in the source electrode into the
minority/ majority carrier bands in the drain electrode. As a result, TMR/JMR values are
strongly dependent on the DOS of the majority/minority carriers in the ferromagnetic
contacts:
2
1L R
L R
P PTMR
P P=
− (2.7a)
2
1L R
L R
P PJMR
P P=
+ (2.7b)
TMR/JMR expressions are often considered to be optimistic/pessimistic depending on the
“ -/+” sign in the denominator.
Julliere’s work has stimulated further research, especially after the first
observation of spin-dependent tunneling effect at room temperature with large JMR ratios
in CoFe/Al2O3/Co MTJ devices by Moodera et al. (Figure 2.2a) [14]. Following their
work, spin-dependent tunneling effect has been successfully shown in a number of
different material systems [17,35-44]. The first accurate theoretical predictions was made
by Slonczewski using a thick barrier limit approximation [27]. By assuming a square
barrier potential and two parabolic bands shifted with respect to each other in electrodes,
he formulated a modified version of the effective spin polarization of tunneling electrons
2
2
maj min maj minF F F F
FM maj min maj minF F F F
k k k kP
k k k k
κκ
− −= =+ +
(2.8)
where:
( )( )22 barr Fm U Eκ = −ℏ (2.9)
Although relatively successful and widely used, predictions of this model tend to
underestimate the measured JMR values due to the thick barrier limit approximation. In
the following sections, Stearns description of ferromagnets and Slonczewski’s model for
MTJ devices are used as a starting point, while the calculations are done using realistic
Hamiltonians through the NEGF formalism. Slonczewki’s results are also presented in
comparison with NEGF calculations as such for the experimental measurements.
The following is a summary of the most significant experimental observations
concerning the nature of the JMR ratios in MTJ devices:
21
1. Conductance depends on the relative magnetization directions of the
ferromagnetic contacts [25-27,31].
2. JMR ratios are sensitive to the details of the barrier, interfaces and the contacts
[14,17,35-44].
3. JMR ratios generally fall as (d) the thickness of the barrier increases [46].
4. JMR ratios rise with increasing barrier height ( barrU ) [46].
5. JMR ratios fall with increasing temperature and bias [29,39].
6. JMR ratios show an earlier decay with increasing temperatures than the Curie
temperatures of the ferromagnetic contacts [47].
In this dissertation, our focus is low bias operation regime of the MTJ devices. In this
regime, observations “1-4” outlined above are relevant with device characteristics.
Temperature and bias dependent measurements involving phonon and magnon scattering
processes are left for future work.
2.2. Method
In our calculations, a real space model Hamiltonian is employed. For the
FM/I/FM model system single band effective mass theory is used following Stearn et al
[31]. It is assumed the effective masses of the tunneling electrons within the tunneling
region are the bare electron masses (*em m= ) as it is in the ferromagnetic contacts.
2.2.1. Choice of Basis
Assuming that throughout the tunneling process parallel momentum (k ) is
conserved [30], the device Hamiltonian can be separated into parallel k and
perpendicular (zk k⊥= ) components with respect to the growth direction (Figure 2.4). In
this dissertation, nomenclature is set as such; longitudinal represents the direction of
current flow while transverse represents the direction perpendicular to current flow. The
22
overall Hamiltonian is then the sum of the longitudinal H and transverse part
zH ( zH H H= + ):
( )
( )
2 2
* 2
2 2 2
* 2 2
2
,2
L c
T
dH E U z
m d z
d dH U x y
m d x d y
= − +
= − + −
ℏ
ℏ (2.10)
For the MTJ devices with large cross section, the transverse confining potential ( ),U x y
can be neglected and the periodic boundary condition can be applied. This enables one to
express the tranverse eigenstates as plane waves:
( ) 1 ik r
kr e
Sϕ ⋅=
� �
��
(2.11)
with:
2 2
*&
2T k kk k
kH
mϕ ε ϕ ε= =
� �ℏ
(2.12)
where *m is the electron effective mass.
Figure 2.4 Energy band diagram for the model MTJ devices considered in this work. Right contact assumed to be the soft ferromagnet whose magnetization direction can be
manipulated with an external magnetic field.
[ ]RΣHamiltonian
a
∆
FE
minoritycE
majoritycE
FE
minoritycE
majoritycE
∆
θ∆
θ∆
0 N+11 N⋯
k
z1k 2k
barrU
[ ]H[ ]LΣ [ ]RΣHamiltonian
a
∆
FE
minoritycE
majoritycE
FE
minoritycE
majoritycE
∆
θ∆
θ∆
0 N+11 N⋯
k
z1k 2k
barrU
[ ]H[ ]LΣ
23
In real space representation for a discrete lattice whose points are located at
jz ja= , j being an integer ( 1,2,3,j N= … ) the Hamiltonian matrix [ ]zH can be
expressed as:
1
2
1
1 2 1
1
2
1
0 0
0 0
0 0
0 0N
N
N N
N
N
H
α ββ α
α ββ α
+
−+
−
−
=
⋯
⋮ ⋮ ⋮ ⋱ ⋮ ⋮ (2.13)
where jα is a the 2x2 on-site matrix:
,
,
2 0
0 2
c j n
j
c j n
E t U
E t Uα
↑
↓
↑ ↓
↑
↓
+ +=
+ +
(2.14)
and tIβ = − is a 2x2 site-coupling matrix with 2 * 22t m a= ℏ and 1 0
0 1I
=
. Due to
the absence of spin-orbit couplings in the barrier region, the on-site and coupling are
diagonal:
( ) ( ), , 0n n nα β α β+ +↑↓ ↑↓ ↓↑ ↓↑= = (2.15)
The overall conduction channel can be decoupled in transverse propagating states (k ) as
in Figure 2.4. Then the device Hamiltonian [ ]H is given by:
[ ]( ) '' ,;z z k k kk kH H H ε δ + = +
� �� � (2.16)
Each transverse mode k can be considered as a separate device. Accordingly, one can
restate the transport problem in one-dimension (1-D) by using two-dimensionally
integrated (2-D) Fermi functions ( )2 ,D z L Rf E µ− . This simplification will be discussed in
detail in section 0.
24
2.2.2. Source Drain Self-Energy Matrices
Ferromagnetic contacts are projected into two independent electron spin channels
(spin-up/spin-down) as shown in Fig. 2.3 according to their magnetic polarization states.
Self-energies for the regular contacts can be expressed as a combination of the two
different spin-channels referring to the majority ( )M and minority ( )m carriers and the
corresponding spin polarization axis ,L Rθ for the related contact as such:
( ) ( ) ( ), , , , , ,L R L R L R L R L R L Rθ θ θ↑ ↓Σ = Σ + Σ (2.17)
Figure 2.5 Source (left) contact self-energy matrix can be partitioned into two independent spin-channels. Self energy matrix of this contact can be written in its
magnetization spin-basis as a diagonal matrix.
From the elementary arguments (the details are given elsewhere [21]), it can be shown
that left contact self-energy is:
( )
1 2
1
2
0 0
0 0 0
0 0 0
L
L z
N
N
E
χ Σ =
⋯
⋯
⋮ ⋮ ⋱ ⋮
⋯
(2.18)
where the Lχ self-energy matrix term is defined within the diagonalizing spin set as:
LΣ↓
Lµ
Lµ
LΣ↑
Channel
[ ]H U+
Sourcez
-z LΣ↓
LµLµ
Lµ
LΣ↑
Channel
[ ]H U+
Sourcez
-z
25
( )
0
0
L
L
ik a
L zik a
teE
teχ
↑
↓
↑ ↓
↑
↓
− = −
(2.19)
with the associated wave-vector ,Lk↑ ↓ obtained from the dispersion relation for the left
contact:
Similarly, right contact self-energy can be projected into two independent spin channels
with a non-zero element only at the right most point N:
( )
1 2
1
2
0 0 0
0 0 0
0 0
R z
R
N
N
E
χ
Σ =
⋯
⋯
⋮ ⋮ ⋱ ⋮
⋯
(2.21)
where the Rχ self-energy matrix term is defined within the diagonalizing spin set as:
( )
0
0
R
R
ik a
R zik a
teE
teχ
↑
↓
↑ ↓
↑
↓
− = −
(2.22)
with the associated wave-vector ,Rk↑ ↓ obtained from the dispersion relation for the right
contact:
The right contact magnetization direction does not necessarily have to be aligned
with that of the left contact. For example, for the anti-parallel polarization state of the
ferromagnetic contacts there is an angular difference =πφ between the contact
magnetization directions. Nevertheless after obtaining the broadening matrices of the
ferromagnetic right contact in its magnetization direction [Eqs. 2.14-2.16], it is possible
to rotate the self-energy matrix of the right contact to the left contact spin basis-set using
unitary transformation operations:
( ), ,2 1 cosz c L LE E U t k a↑ ↓ ↑ ↓ = + + −
(2.20)
( ), ,2 1 cosz c R RE E U t k a↑ ↓ ↑ ↓ = + + −
(2.23)
26
Figure 2.6 Drain (right) contact self-energy matrix can be partitioned into two independent spin-channels. Self energy matrix of this contact can be written in its
magnetization spin-basis as a diagonal matrix.
( )
( )
( )
( )
, ,
†
2 2 2 2
2 2 2 2
cos sin cos sin;
sin cos sin cosR z R zE E
θ θ θ θ
θ θ θ θ
θ θ
χ θ χ↑ ↓ ↑ ↓
∆ ∆ ∆ ∆
∆ ∆ ∆ ∆
ℜ ∆ ℜ ∆
− ∆ = −
ɶ ɶ������� �������
(2.24)
2.2.3. Current and JMR Ratio
Effective masses for the tunneling electrons are assumed to be constant
throughout the MTJ device, and the tunneling current is simply expressed as:
( ) ( ) ( )4 z L z R z zk kk
qI T E f E f E dEε ε
π
∞
−∞
= + − + ∑ ∫
� ��ℏ
(2.25)
where one could work in a decoupled transverse mode space as discussed in section
2.2.1. Within this context, each transverse mode has an extra energy 2 2 2k k mε = ℏ that
should be added to the total energy used in Fermi function (Eq. 1.15). This could be done
analytically by replacing the Fermi functions with ( )2 ,D z L Rf E µ− functions [21]:
( ) ( )( ), ,ln 1 expD z L R s z L R Bf E N E k Tµ µ − = + − −
(2.26)
Rµ
RµRΣ↓
RΣ↑
Channel
[ ]H U+
Drain∆θ
∆θ
Rµ
Rµ
Rµ
RµRΣ↓
RΣ↑
Channel
[ ]H U+
Drain∆θ
∆θ
27
where * 22s BN m k T π= ℏ is defined for per unit area leading to a single variable current
relation:
( ) ( ) ( )2 , 2 ,4c
z D z L R D z L R z
E
qI T E f E f E dE
σ
µ µπ
∞
= − − − ∫ℏ (2.27)
Referring to F AFI I as the current values for the parallel/anti-parallel
magnetizations ( 0θ θ π∆ = ∆ = ) of the ferromagnetic contacts:
( )∑ ==F
zF EII 0;φ (2.28a)
( )∑ ==F
zAF EII πφ; (2.28b)
the JMR is defined as:
F AF
F
I IJMR
I
−= (2.29)
2.3. Results
Coherent tunneling regime features are obtained by benchmarking the experiment
measurements made in impurity free tunneling oxide MTJ devices at small bias voltages
( 1biasV meV= ). The parameters used here for the generic ferromagnetic contacts are the
Fermi energy 2.2FE eV= and the exchange field 1.45eV∆ = [31]. The tunneling region
potential barrier [ ]barrU is parameterized within the band gaps quoted in literature (as low
as 1.8 eV [48] and as high as approximately 7eV [49]), while the charging potential [ ]U
is neglected.
Dependence of theJMR ratios on the barrier thicknesses and the heights are
shown in Figure 2.7 in comparison with experimental measurements. These barrier
thicknesses are taken from the experimental measurements believed to be close to actual
physical barrier thicknesses while the barrier heights are used as adjustable parameters.
28
The barrier heights obtained here may differ from those reported in literature [14,17,35-
44] based on empirical models [50].
JMR values are shown to be improving with increasing barrier heights for all
barrier thicknesses, a theoretically predicted [46] and experimentally observed feature
[14,17,35-44]. On the other hand, thickness independent JMRs (flat lines) in Figure 2.7
obtained by the Slonczewski’s formula seem to conflict with the experimentally observed
JMR ratios in the thin tunneling barrier limit. In the thick barrier limit a convergence with
experimental JMR values is observed. This observation suggests that Slonczewski’s
asymptotically thick limit is a poor approximation for typical experimental devices with
barrier thicknesses are less than 1-nm.
0.5 1 1.5 2 2.50
10
20
30
Thickness [nm] --->
JMR
% -
-->
Ubarrier
=5.3eV
Ubarrier
=2.4eV
Ubarrier
=1.6eV
Figure 2.7 Thickness dependence of theJMR ratios for different barrier heights are shown in comparison with experiments [35-44]. The TMR ratios given in experimental
measurements are converted to JMR ratios when it’s necessary. Slonczewski’s asymptotic limit result is also shown (flat lines).
29
Observation of deteriorating JMR ratios with increasing tunneling barrier
thicknesses (Figure 2.7) can be justified with an energy resolved analysis of the tunneling
currents (Figure 2.8). JMR ratios, defined in Eq. 2.29 (subsection 2.1), can be broken
down into energy resolved components for junction magnetoresistances as in:
( ) ( ) ( )( )
F z AF zz
F z
I E I EJMR E
I E
−= (2.30)
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1.0
JMR
Rat
io ---
>
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1.0
Ez Energy [eV] --->
norm
aliz
ed ω
(Ez)
--->
t=0.7nmt=1.4nmt=2.1nm
Ubarrier
=1.6eV
Figure 2.8 An energy resolved analysis of ( )zJMR E (left-axis) and normalized ( )zEω
distributions (right-axis) are presented for a device with a tunneling barrier height of 1.6eV.
In Figure 2.8, the energy resolved ( )zJMR E ratios are shown to be independent
of the barrier thicknesses. This initially counter intuitive observation can be understood
by considering the redistribution of tunneling electron densities over energies with
changing barrier thicknesses. Defining ( )zEω as a measure of the contributing weight of
the ( )zJMR E , one can show that experimentally measured JMRis a weighted integral of
( )zJMR E ratios over zE energies:
30
( )[ ]( )z z zJMR E JMR E dEω= ∫ (2.31)
where:
( ) ( ) ( )z
z F z F zE
E I E I Eω = ∑ (2.32)
is the energy resolved spin-continuum current component (weighting function). In Eq.
3.31, ( )zJMR E ratios are constant (solid curve Figure 2.8) while the normalized ( )zEω
distributions shifts towards higher energies with increasing barrier thicknesses (dashed
lines in Figure 2.8). Hence, JMR ratios, an integral of the multiplication of the ( )zEω
distributions with the the energy resolved ( )zJMR E ratios, decreases with increasing
barrier heights.
2.4. Summary
Our calculations are able to demonstrate the intrinsic characteristics of MTJ
devices related with barrier properties in agreement with experimental observations. It is
shown that even within the simplest free-electron description, the spin-dependent
tunneling current and the JMR ratios are not determined solely by the characteristics of
the ferromagnetic contacts; they also depend on the properties of the tunneling barriers
such as thicknesses and the barrier heights. Clearly, large barrier thicknesses suppress the
JMR ratios while the higher potential barriers enhance them [46]. In the thin barrier
limit, SPT depends on the barrier thicknesses due to the redistribution of the tunneling
electrons in the k -space. In the asymptotic limit our calculations converges with the
Slonczewski’s formula [27].
31
CHAPTER 3. MTJ DEVICE: INCOHERENT TRANSPORT
In the presence of "rigid" scatterers such as impurities and defects, electron
transport is considered coherent since the phase relationships among different paths are
time independent. Therefore the relevant scattering effects can be incorporated into the
transport problem through the device Hamiltonian[ ]H . However, the situtation is
different when the impurities have an internal degree of freedom (such as the internal
spin states of magnetic impurities). The effect of such scatterers can not be simply
incorporated through the device Hamiltonian. Instead scattering self energy matrices are
needed. An implementation of this self-energy matrix treatment will be discussed in this
part of the dissertation for electron-impurity exchange scattering processes in MTJs.
3.1. Preliminaries
The sensitivity of MTJ devices to magnetic impurities was controllably
investigated by Jansen and Moodera through a series of experiments [5]. Schematics of
the model devices investigated are given in Figure 3.1a. Here the incoherent transport
regime is created by deposition of magnetic impurities during the growth process of
oxide tunneling barriers. The δ-doped magnetic impurities form a diluted magnetic
impurity layer in two dimensions (2-D) with submonolayer thicknesses. Experimental
measurements have revealed that the inclusion of magnetic impurities does not change
the overall conductance in these devices with respect to (impurity free) control junctions.
Therefore, despite the presence of magnetic impurities, the tunneling character of the
electron transport between ferromagnetic contacts in these systems is maintained.
32
Nevertheless measurements have revealed a rapid decay of JMR values as a function of
δ-dopant thickness and concentration (Figure 3.1b).
Figure 3.1 (a) A schematic illustration of the vertical MTJ device with δ-doped magnetic impurities is given. (b) Reduction in MR values are observed with increasing impurity
content [5].
Spin exchange scattering processes are responsible for the incoherent nature of
electron tunneling transport in the model devices considered here. Such processes may or
may not involve energy exchange between the tunneling electrons and the localized
spins, since energy exchange in between depends on the energy difference of the
magnetic impurity spin states ( qωℏ ). The incoherent nature of the spin states arises from
our assumption that unspecified external forces continuously restore the localized spins
into a state of equilibrium (%50 up, %50 down). These external forces, believed to be
present in a closely packed impurity layer, could be due to magnetic dipole-dipole
interactions amongst the magnetic impurities or spin relaxation processes coupled with
phononic excitations. Nevertheless the physical origin of the equilibrium restoring
processes is not of our interest to this discussion (at least from a tunneling electron's point
of view) assuming equilibrium restoring processes are fast enough to maintain the
impurity spins in a thermal equilibrium state. Accordingly, the effect of spin scatterings
from magnetic impurities cannot be included in the Hamiltonian and are included through
an appropriately determined scattering self-energy as described below. In the following
subsection, we discuss the spin-scattering self-energy.
F
F
φ
Hard Layer
Soft Layer
Impurity Layer
Tunneling Oxide
33
3.2. Spin Exchange Interaction and Scattering Tensors
In this chapter, calculations are restricted to the elastic spin exchange interactions
(magnetic impurity scattering), although the NEGF model, presented in chapter 2, can be
easily extended to incorporate other spin scattering mechanisms as well including nuclear
hyperfine, electron-impurity and electron-hole exchange interactions (Bir-Aranov-Pikus).
As discussed in chapter 2, coupling between the number of available
electrons/holes ( n pG G ) at a state and the in/out-flow (in outS S Σ Σ ) to/from that
state is related through the fourth order scattering tensor n pD D in Eqs. 1.28 and
1.36. In general, these scattering tensors can be obtained starting from a spin scattering
Hamiltonian of the form:
( ) ( )int -j
j jRH r J r R Sσ= ⋅∑
��� � � (3.1)
where jr R��
are the spatial coordinates and jSσ��
are the spin operators for the channel
electron / (j-th) magnetic-impurity. For the model systems considered here, the spin-
conserving ("nsf") scattering tensor elements in Eq. 1.38 are neglected due to their minor
effect on the JMRs due to the degeneracy of magnetic impurity spin states ( 0qω =ℏ ).
Accordingly, for large cross-section multilayer devices in a discrete lattice with spacings
" a ":
( ) ( )2 2
2
1'
q
I q IDr r J N J n
aωδ ω− →∑ (3.2)
and the n pD D scattering tensors are given by:
( )
n,p 2Isf 2D
0 0 0,
0 0 0,D 0 n a
0 0 0 0
0 0 0 0
k l
i j
Fu d
Fd uJ
σ σ
σ σ
ω
↑↑ ↓↓ ↑↓ ↓↑
↑↑
↓↓
↑↓
↓↑
→
↓
= =
ℏ
(3.3)
34
where I In N S= is the impurity concentration per unit area (S being the device cross
section), and 2
2DJ is a parameter reflecting the spin scattering strength of the impurity
layer. The 2
2DJ values can be extracted from the available experimental data and are
independent from the device cross section S and the lattice grid spacing a used in the
calculations. Following Eq. 4.3, the spin scattering tensor relationship given in Eqs. 1.28,
1.37 and 1.38 will simplify to:
( ) ( ), , ,; ;i j i j k l k l
k l
in out n p n pS
sfE D G Eσ σ σ σ σ σ σ σ
σ σ
Σ = ∑ (3.4)
A formal derivation of the scattering tensor in Eq. 1.37 is given in the appendix A.
However, the simplified version given in Eq. 3.3 can be understood heuristically from
elementary arguments. For the corresponding lattice site "j" with magnetic impurities, the
in/out-scattering into spin-up component is proportional to the spin-down electron/hole
density times the number of spin-up impurities per unit area, I un F :
( ) ( ), 2 ,; 2 ,,
in out n pI uS D j jj j
J n F G↑↑ ↓↓Σ = (3.5)
Similarly, the in/out-scattering into spin-down component is proportional to the spin-up
electron/hole density times the number of spin-down impurities per unit area, I dn F :
( ) ( ), 2 ,; 2 ,,
in out n pI uS D j jj j
J n F G↑↑ ↓↓Σ = (3.6)
Here the n pG G correlation function is full matrices of the form:
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
, , , ,
1,1 1,2 1, 1 1,
, , , ,
2,1 2,2 2, 1 2,
,
, , , ,
1,1 1,2 1, 1 1,
, , , ,
,1 ,2 , 1 ,
n p n p n p n p
N N
n p n p n p n p
N N
n p
n p n p n p n p
N N N N N N
n p n p n p n p
N N N N N N
G G G G
G G G G
G
G G G G
G G G G
−
−
− − − − −
−
=
⋯
⋯
⋮ ⋮ ⋱ ⋮ ⋮
⋯
⋯
(3.7)
with the (2x2) block diagonal elements:
35
( )( ) ( )( ) ( )
, ,
, ,,
, , ,
, ,
n p n p
j j j jn p
j j n p n p
j j j j
G GG
G G
↑↑ ↑↓
↓↑ ↓↓
↑ ↓
↑
↓
=
(3.8)
While for point like spin exchange scatterings, in outS S Σ Σ in/out-scattering matrix is
block diagonal:
( )( )
( )( )
,
1,1
,
2,2
,
,
1, 1
,
,
0 0 0
0 0 0
0 0 0
0 0 0
in outS
in outS
in outS
in outS N N
in outS N N
− −
Σ
Σ
Σ =
Σ Σ
⋯
⋯
⋮ ⋮ ⋱ ⋮ ⋮
⋯
⋯
(3.9)
with the (2x2) block diagonal elements:
( )( ) ( )( ) ( )
, ,; ;, ,,
, , ,; ;, ,
in out in outS Sj j j jin out
S j j in out in outS Sj j j j
↑↑ ↑↓
↓↑ ↓↓
↑ ↓
↑
↓
Σ Σ Σ = Σ Σ
(3.10)
3.3. Solution of NEGF Equations
In the section, a real space approach is employed to analyze the spin-dephasing
effects of magnetic impurity layers (with device parameters benchmarked in the coherent
regime in chapter 2). The channel Hamiltonian and contact self-energy matrices are
directly taken from coherent regime calculations. The model system is illustrated in
Figure 3.2. δ-doped magnetic impurities are incorporated (Figure 3.2) through single grid
point defined scattering self-energy and in/out-scattering matrices. These matrices are
non-zero only at the single block diagonal ‘j,j” where the grid point for the delta doping
is assumed to be:
36
( ),
0 0 0
0 0
0 0 0
SS j j
ΣΣ =
⋯ ⋯
⋮ ⋱ ⋮ ⋰ ⋮
⋯ ⋯
⋮ ⋰ ⋮ ⋱ ⋮
⋯ ⋯
(3.11a)
( ) ,,
,
0 0 0
0 0
0 0 0
in outin outS S j j
Σ = Σ
⋯ ⋯
⋮ ⋱ ⋮ ⋰ ⋮
⋯ ⋯
⋮ ⋰ ⋮ ⋱ ⋮
⋯ ⋯
(3.11b)
Figure 3.2 Energy band diagram for the model MTJ devices considered in this work. Right contact assumed to be the soft ferromagnet whose magnetization direction can be
manipulated with an external magnetic field.
3.3.1. Calculation Scheme for the Green’s Function and Scattering Self-Energies
Here, it is assumed that the spin-up and spin-down populations of magnetic
impurities are equally distributed with 0.5u dF F= = . This can be justified by considering
[ ]RΣHamiltonian
a
∆
FE
minoritycE
majoritycE
FE
minoritycE
majoritycE
∆
θ∆
θ∆
0 N+11 N⋯
k
z1k 2k
barrU
[ ]H[ ]LΣ
Impurity Layer [ ]SΣ
[ ]RΣHamiltonian
a
∆
FE
minoritycE
majoritycE
FE
minoritycE
majoritycE
∆
θ∆
θ∆
0 N+11 N⋯
k
z1k 2k
barrU
[ ]H[ ]LΣ
Impurity Layer [ ]SΣ
37
the relatively fast spin relaxations of magnetic impurity spin-states in the presence of
magnetic dipole-dipole interactions. Under these conditions, the n
sfD and
p
sfD scattering tensors are equal:
n p
sf sfD D D = = (3.12)
Through the definition of the spectral energy function ( )A E , it can be shown that there
exists a simple relationship between device Green’s function and the spin scattering self
energy:
( ); ; i j i k k l k l
i j
S D G Eσ σ σ σ σ σ σ σσ σ
Σ = ∑ (3.13)
This assumption ( n pD D= ) simplifies the treatment of scattering processes
significantly for two distinct reasons. Firstly, Eq. 3.13 allows us to obtain [ ]SΣ without
using Hilbert transformations [Eq. 1.18]. More importantly, it decouples the solution of
[ ] [ ]SG Σ from the solution of n inSG Σ making it possible to use ( )2 ,D z L Rf E µ−
functions (see Eq. 2.26) to represent the sum over the transverse momentum as in the
coherent regime. The overall procedure can now be summarized in two steps:
i. The device Green's function [ ]G and the scattering self energy matrix [ ]SΣ are
calculated in a self consistent manner using Eq. 3.13 with:
( ) ( ) ( ) ( ) 1
z z L z R z S zG E E I H U E E E−
= − − − Σ − Σ − Σ (3.14a)
( ) ( ),S z S zE k EΣ = Σ (3.14b)
where due to the decoupling of the 2-D translational modes, operators are
independent from the transverse energy 2 2 2k k mε = ℏ of the tunneling electrons.
ii. Electron correlation function nG and in-scattering matrix inS Σ can be
obtained non-iteratively from Eqs. 1.28 and 3.4 and using the Green’s function
( )zG E obtained in the previous self-consistent loop (appendix B):
38
Figure 3.3 Flowchart for the iterative calculation of Green’s function G and scattering self-energy ΣS.
10
L R SG EI H
(Eq. 3.14)
− = − − Σ − Σ − Σ
CONVERGED
( ) ( .3
s E D G
Eq .15)
Σ = ⊗
( )0 0 2S S SΣ = Σ + Σ
& SGET FINAL G Σ
Yes
No
0 0 S INITIAL GUESSΣ =
( ) 12 , 2 ,
D n p D n pG I P S
(Eqs. B.9 - B.11)
−= −ɶ ɶɶ
Direct
Solution
GET FINAL G
39
( ) ( )2 2D; ;
i j i k k l k l
i j
D in nS z zE D G Eσ σ σ σ σ σ σ σ
σ σ
Σ = ∑ (3.15)
where the 2-D integrated versions of the in-scattering and correlation functions are
defined as:
( ) ( )2 , k l
k
D nz z k
G E G Eσ σε
ε=∑
�
� (3.16a)
( ) ( )2 , k
D in inS z S z k
E Eε
εΣ = Σ∑
�
� (3.17a)
One important point to note here is that for elastic spin scattering processes there
is no need to calculate 2-D integrated pG hole correlation function and outS Σ out-
scattering matrix self-consistenly. The spectral function ( )zA E used in the current
relation Eq. 1.29 can be directly obtained from Eq. 1.19 using the device Green's function
( )zG E obtained in the previous self-consistent loop.
3.3.2. Current Equations:
Now using the 2-D versions of the necessary operators one can proceed to the
current calculation Eq. 1.29:
( ) ( ) ( ) ( )( ),
, , , , z k
in nL L z k z k L z k z k z k
E E
qI tr E E A E E tr E E G E E dE dE
h = Σ − Γ ∫∫ (3.18)
which eventually simplifies to a decoupled version;
( ) ( ) ( ) ( )( )2 2 z
D in D nL L z z L z z z
E
qI tr E A E tr E G E dE
h = Σ − Γ ∫ (3.19)
Similarly, the current density relation given in Eq. 1.30 can be expressed in a 2-D
integrated form:
40
( ) ( )( )( )
2, 1 1,
1,
2 Re D nL L L z L L z z
L L zG E
qJ tr H E i G E dE
h
∞
+ +−∞
<+−
=
∫ ����� (3.20)
The method outline here is applied to magnetic impurity doped MTJ devices (Figure 3.1)
and compared with experimental observations in the following subsections.
3.4. Results
Figure 3.4 For MTJ devices with impurity layers, variation of JMR ratios for varying barrier thicknesses and interaction strengths (0.6-0.3-0 eV) are shown. Normalized JMR
values are proved to be thickness independent as displayed in the inset.
Incoherent tunneling regime device characteristics in the presence of magnetic
impurities are presented below for a fixed barrier height of 1.6barrU = (Figure 3.4-Figure
3.5), with varying barrier thicknesses and electron-impurity spin exchange interactions
41
( 2 2
20 / 3/ 6 eVID
J n nm= − ). Nonlinear decreasingJMRs with increasing spin-
exchange interactions are observed at all barrier thicknesses due to the mixing of
independent spin-channels [35] while the normalized JMRs are shown to be thickness
independent (inset). This observation is attributed to the elastic nature of the spin
exchange interactions yielding a total drop in ( )zJMR E values at all zE energies in Eq.
3.30 while preserving the normalized ( )zEω carrier distributions (Figure 3.5).
Figure 3.5 For MTJ devices with impurity layers a detailed energy resolved analysis is shown. Nomalized ( )zEω distributions are unaffected by exchange interactions
reflecting the inelastic nature of the scattering processes.
Another interesting feature is that “normalized” JMRs deteriorate with increasing
spin-dephasing strengths (22 ID
J n ) independently from the tunneling barrier heights
(Figure 3.6). This general trend can be shown by mapping the “normalized” JMRs into a
single universal curve using a tunneling barrier heights dependent constant ( )barrc U
(inset in Figure 3.6).
42
This allows us to choose a particular barrier height value ( 1.6barrU eV= in this
case) and adjust the single parameter 2
2DJ to fit our NEGF calculations (Figure 3.8-
Figure 3.10) with experimental measurements obtained from δ-doped MTJs [5]. Sub-
monolayer impurity thicknesses (t ) given in the measurements are converted into
impurity concentrations per unit area (In ) using I bulkn t n= × where bulkn is the bulk
material density of Pd/Ni/Co metals [5].
Figure 3.6 JMR values for different barrier heights are shown.
Close fitting to experimental data is obtained at 77 K (Figure 3.8-Figure 3.10)
(solid line) using physically reasonable coupling constants 2
2DJ for devices with
Pd/Co impurities. However, the experimentally observed temperature dependence of
normalized JMR ratios can not be accounted for by our model calculations. The
normalized JMR ratios vary within a line width as the temperatures is raised from 77 K
to 300 K. As a result, different 2
2DJ couplings are used in order to match the
experimental data taken at 300K.
43
Figure 3.7 Using a barrier height dependent constant ( )barrc U , one can shown that JMR
values for different barrier heights scale to a universal curve.
For Pd and Ni doped MTJs, a relatively small variation in exchange couplings is
needed 2 2300 772 2
1.32K KD DJ J = (for Pd) and 2 2
300 772 21.25K KD D
J J = (for Ni) in
order to match the experimental data at 300K (dashed lines). These small temperature
dependences could be due to the secondary mechanisms not included in our calculations.
One such mechanism reported in the literature includes, the presence of impurity assisted
conductance contribution through the defects (possibly created by the inclusion of
magnetic impurities within the barrier) which is known to be strongly temperature
dependent [45]. In fact, the contribution of impurity assisted conductance is proportional
to impurity concentrations in accordance with experimental measurements.
A similarly interesting feature observed in calculations is the comparable 2
2DJ
couplings for Pd and Ni impurities at temperatures of 77K and 300K. Accordingly, a
possible estimate of the impurity spin states can be made by considering the most
commonly encountered oxidation states of the Pd and Ni impurities. Closed-shell
elemental Pd is only known to be in a magnetic oxidized state of S=1 in octahedral
oxygen coordination according to the Hund's rules. Similarly, one can attribute the
comparable 2
2DJ couplings for Ni impurities due to the S=1 spin state of the Ni+2
which is known to be a frequently observed ionized state in oxygen environment.
44
Figure 3.8 Experimental data taken at 77K/300K is compared with theoretical analysis in the presence of palladium magnetic impurities with increasing impurity concentrations.
On the contrary, for Co doped MTJs, there is a clear distinction between
normalized JMR ratios at different temperatures (Figure 3.10), which can not be justified
by the presence of secondary mechanisms. Fitting these large deviations requires larger
variations in the 2
2DJ exchange coupling parameters (2 2
300 772 22.69K KD D
J J = ). We
propose this to be a result of thermally driven low-spin/high-spin phase transition [53],
since the oxidation state of the cobalt atoms can be Co+2 (S= 3/2, high-spin) or Co+3
(S= 0, low-spin) state or partially in both of the states depending on the oxidation
environment. Such thermally driven low-spin/high-spin phase transitions for metal-
oxides have been predicted by theoretical calculations and observed in experimental
studies [53,54]. These phase transitions have not been discussed in the MTJ community
in connection with possible scattering factors determining the temperature dependence of
JMRs. Although from the available experimental data it is not possible to make a
decisive conclusion in this direction, given the non-linear dependence of JMRson
45
magnetic impurity states in our calculations, we believe it is important to point out this
possibility here.
Figure 3.9 Experimental data taken at 77K/00K is compared with theoretical analysis in the presence of nickel magnetic impurities with increasing impurity concentrations.
An order of magnitute analysis of the 22D
J exchange interaction parameters
obtained in this article can be done using 2 3 20 2D
J a J= , where 30a is a normalization
volume related to the wavefunction overlap. For our purposes, it is good enough to
assume 30a equal to the Bohr radius. Accordingly, 2J values are within a physically
reasonable range of 31.3 2.9 -meV nm− in accordance with the ab-initio calculations
[55].
46
Figure 3.10 Experimental data taken at 77K/300K is compared with theoretical analysis in the presence of cobalt magnetic impurities with increasing impurity concentrations.
3.5. Summary
A NEGF-based quantum transport model incorporating spin-flip scattering
processes within the self-consistent Born approximation is presented. Spin-flip scattering
and quantum effects are simultaneously captured. Spin scattering operators are derived
for the specific case of electron-impurity spin-exchange interactions and the formalism is
applied to spin-dependent electron transport in MTJs with magnetic impurity layers. The
theory is benchmarked against experimental data involving both coherent and incoherent
transport regimes. JMRsare shown to decrease both with barrier thickness and spin-flip
scattering but our unified treatment clearly brings out the difference in the underlying
physics. Our numerical results show that both barrier height and the exchange interaction
constant can be subsumed into a single parameter (2
2DJ ) that can explain a variety of
experiments (Figure 3.8-Figure 3.10). Small differences in spin-states and concentrations
of magnetic impurities are shown to cause large deviations in JMRs. Interesting
47
similarities and differences among devices with Pd, Ni and Co impurities are pointed out,
which could be signatures of the spin states of oxidized Pd and Ni impurities and low-
spin/high-spin phase transitions for oxidized Co impurities. This is work has been
published in Physical Review B [56].
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48
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APPENDICES
53
Appendix A. Spin Exchange Scattering Tensors
In the following, the scattering tensors stated in Eqs. 2.37 and 2.38 are obtained
starting from standard expressions obtained in the self-consistent Born approximation
from the NEGF formalism. Here we start from the formulation in [20] (see Sections 10.4
and A.4) which represents a generalization of the earlier treatments [52,57 and 58]. For
spatially localized scatterings, we have:
( ) ( ) ( )
( ) ( )
*; ; ;
,
†int; int;
: ' '
'
i j k l i k j l
i k l j
ns s s s
s s
s s
D
s H s s H s
α β α βα β
α β
σ σ σ σ σ σ σ σ
α σ σ β β σ σ α
ξ ξ τ ξ τ ξ
ξ ξ
=
=
∑
∑
(A.21a)
( ) ( ) ( )
( ) ( )
*; ; ;
,
†int; int;
: ' '
'
i j k l l j k i
l j i k
ps s s s
s s
s s
D
s H s s H s
β α β αα β
α β
σ σ σ σ σ σ σ σ
β σ σ α α σ σ β
ξ ξ τ ξ τ ξ
ξ ξ
=
=
∑
∑
(A.1b)
where sα and sβ are impurity spin subspace states and intH is the interaction
Hamiltonian (Eq. 2.35) defined within the channel electron spin subspace as:
( ) ( )int; int ,i k l j i l k jH H r tσ σ σ σ ξ σ σ σ σ= (A.2a)
( ) ( )†int; int ', '
i k i k i l k jH H r tσ σ σ σ ξ σ σ σ σ= (A.2b)
For an uncorrelated impurity spin ensemble with:
s ss s
s s s sα β
α β
α α β βρ ω ω= =∑ ∑ (A.3)
averaging in Eqs. (A.1a) and (A.1b) can be done through a weighted summation of spin
scattering rates of magnetic impurities:
54
( ) ( ) ( )
( ) [ ] ( )
( ) ( )( )
†; int; int;
,
†int; int;
†int; int;
, ; ' ' '
, ', '
', ' ,
i j k l i k l j
i k l j
l j i k
ns
s s
s
D r t r t s H t s s H t s
s H r t H r t s
tr H r t H r t
βα β
α
σ σ σ σ α σ σ β β σ σ α
α σ σ σ σ α
σ σ σ σ
ω
ρ
ρ
=
=
=
∑
∑ (A.4a)
( ) ( ) ( )
( ) [ ] ( )
( )
†; int; int;
,
†int; int;
†int; int;
, ; ' ' , ', '
, ', '
', '
i j k l l j i k
l j i k
i k l
ns
s s
s
D r t r t s H r t s s H r t s
s H r t H r t s
tr H r t H
αα β
β
σ σ σ σ β σ σ α α σ σ β
β σ σ σ σ β
σ σ σ
ω
ρ
ρ
=
=
=
∑
∑
( )( ),j
r tσ
(A.4b)
The trace ( )tr Aρ for any operator [ ]A is independent of representation. Accordingly,
n pD D scattering tensors can be evaluated using any convenient basis for magnetic
impurity spin states.
Through Jordan-Wigner transformation, single spins can be thought as an empty
or singly occupied fermion state (as in Figure A.1):
† 0a↑ = (A.5a)
0↓ = (A.5b)
with †a and a spin creation and annihilation operators for the channel electrons:
( ) ( )† 0
0 0
ei tet a t
ω
σ + = =
(A.6a)
( ) ( ) 0 0
0ei tt a te ωσ −
−
= =
(A.6b)
For degenerate electron spin states ( 0eω =ℏ ), there is no time dependence as such
( )† †a t a→ and ( )a t a→ . In this representation, the Pauli spin-matrices are:
( ) ( )†1 1
2 2x a aσ σ σ+ −= + ≡ + (A.7a)
55
( ) ( )†1 1
2 2y a ai i
σ σ σ+= − − ≡ − (A.7b)
† 1
2z a aσ = − (A.7c)
Accordingly the ( ) ( ) ( )int , H r t J r R S tδ σ= − ⋅��� �
interaction Hamiltonian is:
( ) ( ) ( ) ( ) ( )�1 1 1,
2 2 2int zH r t J r R aS t a S t a a S tδ + − = − + + −
�� � (A.8)
where S is the spin operator for the localized magnetic impurity.
1fn =
a †a
spin state
ωℏ
0fn =
1fn =
a †a
spin state
ωℏ
0fn = 0fn =
Figure A.1 Up/down spin states can be treated as one particle state which is either full or empty (Jordan-Wigner representation).
Substituting these terms into matrix elements will yield:
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
2
, ; ', ' '
' ' ' '
' '
n
z z z z
z z z
D r t r t r r J
S t S t S t S t S t S t S t S t
S t S t S t S t S t S
δ
↑↑ ↓↓ ↑↓ ↓↑
+ − + −↑↑
↓↓ − +
↑↓
↓↑
= −
− ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
' '
' ' ' '
' ' ' '
z
z z z z
z z z z
t S t S t
S t S t S t S t S t S t S t S t
S t S t S t S t S t S t S t S t
+ −
− − − −
+ + + +
−
− − − −
(A.9a)
56
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
2
, ; ', ' '
' ' ' '
' '
p
z z z z
z z
D r t r t r r J
S t S t S t S t S t S t S t S t
S t S t S t S t S t S
δ
↑↑ ↓↓ ↑↓ ↓↑
− + + −↑↑
↓↓ + − +
↑↓
↓↑
= −
− ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
' '
' ' ' '
' ' ' '
z z
z z z z
z z z z
t S t S t
S t S t S t S t S t S t S t S t
S t S t S t S t S t S t S t S t
−
− − − −
+ + + +
−
− − − −
(A.9b)
The localized magnetic impurity spin-operators can be written in its diagonalized
impurity spin subspace as:
0
0 0
Ii teS d
ω+ +
= =
(A.10a)
0 0
0Ii tS de ω
−−
= =
(A.10b)
1 01 1
0 12 2zS d d+ = − = −
(A.10c)
For an impurity density matrix of form ( 1u dF F+ = ):
( ) 0
0u
I qd
FN
Fρ ω
=
(A.11)
with IN being total number of impurities at that location, the desired quantities
n pD D can be obtained by evaluating the expectation values of the operators in
Eqs. A.9a-b. Here the only non-zero elements are:
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
1
1 2
2
2
1
0 1 2 0 1 2 0 1'
0 0 1 2 0 1 2 4
0 0 00'
0 00 0
0 0 0 0'
0 0 0 0
uz z I q
d
iEtiE t tu
I q uiEtd
i tu
i td
FS t S t trace N
F
F eS t S t trace N F e
F e
F eS t S t trace
F e
ω
ω
ω
ω −+ − −
− + −
= = − −
= =
=
ℏℏ
ℏ
( ) ( )2 1i t tI q dN F eωω −
=
(A.12)
57
Finally, n pD D scattering tensors are obtained as:
( ) ( ) ( )
( )
( )
2
'
'
, ; ', ' '
1 4 0 0
1 4 0 0 0 0 1 4 0
0 0 0 1 4
q
q
q
nqI
i t tu
i t td
D r t r t r r J N
F e
F e
ω
ω
ω
δ ω
↑↑ ↓↓ ↑↓ ↓↑
− −↑↑
−↓↓
↑↓
↓↑
= −
− −
∑
(A.13a)
( ) ( ) ( )
( )
( )
2
'
'
, ; ', ' '
1 4 0 0
1 4 0 0 0 0 1 4 0
0 0 0 1 4
q
q
q
pqI
i t td
i t tu
D r t r t r r J N
F e
F e
ω
ω
ω
δ ω
↑↑ ↓↓ ↑↓ ↓↑
− −↑↑
−↓↓
↑↓
↓↑
= −
− −
∑
(A.13b)
It is convenient to work with the Fourier transformed functions as such ( )'t t ω− → ℏ :
( ) ( )' 'qi t t t tqe eω η δ ω ω− − − − → −ℏ
ℏ ℏ (A.14)
where η is a positive infinitesimal. With Fourier transforming Eqs. A.13a-b will simplify
to Eqs. 2.37 and 2.38. For the calculations reported in this article, diagonal elements not
leading to spin-dephasing are omitted due to their negligible effect on JMR ratios. In this
case, n pD D scattering tensors simplifies to a form which can be understood from
simple common-sense arguments (Eqs. 3.5-3.6).
58
Appendix B. Direct Calculation Scheme For In-scattering and Correlation Matrices
In the following a non-iterative solution scheme for inS Σ in-scattering matrix
and nG electron correlation function in the presence of an elastic spin scatterings
( 0=qωℏ ) is summarized. It is shown that for decoupled transverse modes, 2-D
integrated 2D inS Σ in-scattering matrix and 2D nG electron correlation function can be
obtained from ( )2 ,D z L Rf E µ− contact Fermi functions (Eq. 2.26) through a tensor to
matrix transformation.
We start our derivation by redefining Eq. 3.4 in an energy grid defined in
longitudinal ( zE ) and transverse (k
ε
� ) directions:
( ) ( ), ;, ,k l k l m n m n
m n
in nS z zk k
E D G Eσ σ σ σ σ σ σ σσ σ
ε εΣ = ∑
� � (B.1)
and using Eqs. 2.26 and 2.27, we obtain:
( ) ( ) ( ) ( );, , ,i j i j i j m n m n
m n
n n nz z z zk k k
G E S E P E G Eσ σ σ σ σ σ σ σ σ σσ σ
ε ε ε = + ∑
� � � (B.2)
where ( ),i j
nz k
S Eσ σ ε
� and ( );i j m n zP Eσ σ σ σ are defined as:
( ) ( ) ( ) ( ) ( ), ,, , ,i j i k k l k l l j
n in inz z L z R z zk k k
S E G E E E G Eσ σ σ σ σ σ σ σ σ σε ε ε + = Σ + Σ
� � � (B.3a)
( ) ( ) ( ); ;i j m n i k i j m n l j
k l
z z zP E G E D G Eσ σ σ σ σ σ σ σ σ σ σ σσ σ
+= ⋅ ⋅∑ (B.3b)
Eq. B.2 can be integrated over transverse modes using 2-D integrated Fermi functions
(see Eqs. 2.26) and replacing ( ), ; ,i j
inL R z k
Eσ σ εΣ
� with ( )2, ; i j
D inL R zEσ σΣ :
( ) ( ) ( )2, ; 2 , ,i j
D inL R z D z L R L R zE f E Eσ σ µΣ = − Γ (B.4)
yielding:
59
( ) ( ) ( ) ( )2;, ,
i j i j m n m n i j
m nk
n n D nz z z zk k
G E P E G E S Eσ σ σ σ σ σ σ σ σ σε σ σ
ε ε
− =
∑ ∑
�
� � (B.5)
Here ( )2
i j
D nzS Eσ σ
is the 2-D integrated sum of the matrix defined in Eq. B.3a. An
index transformation as such i j Jσ σ → and m n Mσ σ → will convert this tensor
relationship into:
( ) ( ) ( ) ( )2, ,k
n n D nJ z JM z M z J zk k
M
G E P E G E S Eε
ε ε − =
∑ ∑
�
� �ɶ ɶ ɶɶ (B.6)
This is simply a matrix multiplication in the transformed basis:
( ) ( )( ) ( )1 2,k
n D nz z zk
G E I P E S Eε
ε− = − ∑
�
�ɶ ɶɶ (B.7)
yielding a only longitudinal energy dependent relation when summed over 2-D
translational energies:
( ) ( ) ( )( ) ( )12 2,k
D n n D nz z z zk
G E G E I P E S Eε
ε−
= = −∑
�
�ɶ ɶ ɶɶ (B.8)
Electron correlation functions defined in this new basis set (Eq. B.9) can be transformed
back to the real space matrix representation after reindexing with i jJ σ σ→ and
m nM σ σ→ wil:
( ) ( )2 2
D n D nz z
re-indexing
vector form matrix form
G E G E→ɶ��� ���
(B.9)
Accordingly, 2-D in-scattering function 2D inS Σ can be obtained using Eqs. 3.4 and B.9
leading to:
( ) ( )2 2, ;k l k l m n m n
m n
D in D nS z zE D G Eσ σ σ σ σ σ σ σ
σ σΣ = ∑ (B.10)
60
Appendix C. Source Code
% TMR_DEPHASING_DETAILED.m % 09/14/2005 % written by Ahmet Ali Yanik <[email protected]> % DESCRIPTION OF THE CODE========================== ==== % This code calculates energy dependence of TMR and I_F/I_AF currents for a % changing thickness of the tunneling barriers of a MTJ device with impurity layers. % ================================================ clear all profile on tic % constants (all MKS, except energy which is in eV) hbar=0.658217; % [eV-fs] Planck's constant q=1.602e-19; % [coulombs] electron charge mass=9.109*(10^-31); % electron mass [kg] kT=k*300; % [eV] at 300K inu=0*i*1e-5; % eta [eV] a=0.1; a_ct=a; a_br=a; % spacing [nm] %================================================== == m_ct=1; m_br=1; t_ct=tfactor/a_ct^2; t_br=tfactor/a_br^2; %================================================== == % contant for f2D integral Nf=((mass*m_ct)*kT/(2*pi*hbar^2*q*(1e-15)^2)); %================================================== === % Energy-Band profile============================== ========== %================================================== === % Cobalt Parameters================================ ========= E_exc=1.45; % [eV] Ef=2.2; % [eV] E_off=1.6; % [eV] U_barrier=Ef+E_off; % [eV] [adjustable parameter] E0=0; % [eV] %================================================== === % voltage bias & energy grid for z-energy============================
61
V_bias=1e-3; N_bias=1; NE_Vbias=1; dE = 0.025*kT; %at room temperature %================================================== === % barrier thickness================================ ========== Nt=1;L=linspace(7,7,Nt); % [angstrom] %================================================== === % scattering energy================================ ========== Nj=1;J2N=linspace(14,14,Nj); % [eV] %================================================== === for tt=1:Nt % barrier thickness N_source=2; N_barr=L(tt); N_drain=2; Np=N_source+N_barr+N_drain; Np2=2*Np; % conduction band================================== ======= Ec_source_M=zeros(N_source-1,1); % majority electrons Ec_source_m=E_exc*ones(N_source-1,1); % minority electrons Ec_barrier=U_barrier*ones(N_barr,1); Ec_drain_M=zeros(N_drain-1,1); % majority electrons Ec_drain_m=E_exc*ones(N_drain-1,1); % minority electrons %========================================= for jn=1:Nj % impurity concentration D(1,4)=J2N(jn); % Fu D(4,1)=J2N(jn); % Fd %================================================== =======% Hamiltonian======================================== ========= %================================================== ======= Ec_MM=[Ec_source_M;0.5*(E0+U_barrier);Ec_barrier;0.5*(E0+U_barrier);Ec_drain_M]'; Ec_Mm=[Ec_source_M;0.5*(E0+U_barrier);Ec_barrier;0.5*(E_exc+U_barrier);Ec_drain_m]'; Ec_mM=[Ec_source_m;0.5*(E_exc+U_barrier);Ec_barrier;0.5*(E0+U_barrier);Ec_drain_M]'; Ec_mm=[Ec_source_m;0.5*(E_exc+U_barrier);Ec_barrier;0.5*(E_exc+U_barrier);Ec_drain_m]'; Ec_F=[Ec_MM; Ec_mm]; Ec_F=reshape(Ec_F,1,2*Np); Ec_AF=[Ec_Mm; Ec_mM]; Ec_AF=reshape(Ec_AF,1,2*Np); alpha_source= [2*t_ct 2*t_ct]; beta_source= [-t_ct -t_ct]; alpha_barr = [2*t_br 2*t_br]; beta_barr = [-t_br -t_br]; alpha_drain = [2*t_ct 2*t_ct]; beta_drain = [-t_ct -t_ct]; alpha_int = [t_ct+t_br t_ct+t_br]; beta_int = [-t_br -t_br];
62
KE_diag=[repmat(alpha_source,[1,N_source-1])' alpha_int' repmat(alpha_barr,[1,N_barr])' alpha_int' repmat(alpha_drain,[1,N_drain-1])']'; KE_offdiag=[repmat(beta_source,[1,N_source-1])' beta_int' repmat(beta_barr,[1,N_barr-1])' beta_int' repmat(beta_drain,[1,N_drain-1])']'; KE=diag(KE_diag) + diag(KE_offdiag,2) + diag(KE_offdiag,-2); H_F=KE+diag(Ec_F); H_AF=KE+diag(Ec_AF); %================================================== ======= % self energy and gamma matrices======================= sig_L_F=zeros(Np2); sig_R_F=zeros(Np2); sig_L_AF=zeros(Np2); sig_R_AF=zeros(Np2); sig_F=zeros(Np2); sig_AF=zeros(Np2); gam_L_F=zeros(Np2); gam_R_F=zeros(Np2); gam_L_AF=zeros(Np2); gam_R_AF=zeros(Np2); gam_F=zeros(Np2); gam_AF=zeros(Np2); %================================================== ==== % applied potential profile======================== =========== mu_source=Ef+(V_bias/2); mu_drain=Ef-(V_bias/2); Udiag=V_bias*[0.5*ones(1,N_source) linspace(0.5,-0.5,N_barr) -0.5*ones(1,N_drain)]; U=kron(diag(Udiag),eye(2)); %================================================== ======= % energy grid============================== =================== NDE=400; Ez=[0:dE:mu_source+(NDE*dE)]; NEz=size(Ez,2); %=========================================== ============== %================================================== ======= II_F=zeros(1,NEz); II_AF=zeros(1,NEz); II_F_curr_density=zeros(1,NEz); II_AF_curr_density=zeros(1,NEz); T_F=zeros(1,NEz); T_AF=zeros(1,NEz); %================================================== ======= II_F_curr_density_lenght=zeros(1,Np); II_AF_curr_density_lenght=zeros(1,Np); % fermi functions================================== ============= f2D_L=Nf.*log(1+exp((mu_source-Ez)./kT)); f2D_L_p=1-f2D_L; f2D_R=Nf.*log(1+exp((mu_drain-Ez)./kT)); f2D_R_p=1.-f2D_R; %================================================== ======= for jz=1:NEz % loop for z-kinetic energy
63
% self energies==================================== ===== % left contact majority electrons================ ck=1-((Ez(jz)+inu-Ec_MM(1)-Udiag(1))/(2*t_ct)); ka=real(acos(ck)); sig_L_maj=-t_ct*exp(i*ka); % right contact majority electrons=============== ck=1-((Ez(jz)+inu-Ec_MM(Np)-Udiag(Np))/(2*t_ct)); ka=real(acos(ck)); sig_R_maj=-t_ct*exp(i*ka); % left contact minority electrons================ ck=1-((Ez(jz)+inu-Ec_mm(1)-Udiag(1))/(2*t_ct)); ka=real(acos(ck)); sig_L_min=-t_ct*exp(i*ka); % right contact minority electrons=============== ck=1-((Ez(jz)+inu-Ec_mm(Np)-Udiag(Np))/(2*t_ct)); ka=real(acos(ck)); sig_R_min=-t_ct*exp(i*ka); % parallel polarization============================ == sig_L_F(1:2,1:2)=[sig_L_maj 0; 0 sig_L_min]; sig_R_F(Np2-1:2*Np,Np2-1:Np2)=[sig_R_maj 0; 0 sig_R_min]; gam_L_F=i*(sig_L_F-sig_L_F'); gam_R_F=i*(sig_R_F-sig_R_F'); % anti-parallel polarization======================= == sig_L_AF(1:2,1:2)=[sig_L_maj 0; 0 sig_L_min]; sig_R_AF(Np2-1:Np2,Np2-1:Np2)=[sig_R_min 0; 0 sig_R_maj]; gam_L_AF=i*(sig_L_AF-sig_L_AF'); gam_R_AF=i*(sig_R_AF-sig_R_AF'); %================================================== == % self energy====================================== ====== sig_F=sig_L_F+sig_R_F; sig_AF=sig_L_AF+sig_R_AF; % gamma coupling=================================== ====== gam_F=gam_L_F+gam_R_F; gam_AF=gam_L_AF+gam_R_AF; % in-scattering matrices=========================== =========== sigin_L_F=gam_L_F*f2D_L(jz); sigin_R_F=gam_R_F*f2D_R(jz); sigin_L_AF=gam_L_AF*f2D_L(jz); sigin_R_AF=gam_R_AF*f2D_R(jz); sigin_F=sigin_L_F+sigin_R_F; sigin_AF=sigin_L_AF+sigin_R_AF; % out-scattering matrices========================== =========== sigout_L_F=gam_L_F*f2D_L_p(jz); sigout_R_F=gam_R_F*f2D_R_p(jz); sigout_L_AF=gam_L_AF*f2D_L_p(jz); sigout_R_AF=gam_R_AF*f2D_R_p(jz); sigout_F=sigout_L_F+sigout_R_F; sigout_AF=sigout_L_AF+sigout_R_AF; %================================================== == %================================================== == % parallel polarization============================ =========== %================================================== ==
64
% calculate Green’s function======================= ============ [G_F,sig_scat,it_GF(jz)]=… greens_func_delta(H_F,U,Ez(jz),D,sig_F,N_source,N_barr,N_drain); % calculate correlation function=================== ============== [Gn_F] = corr_func_delta_direct(G_F,D,sigin_F,N_source,N_barr,N_drain); A_F=-2*imag(G_F); % current ========================================= ===== II_F(jz)=real(trace(sigin_L_F*A_F)-trace(gam_L_F*Gn_F)); % current density method=========================== ========= for jj=1:Np-2 ll=2*jj-1; II_F_curr_density_lenght(jj)=II_F_curr_density_lenght(jj)… +2*real(trace(H_F(ll:ll+1,ll+2:ll+3)*i*Gn_F(ll+2:ll+3,ll:ll +1))); end % coherent transport regime======================== =========== G_F_c=inv((Ez(jz)+inu)*eye(Np2)-H_F-U-sig_F); % transmission parallel configuration============== ============= T_F(jz)=real(trace(gam_L_F*G_F_c*gam_R_F*G_F_c')); A_F_c=i*(G_F_c-G_F_c'); II_F_c2(jz)=T_F(jz)*(f2D_L(jz)-f2D_R(jz)); %================================================== == % parallel polarization============================ =========== %================================================== == % calculate Green’s function======================= ============ [G_AF,sig_scat,it_GAF(jz)]=… greens_func_delta(H_AF,U,Ez(jz),D,sig_AF,N_source,N_barr,N_drain); % calculate correlation function=================== ============== [Gn_AF] = corr_func_delta_direct(G_AF,D,sigin_AF,N_source,N_barr,N_drain); A_AF=-2*imag(G_AF); % current ========================================= ===== II_AF(jz)=real(trace(sigin_L_AF*A_AF)-trace(gam_L_AF*Gn_AF)) % current density method=========================== =========
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for jj=1:Np-2 ll=2*jj-1; II_AF_curr_density_lenght(jj)=II_AF_curr_density_lenght(jj)… +2*real(trace(H_AF(ll:ll+1,ll+2:ll+3)*i*Gn_AF(ll+2:ll+3,ll:ll+1 ))); end % coherent transport regime======================== =========== G_AF_c=inv((Ez(jz)+inu)*eye(Np2)-H_AF-U-sig_AF); % transmission parallel configuration============== ============= T_AF(jz)=real(trace(gam_L_AF*G_AF_c*gam_R_AF*G_AF_c')); A_AF_c=i*(G_AF_c-G_AF_c'); II_AF_c2(jz)=T_AF(jz)*(f2D_L(jz)-f2D_R(jz)); end % loop for z-kinetic energy ends %========================================= ========= TMR_Ez=(II_F-II_AF)./II_F; % tunneling magnetoresistance energy resolved=== % ================================================= = I_F=sum(II_F); % Parallel Configuration I_AF=sum(II_AF); % Anti-Parallel Configuration % ================================================= == TMR(tt,jn)=(I_F-I_AF)/I_F; % tunneling magnetoresistance============== % ================================================= = % ================================================= == TMR_c_Ez=(II_F_c-II_AF_c)./II_F_c; % TMR_coherent_Ez=============== % ================================================= = I_F_c=sum(II_F_c); % Parallel Configuration I_AF_c=sum(II_AF_c); % Anti-Parallel Configuration % ================================================= ==== TMR_c(tt,jn)=(I_F_c-I_AF_c)/I_F_c; % tunneling magnetoresistance=========== % ================================================= ==== % ================================================= ==== TMR_curr_density_lenght=(II_F_curr_density_lenght-II_AF_curr_density_lenght)… ./II_F_curr_density_lenght; % tunneling magnetoresistance======= % ================================================= ====
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TMR_curr_density(tt,jn) = TMR_curr_density_lenght(1); end % impurity concentration loop ends end % barrier thickness loop ends toc profile viewer save output.mat
VITA
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VITA
Ahmet Ali Yanik was born in Tekirdağ, Turkey in June, 1979. He received his B.Sc.
degree in Physics in 1999 from Orta Doğu Teknik Üniversitesi, Turkey. In August, 2000,
he started his Ph. D. study in the Department of Physics at Purdue University, West
Lafayette, IN. Since June 2002, he has been working under the supervision of Prof.
Supriyo Datta for NSF Network for Computational Nanotechnology. His current research
focuses on device physics and spin based electronics in nanostructures. His previous
research includes semiconductor device fabrication and high speed nonlinear optics.
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