Nonvolatile Spin Memory based on Diluted Magnetic Semiconductor and Hybrid Semiconductor Ferromagnetic Nanostructures

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ENAYA, HANI A. Nonvolatile Spin Memory Based on Diluted Magnetic Semiconduc-tor and Hybrid SemiconducSemiconduc-tor Ferromagnetic Nanostructures. (Under the direction of Professor Ki Wook Kim).

The feasibility of two nonvolatile spin-based memory device concepts is explored.

The first memory device concept utilizes the electrically controlled

paramagnetic-ferromagnetic transition in a diluted magnetic semiconductor layer (quantum well or

dot) when the ferromagnetism in the diluted magnetic semiconductor is mediated

by itinerant holes. The specific structure under consideration consists of a diluted

magnetic semiconductor quantum well (or quantum dot) that exchanges holes with a

nonmagnetic quantum well, which acts as a hole reservoir. The quantitative analysis

is done by calculating the free energy of the system. It takes into account the energy

of holes confined in a nanostructure and the magnetic energy. Formation of two

stable states at the same external conditions, i.e., bistability, is found feasible at

temperatures below the Curie temperature with proper band engineering. The effect

of scaling the magnetic quantum well to a quantum dot on bistability is analyzed.

The bit retention time, i.e., lifetime, with respect to spontaneous leaps between the

two stable states is calculated. The write/erase and read operations as well as the

dissipation energy are discussed. Also, potential logic operations are proposed. In the

second memory concept, the active region is a semiconductor quantum dot sharing

an interface with a dielectric magnetic layer. The operating principle of the device is

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the quantum dot. The quantitative analysis considers the holes thermal distribution

over the energy spectrum in the quantum dot, hole-hole interaction, exchange

inter-action between the holes and the magnetic ions, and magnetic energy of the magnetic

insulator. Room temperature operation is possible given the availability of

insulat-ing ferromagnetic or antiferromagnetic materials whose Curie or N´eel temperature is

above room temperature. The specific range of material parameters where bistability

is achieved is found. Analysis is extended to different quantum dot and magnetic

dielectric materials and designs. The influence of material choice and design on the

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by Hani A. Enaya

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

in partial fullfillment of the requirements for the Degree of

Doctor of Philosophy

Electrical Engineering

Raleigh, North Carolina

2008

APPROVED BY:

Dr. Ki Wook Kim Chair of Advisory Committee

Dr. Salah Bedair Dr. Veena Misra

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DEDICATION

To

My parents Abdulahad and Noura

My wife Samah

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BIOGRAPHY

Hani Enaya graduated summa cum laude as a valedictorian with a B.S. degree

in electrical engineering and physics from North Carolina State University in 2002.

Hani joined the theoretical nanoscale quantum engineering group at NCSU as a

re-search assistant and pursued his graduate studies in Electrical Engineering under the

direction of Dr. Ki Wook Kim. He received his M.S. degree in 2003 and Ph.D. degree

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ACKNOWLEDGMENTS

This work could not have been completed without the support and guidance of

many people. First, I would like to express my sincere gratitude to Dr. Ki Wook

Kim for giving me the opportunity to be part of his research group. This work would

have never materialized without Dr. Kim’s guidance, critique, and patience. I am

also grateful to Dr. Yuriy Semenov who was always available to discuss my work in

length and provided me with unique insight.

I extend my sincere thanks to Dr. Salah Bedair, Dr. Jacqueline Krim, Dr. Veena

Misra, and Dr. John Zavada, whom I am deeply honored to have in my Ph.D.

committee. Thank you Dr. Bedair and Dr. Misra for all the fruitful discussions.

Thank you Dr. Krim for the support and encouragement. Thank you Dr. Zavada for

the research support and for closely following my research progress over the past few

years.

During my graduate years, my group members have always been there when

needed. I thank Kostyantyn, Byoung-Don, and Ned for all the help and

encour-agement. I also thank Ms. Kaye Baliy for her help with various administrative tasks

and handling cumbersome paperwork.

My deepest thanks go to my parents, Abdulahad and Noura, whose love and

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encouragement always keep me going. I am grateful to my wife, Samah, who has

always been there for me and sacrificed her time after her graduation to stay home

with our daughters, Noura and Reema. Her commitment to the family and the girls

during my Ph.D. years is something that I will always remain in debt to. Thank you

Noura and Reema for the pleasure you bring to my life. You always inspire me to be

a better person.

I am blessed with many close friends who were my extended family and made all

these years possible. I thank my friends Wael, Ahmed, Joud, Bader, Mohammad,

and Tariq. Our gatherings and discussions will always bring nice memories, and they

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LIST OF TABLES

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LIST OF FIGURES

Figure 2.1 Schematic energy diagram (valence band) of the structure in the two coexisting stable states. The first stable state corresponds to a depopu-lated diluted magnetic semiconductor (DMS) quantum well (QW), which would be paramagnetic (PM). In this state, the holes reside in the non-magnetic (NM) QW. The second stable state corresponds to a populated DMS QW, which would be ferromagnetic (FM). The mutual alignment of the hole spins (large arrows) and localized spins (small arrows) reduces the energy in the FM QW byEexch'2FM/n0hdue to their exchange interaction.

The switching between these states (arch arrows) is achieved by applying an appropriate bias pulse on the gate electrode. . . 25

Figure 2.2 Free energy F(M) below and above Curie temperature TC (Landau

mean field theory). . . 26

Figure 2.3 Magnetization of a ferromagnet as a function of temperature (Landau mean field theory). . . 27

Figure 2.4 Free energy trial functionF(η) at different values ofu(= ∆U/kBTC0).

Three different scenarios are shown: curve 1 (u = 29) - a monostable case with holes occupying the NM QW (η = 0), while the DMS QW is in a PM phase; curve 2 (u= 5) - a bistable case where the PM and FM phases coexist; curve 3 (u = −9) - a monostable case with holes populating the DMS QW (η = 1) in the FM phase. . . 28

Figure 2.5 Phase diagram of the parameter space indicating the potential bista-bility region.. . . 29

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state energyU (defined with no magnetic contribution) is near or above the chemical potential µ0 of the hole reservoir. Right: Another thermodynam-ically stable state (at the same external conditions) is possible when the magnetic ions are in the ferromagnetic (FM) phase. Magnetic interactions can decrease the hole potential so that the bottom of the DMS QD is now substantially belowµ0 ; i.e., the equilibrium hole population is high enough

to stabilize the FM phase. Switching between the PM and FM states can be achieved by applying a gate bias. Vg can proportionally shift the DMS QD

potential energy profile up/down populating or depopulating the structure. NQW stands for non-magnetic quantum well. . . 51

Figure 3.2 Radius of the hole localization area in the FM state of the DMS QD vs. confinement energy at T = 70 K. The material parameters of Ga0.95Mn0.05As are assumed with TC0 = 110 K. . . 52

Figure 3.3 Radius of the hole localization area in the FM state of the DMS QD vs. temperature for three different confinement energies: (1) ~ω = kBTC0;

(2) ~ω= 1.3kBTC0; (3) ~ω= 1.6kBTC0. Other parameters are the same as in

Fig. 3.2 . . . 53

Figure 3.4 Chemical potential of the DMS QD (in reference toU) calculated as a function of the numbers of trapped holes. The parameters of Ga0.95Mn0.05As

are assumed withT = 70 K andT0

C = 110 K. The QD dimensions are taken

to be 5 nm (thickness) and ~ω = kBTC0 (lateral confinement energy). The

solutions of equations (3.16) and (3.18) can be found as the intersections of the solid curve with a horizontal line corresponded to a certain value of µ0.

Two cases of µ0−U (−0.5kBTC0 - line 1; 5kBTC0 - line 3) correspond to the

monostable PM and FM, while line 2 in the middle (µ0 −U = 2.3kBTC0)

depicts the bistable state. Stable solutions are indicated by single head arrows and the unstable one by the horizontal double-head arrow.. . . 54

Figure 3.5 Phase diagram of the parameter space indicating the potential bista-bility region. The PM and FM labels denote the monostable areas corre-sponding to the PM and FM QD states, respectively. The same parameters as in Fig. 3.4 are assumed. . . 55

Figure 3.6 Free energy of the QD calculated as a function of hole population for three different values of (µ0−U)/kBTC0 =−0.5 (curve 1); 2.3 (curve 2);

5 (curve 3). The single minima of curves 1 (PM phase) and 3 (FM phase) correspond to the vicinities of the left and right boundaries of the bistable area in Fig. 3.5; curve 2 represents the bistable case with the optimal free energy barrier height separating the two local minima. The same parameters as in Fig. 3.4 are assumed (T = 70 K, T0

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Figure 3.7 Bistability lifetime vs. temperature. Three different parabolic QD confinement energies are considered: (1)~ω=kBTC0; (2)~ω = 1.3kBTC0; (3)

~ω= 1.6kBTC0. The mean timeτ0 of particle exchange between the QD and

the reservoir via thermal activation is assumed to be 1 ns. Other parameters are the same as in Fig. 3.4. . . 57

Figure 3.8 Hysteresis loop showing the variation of hole population in the QD as a function of applied bias. The upper (lower) solid curve represents FM (PM) state of the QD. At the zero applied voltage, the system can be found either in the ”0” (PM) or ”1” (FM) state. ∆UW is a writing signal of large

enough amplitude and duration τ0. Applying ∆UW will cause the system

to progress to point F. The system will evolve to the ”1” state after the voltage is turned off. ∆URis a readout signal of small amplitude and period

f_R−1 that will not alter the state of the memory cell. If the system is in state ”0” (”1”), ∆URinduces a currentI0 (I1). The inequalityI1 > I0 takes place

because the slope of the upper portion of the loop is larger than that of the lower portion. The same parameters, as case 2 in Fig. 3.4, are assumed. . . 58

Figure 3.9 Circuit representation of the proposed memory cell. CQD(RQD)

de-notes the capacitance (resistance) of the hole transfer between the QD and the reservoir. CD is the capacitance between the gate electrode and the hole

reservoir, and R0 accounts for the remaining contributions to the circuit

resistance. . . 59

Figure 3.10 Ratio of FM to PM readout current (or conductance) as a function of the small signal frequency at two different values ofτ0. The same parameters

as in Fig. 3.4 are assumed (T = 70 K, T0

C = 110 K, etc.). . . 60

Figure 3.11 Hysteresis loop showing the operation of logic gates. The logic oper-ation AND is realized at the gate bias points where the ”0” to ”1” transition is allowed only for the sum of two simultaneous ”High” (H) input pulses. The logic operation OR is realized at the gate bias points where the sum of ”Low” (L) and ”High” pulses is sufficient to trigger the ”0” to ”1” transition. The same parameters as in Fig. 3.4 are assumed. . . 61

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along the growth direction (state ”1”) (collective spin polaron formation). Hole exchange between the QD and a nonmagnetic quantum well (NM QW), which acts as a hole reservoir, is controlled by applied bias pulses Vg. Note

that the holes in the QD and the corresponding magnetic ions in the FM layer can equivalently be polarized in the opposite directions compared to the case shown for state ”1”. . . 82

Figure 4.2 An enlarged view of the two memory cells in different states high-lighting an unperturbed FM layer with nearly empty QD (state ”0”) and collective spin polaron formation in the FM layer with sufficiently populated QD (state ”1”).. . . 83

Figure 4.3 Magnetic rotation in the FM layer vs. hole occupancy right above the center of the QD.θ0 = 0 corresponds to an unperturbed magnetic layer

(hole spin is perpendicular to the layer magnetization), while θ0 = π/2

corresponds to maximal magnetic rotation (hole spin is antiparallel to the layer magnetization). . . 84

Figure 4.4 Variational parameters b1 and b2 (normalized by QD radius RQD)

vs. hole occupancy. A larger b1 and b2 indicate a larger polaron area right

above the QD. . . 85

Figure 4.5 Collective spin polaron formation showing the magnetic orientation vs. distance from the center of the QD when the memory cell is in the second stable state, where the QD is sufficiently populated (j ∼ 18). The FM layer thickness Lm is 5 nm. mz = 1 corresponds to maximal rotation in

the FM layer.. . . 86

Figure 4.6 Chemical potential of the QD with a 2 nm thickness and a cross section of 30 × 30 nm2_{. Typical parameters of GaAs are used at room}

temperature. The intersections of the solid curve with the horizontal lines are the solutions of equation (4.14). The horizontal lines correspond to certain values of =µ0−U. The two extreme cases ofµ0−U (−3kBT - line

1; −7kBT - line 3) correspond to the monostable states, while a moderate

value of(µ0 − U = −5.7kBT - line 2) depicts the bistable state. Stable

solutions are indicated by single head arrows and the unstable ones by the horizontal double-head arrow. . . 87

Figure 4.7 Diagram of the magnetic parameter space showing the potential bistability region. The same parameters as in Fig. 4.6 are used. . . 88

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reservoir via thermal activation is assumed to be 1 ns. Other parameters are the same as in Fig. 4.6. . . 89

Figure 4.9 Schematic illustration of different material and design choices for the presented memory concept. . . 90

Figure 4.10 Free energy vs. the hole occupancy for a NM QD and a diluted magnetic semiconductor (DMS) QD. The minima indicate stable states, while the maxima indicate unstable states. The higher energy difference between the stable states and the unstable state in the case of the DMS QD is indicative of a higher bit retention time. Parameters are the same as in Fig. 4.6. . . 91

Figure 4.11 Stable states lifetime vs. the lateral dimension of a NM QD (GaAs) and a DMS QD (Ga0.8Mn0.2As). The mean time τ0 of particle exchange

between the QD and the reservoir via thermal activation is assumed to be 1 ns. Other parameters are the same as in Fig. 4.6. . . 92

Figure 4.12 Collective spin polaron formation showing the magnetic orientation vs. distance from the center of the QD when the memory cell is in the second stable state, where the QD is sufficiently populated (j ∼ 21). The FM layer thickness Lm is 20 nm. mz = 1 corresponds to maximal rotation

in the FM layer. . . 93

Figure 4.13 Stable states lifetime vs. the lateral dimension of a NM QD for three different designs: a 5 nm FM layer (curve 2), 20 nm FM layer (curve 3), and 20 nm FM layer with the QD embedded in it (curve 1). The mean timeτ0 of

particle exchange between the QD and the reservoir via thermal activation is assumed to be 1 ns. Other parameters are the same as in Fig. 4.6.. . . 94

Figure 4.14 Stable states lifetime vs. the lateral dimension of a NM QD for two different magnetic layer materials: NM QD adjacent to a 5 nm NiO layer with exchange energy Eex−1 = 9kBT (curve 3), NM QD adjacent to a 5

nm FM layer (Eex−1 = 9kBT) (curve 2), NM QD adjacent to a 5 nm NiO

at Eex−2 = 1.25·Eex−1 (curve 1). The mean time τ0 of particle exchange

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Chapter 1 Introduction

Semiconductor electronics and the memory technology have been the backbone

of the information revolution. The continuing scaling of semiconductor devices leads

to reaching a limit where quantum effects like tunneling become unavoidable. Even

though technologists were very successful in avoiding the quantum limit, a promising

field, spintronics or spin-based electronics, has the potential to provide unique

ad-vantages over conventional electronics. While conventional electronics are based on

utilizing the electric charge to store and manipulate information, spintronics function

by taking advantage of the spin state of the carrier (electron or hole) in addition or

instead of its electric charge. The term was first introduced in 1994 by S. A. Wolf

as a U.S. Defense Advanced Research Project Agency (DARPA) program [1]. The

program main purpose was to develop advanced magnetic memory and sensors based

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A spintronic device based on a metal-insulator-semiconductor structure was

theo-retically proposed in 1990 by Datta and Das [2]. In the conventional

metal-insulator-semiconductor field-effect transistor, electric charges (electron or holes) are introduced

in the channel via the source terminal and collected at the drain terminal. The third

terminal, the gate, produces a vertical electric field that effectively controls the

car-riers flow in the channel. In the spin-based counterpart, carcar-riers enter the channel

after passing through a ferromagnetic electrode (source). As a result, the spins of the

carriers are oriented along the direction of the source magnetization. The electric field

generated by the gate controls the precession of the carrier spins in the channel. On

the other side of the channel, a ferromagnetic electrode (drain) acts like a spin filter,

accepting only carriers with the same spin as the direction of the drain magnetization.

Hence, in order to realize the device, it is a necessity to be able to inject spin

polar-ized carriers at the source, modify the carrier spins via gate voltage, and selectively

collect the carriers based on their spin orientation at the drain. Researchers in the

spintronics field have focused mainly on the mentioned challenging requirements. A

further review of the subject matter can be found in [1], [3]–[7].

A promising application of spintronics is spin-based memory devices [1], [7], [8].

The ultimate goal in the memory technology field is to have a universal

semiconductor-based memory that is capable of delivering the speed of static and dynamic random

access memories and nonvolatility of FLASH memory. Random access memories are

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off. On the other hand, FLASH memory is more expensive and much slower (∼ms), but it retains data even after the power is switched off (nonvolatile).

As a matter of fact, a success story in spintronics research was the development

of the hard disk memory technology. A spintronic technology that had an enormous

commercial success is the giant magnetoresistive (GMR) structure, whose resistance

depends on the current and the magnetic orientation of the respective magnetic layers

in the structure. Gr¨unberg et al. [9] discovered in 1986 the antiferromagnetic

cou-pling between ferromagnetic Fe layers across a nonmagnetic Cr layer. In 1988, Fret

group [10] observed the GMR effect in Fe/Cr superlattices. These discoveries lead to

the development of sensitive hard disk read heads consisting of magnetic layers. The

operation mechanism in reading hard disk data relies entirely on the GMR effect.

The result is a tremendous increase in the storage capacity of hard disks.

This dissertation aims to theoretically explore the feasibility of two distinct

spin-based memory concepts. The memory concepts discussed here rely on spin correlation,

where the interplay between spin order and spin disorder in a semiconductor

struc-ture is utilized. Such an approach does not require the challenging task of injecting

polarized carriers and dealing with issues surrounding spin transport.

Chapter 2 discusses the first memory concept, where the active layer is composed

of a diluted magnetic semiconductor quantum well separated from a nonmagnetic

quantum well. The mechanism of operating the cell is based on altering the magnetic

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quantum well by controlling the hole distribution in the quantum wells via a gate

bias. A central requirement for the diluted magnetic semiconductor is that

ferromag-netism to be mediated by free carriers, e.g., GaMnAs. The parameter space including

the operating temperature, where the memory effect is expected to be observed, is

demonstrated. A major challenge to achieving room temperature operation is the

availability of a diluted magnetic semiconductor that is ferromagnetic at

tempera-tures higher than room temperature.

Chapter 3 analyzes the effects of scaling the memory device discussed in Chapter

2 by reducing the lateral dimensions of the diluted magnetic semiconductor quantum

well, i.e., going from a quantum well to quantum dot. The formation of bistability,

i.e., memory effect, is investigated as well as the conditions necessary for bistability.

To quantify the effect of scaling, the lifetime of the memory states, i.e., bit retention

time, is estimated as function of the quantum dot size and temperature. Moreover,

the write/erase and readout schemes, energy dissipation, potential logic operations

are discussed.

Chapter 4 explores a distinct memory device concept from Chapter 2 and 3. The

active layer of the proposed memory device consists of a nonmagnetic quantum dot

sharing an interface with an insulating ferromagnetic (ferrimagnetic or

antiferromag-netic) layer whose Curie or N´eel temperature is much higher than room temperature.

The operation is based on manipulating the direction of the magnetization of the

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quantum dot. The effect of different designs and materials on the memory lifetime is

discussed as well.

The work discussed in Chapter 2 and 3 was partially published in the Applied

Physics Letter [11] and IEEE Transactions on Electron Devices [12], respectively.

The work presented in Chapter 4 has been accepted for publication in the IEEE

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Chapter 2 Memory Effect Based on Electric

Tuning of Ferromagnetism in a

Diluted Magnetic Semiconductor

Quantum Well

2.1 Diluted Magnetic Semiconductors

Ferromagnetic semiconductors can be categorized into two groups: magnetic

conductors and diluted magnetic semiconductors. Magnetic semiconductors are

semi-conductors that have a periodic array of magnetic ions in their crystal structure. Most

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and chromium based spinels like CdCr2Se4 [15].

In diluted magnetic semiconductors, magnetic ions are introduced into the crystal

of the nonmagnetic semiconductor host. Only a fraction of the host ions is substituted

by the magnetic ions, which introduce local magnetic moments. Early studies in the

1980s of diluted magnetic semiconductors were based on Mn-doped II-VI

semicon-ductor compounds (e.g., CdMnTe) [16], [17]. The manganese ions easily substitute

the cation sites since they both share the same valence. Nonetheless, most of II-VI

compounds exhibit a paramagnetic state, and ferromagnetic ordering only occurs at

very low temperatures, i.e., Curie temperature is low (TC ≈1.8 K in CdMnTe [16]).

A breakthrough occurred to the research of diluted magnetic semiconductors in

1992 when H. Ohno et al. [18] demonstrated ferromagnetism in Mn-doped InAs

(In-MnAs) with TC ≈ 8 K. This discovery was followed by observing ferromagnetism

in GaMnAs with TC ≈ 60 K in 1996 [19]. InMnAs and GaMnAs to date are the

most studied diluted magnetic semiconductors, where the mechanism causing the

ferromagnetic ordering is well established as discussed below.

2.1.1 Progress on GaMnAs

In GaMnAs, Mn substitutes Ga and acts as a shallow acceptor. Hence, Mn

intro-duces local magnetic moments to the host material and provides holes, which mediate

the ferromagnetic interaction between the local magnetic moments of the Mn ions.

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correla-tion between the Curie temperature and the hole concentracorrela-tion [20] and the Curie

temperature and the Mn concentration [17]. As the Mn concentration increases, both

the hole concentration and the Curie temperature increase. However, after the Mn

concentration x reaches a critical value (e.g., x = 0.053 with TC = 110 K in [17]),

the Curie temperature decreases steadily as more Mn ions are introduced to the host

material.

Since incorporating a large concentration of Mn in the GaAs crystal can only

be accomplished by low temperature molecular beam epitaxy, high density of point

defects are created [21]. The relevant point defects are As antisites and Mn

intersti-tials that act as double donors. This results in compensating a fraction of the free

holes introduced by substitutional Mn [21], [22]. The result is a drop in the Curie

temperature.

However, post-growth low temperature annealing of GaMnAs samples leads to

a drastic enhancement of the Curie temperature [20], [23]–[27]. Annealing

experi-ments demonstrated that the Curie temperature of samples with initial low Curie

temperature can be increased for a wide range of sample compositions and growth

conditions. The reported strong correlation between the location of the Mn sites in

ferromagnetic GaMnAs samples and their Curie temperature [22] strongly supports

the picture that annealing reduces the population of the Mn interstitials.

Further-more, both the GaMnAs layer thickness and the capping of the GaMnAs layer have a

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Curie temperature increases as the GaMnAs layer thickness decreases [20], while the

capping of the GaMnAs layer reduces the Curie temperature [27]. The Curie

tem-perature enhancement or reduction is related to the diffusion of Mn interstitials [21].

Thermal annealing increases the Curie temperature by driving the Mn interstitials

to the free surface. Hence, the free hole concentration increases and the Curie

tem-perature subsequently increases [28]. The current record for Curie temtem-perature of

GaMnAs is ∼170 K, which is achieved by low temperature annealing and careful design of the sample heterostructure [26].

2.1.2 Theoretical Picture

It is widely accepted that ferromagnetism is carrier mediated in many diluted

magnetic semiconductors like InMnAs, GaMnAs, and GeMn. Carrier mediated

fer-romagnetism can be explained in the framework of the mean field theory. The theory

predicts that the ferromagnetic ordering is strongly correlated to the hole-ion

ex-change interaction constant and the hole population.

The mean field approximation of ferromagnetism in diluted magnetic

semicon-ductors has been sufficient to explain the hole-induced ferromagnetism in GaMnAs,

the most studied diluted magnetic semiconductor. The theory is based on the

self-consistent mechanism between the magnetic ions polarization and carriers

polariza-tion. A brief description is given below following [6], [29], [30]. A comprehensive

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elsewhere [31]–[39].

In the mean field theory, the strong kinetic exchange energy between the free

carriers (holes) and magnetic ion spins is responsible for the observance of carrier

mediated ferromagnetism. The kinetic exchange Hamiltonian reads

HKE =

Z

d3_rX

j

JpdSj·s(r)δ(r−Rj) (2.1)

where Jpd characterizes the exchange interaction strength between the magnetic ion

(Mn) spin Sj located at Rj and the local hole spin density s(r). In the mean field

approximation, the action of the spin hole density on a single magnetic moment

and the action of the total magnetic moments on a local hole spin are expressed

in terms of effective fields. This approach leads to the important finding that the

Curie temperature is proportional to the hole densitypand the coupling constantJpd

(TC ∝Jpdp1/3 for the bulk case).

An opposing mechanism to the ferromagnetic kinetic exchange interaction is the

superexchange interaction, which is the direct antiferromagnetic exchange interaction

among the magnetic ions. It can be described by the following Hamiltonian

HAF =

X

ij

J_ijAFSi·Sj (2.2)

whereJAF

ij is the antiferromagnetic exchange constant. Accounting for this interaction

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2.1.3 Electric Control of Ferromagnetism

Manipulating the magnetic phase in a diluted magnetic semiconductor layer by

external means like electric bias at a fixed temperature is an important step toward

device application. In addition, electric control of ferromagnetism provides a definite

proof of carrier mediated ferromagnetism. A number of key experiments have already

demonstrated such control [40]–[45].

In 2000, H. Ohno et al. [40] demonstrated electric control of the magnetic phase

of a 5 nm thick InMnAs layer by application of gate voltage. The InMnAs layer is

incorporated as a channel layer in a metal-insulator-semiconductor field-effect

tran-sistor structure. The device is processed into a Hall bar in order to measure the Hall

resistance RHall, which is proportional to the magnetization of the sample [40]. Then

if theRHall versus magnetic field H curve shows a hysteresis below a critical

temper-ature TC, InMnAs would be in the ferromagnetic phase. At a temperature slightly

below TC, a weak hysteresis appears when no gate voltage is applied. However,

ap-plication of a negative gate voltage enhances the hysteresis, whereas apap-plication of

a positive gate voltage reduces or destroys the hysteresis (paramagnetic response).

Note that the negative gate voltage populates the InMnAs channel with holes, and

the positive gate voltage depletes the channel from holes. In addition, a change in the

Curie temperature is deduced when applying gate voltage. This demonstrates that

ferromagnetism in this material is indeed hole mediated.

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in a Mn δ-doped GaAs/AlGaAs heterostructure. Electric control of ferromagnetism

in a GaMnAs layer was later achieved in 2006 by D. Chiba et al. [44]. In all the

aforementioned experiments, the investigators have shown the modulation of Curie

temperature when applying external electric field. Moreover, electric control of

ferro-magnetism in a group IV diluted magnetic semiconductor, namely GeMn, has been

achieved by Chen et al. [45]. In this experiment, a Ge layer was doped with Mn via

a nanopatterned mask forming in effect GeMn quantum dots.

2.1.4 Room Temperature Ferromagnetism

The theoretical prediction in 2000 by Dietl et al. [31] that room temperature

ferromagnetism is feasible in some diluted magnetic semiconductors and oxides has

resulted in tremendous experimental efforts to achieve room temperature

ferromag-netism in wide band gap semiconductors and oxides [46]–[54]. The calculations were

based on the mean field model of ferromagnetism. The model predicted room

tem-perature ferromagnetism in 5% Mn-doped p-GaN and p-ZnO [31].

Since then, room temperature ferromagnetism has been observed in Mn-doped

GaN [46]–[48] and Cr-doped GaN [49]. However, the position of Mn in GaN is reported

to be very deep (Ev+ 1.4eV [55], [56]). Hence, Mn would not be an acceptor dopant,

and pd hybridization would not be feasible. These observations in addition to the

insulating nature of GaMnN do not agree with the mean field model. The mechanism

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Nonetheless, M. Reed et al. [48] reported a strong correlation between the

mag-netic properties of GaMnN and intentional doping with shallow impurities.

Specifi-cally, the hysteresis strength and the saturation magnetization at room temperature

show a remarkable dependence on co-doping with Si donors and Mg acceptors. The

investigators attributed this dependence to the position of the Fermi level, which is

affected by co-doping, relative to the magnetic impurity band.

For group IV semiconductors, ferromagnetism has been observed in GeMn with

TC ∼ 250 K [57] and CoMnGe [58]. Room temperature ferromagnetism has been

observed in thin GeMn layers [59] and GeMn quantum dots [45]. The hole mediated

ferromagnetism picture in GeMn is supported by the successful demonstration of the

electric control of the magnetic phase [45].

2.2 Nonvolatile Memory Based on Diluted

Mag-netic Semiconductor Quantum Wells

2.2.1 Device Concept

This section discusses an innovative spin-based memory cell whose active region

is a diluted magnetic semiconductor quantum well. The basic operation principle is

based on controlling the paramagnetic-ferromagnetic transition via electric control of

the hole population in the diluted magnetic semiconductor quantum well.

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mag-netic quantum well and a nonmagmag-netic quantum well of widthsLwM andLwN,

respec-tively, separated by a barrier of widthLb. The total two-dimensional hole

concentra-tion n0

h is assumed to be a constant. The paramagnetic-ferromagnetic transition in

the diluted magnetic semiconductor quantum well is assumed to be mediated by free

holes. In addition, the width and the height of the barrier separating the

nonmag-netic and magnonmag-netic layer should be large enough to form noncoherent single quantum

well states and yet not too large in order to enable hole redistribution between the

magnetic and nonmagnetic layer when a gate bias is applied.

When the hole energy in the nonmagnetic quantum well is lower than that in

the magnetic quantum well, it is expected that the stable state will correspond to

hole localization primarily in the nonmagnetic quantum well with a small leakage in

the magnetic quantum well, which is in a paramagnetic phase due to the low hole

population (see the schematic on the left in Fig. 2.1). When a proper bias is applied,

holes from the nonmagnetic quantum well are transferred to the magnetic quantum

well via tunneling or over-barrier injection. As the hole population in the magnetic

quantum well increases and surpasses a certain threshold at a given operating

tem-perature, the layer undergoes the paramagnetic-ferromagnetic transition. When the

hole exchange interaction with the ferromagnetic-ordered ion spins in the magnetic

quantum well is strong, it can reduce the total free energy below that of the initial

state with the paramagnetic phase even after the bias is switched off. Then, the holes

(30)

be maintained (the schematic on the right in Fig. 2.1). When a reverse bias pulse is

applied, the holes are drained out of the magnetic quantum well into the nonmagnetic

quantum well, and the magnetic quantum well will return to the paramagnetic state.

Hence, if realized, these two stable states can coexist under the same external

condi-tion (i.e., bistability) and the switching between them is mediated by the electrically

controlled paramagnetic-ferromagnetic phase transition [40]–[45]. It is expected that

the structure can operate up to a temperature slightly below the saturated maximum

of the Curie temperature TC, which can reach room temperature or higher in some

material systems. Bistability is expected also when a magnetic quantum dot is used

instead of a magnetic quantum well. The principal difference between the two cases

is the finiteness of the hole population when considering a quantum dot. As a result,

lifetime calculations becomes a necessity as discussed in Chapter 3.

2.2.2 Theoretical Model

To analyze this problem, one would typically derive the magnetic HamiltonianHm,

which describes the effect of the effective ferromagnetic inter-ion spin-spin interaction

in the presence of free holes. The calculation of the free energy F of the total system

consisting of magnetic ions and free carriers, which occupy the magnetic quantum well

with a concentration of nhM and the nonmagnetic quantum well with nhN (nhM +

nhN = n0h), leads to the carrier population factor η = nhM/n0h and 1−η = nhN/n0h

(31)

with respect to η demonstrates the bistability of the system under consideration.

Although conceptually correct, this approach faces the difficulty of specifying the

mechanisms responsible for ferromagnetic ordering in the magnetic quantum well

(for details on various mechanisms, refer to those cited in [29]–[39]). To

circum-vent this problem, a semiphenomenological approach is developed, which utilizes the

data extracted from routine experimental measurements of magnetism. Namely, the

magnetic part of the free energy FM is expanded with the magnetization M (order

parameter) according to Landau theory. This expansion approximates satisfactorily

the veritable dependence in the whole temperature interval under consideration.

Magnetic Energy

According to Landau theory, the free energy of a ferromagnet is written as an

even power series in M, the magnetization. Assuming the easy magnetization axis is

directed along the growth axis of the sample and an applied magnetic fieldBparallel

toM, the FM expansion in the most general form reads

FM =−a(TC −T)M2+bM4−

1

2MB. (2.3)

Note that as the temperature T increases (decreases) above (below) the Curie

tem-perature of the ferromagnet TC, a phase transition occurs as suggested by the sign

change of the quadratic coefficient. Fig. 2.2 shows the free energy of a ferromagnet

(32)

distribution over the magnetic and nonmagnetic quantum wells at B = 0, the free

energy is minimized in the absence of applied magnetic field by solving ∂F/∂M = 0

to obtain

−2M[a(TC −T) + 2bM2] = 0. (2.4)

Consequently, the solutions that yield the ground state of the system are (see Fig. 2.3)

M = 0

M = ±

r

a(TC−T)

2b . (2.5)

The first solution seems at first to be valid for all temperatures above or below TC.

However, evaluating ∂2_F

M/∂M2 at M = 0 indicates that this solution (M = 0) is

unstable for T < TC and stable for T ≥TC. The second solution apparently is valid

only when T < TC. Therefore, the magnetization is nonzero for temperatures below

the Curie temperature and zero otherwise. Plugging the solutions inFM in equation

(2.3) yields

FM = −

a2₍_T

C −T)2

4b , T < TC

FM = 0, T ≥TC. (2.6)

It can be shown that the parametersaandbof Landau expansion in equation (2.3)

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Curie-Weiss law, which describes the magnetic susceptibilityχ of a ferromagnetic material

in the paramagnetic phase above the Curie temperature

χ=C0/(T −TC), T > TC (2.7)

where C0 is the Curie constant. Minimizing the free energy at T > TC by solving

∂FM/∂M = 0 yields

a= 1/4C0. (2.8)

Also, the saturation spontaneous magnetizationMs = [a(TC)/2b]1/2 forT < TC (that

minimizes FM at B = 0) provides

b =a(TC)/2Ms2. (2.9)

By these relations in equations (2.8) and (2.9), all parameters inFM in equation (2.6)

are determined. It is important to note that equation (2.6) includes the dependence on

the free hole concentration in the magnetic quantum well via the critical temperature

TC =TC(η).

Assuming that the total magnetization of the magnetic quantum well stems mainly

from the magnetic ions and n0

h ¿nmLwM (nm is the three-dimensional magnetic ion

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can be expressed as

M =nmgµBhSi (2.10)

where hSi = S(S + 1)gµBB/3kBT. g is the magnetic ion g-factor with spin S, µB

denotes the Bohr magneton, and kB is the Boltzmann constant. One can then find

the parametersa = 3/[4S(S+1)g2_µ2

Bnm] andb= 3kBTC/[8S3(S+1)g4µ4Bn3m]. Hence,

FM becomes

FM(η) = −

3 8

S S+ 1nm

kB

TC

(TC −T)2. (2.11)

Energy of the Two-Dimensional Hole Gas

The total free energy of the system can be obtained if equation (2.11) is added to

the free energy of the two-dimensional hole gas

Fh =F2D(η) +U(η) +C(η). (2.12)

F2D(η) accounts for the kinetic energy of the hole gas in both quantum wells,U(η)

ac-counts for the energy shift between magnetic quantum well and nonmagnetic quantum

well, and C(η) is the energy of the Coulomb interaction between the two quantum

wells. Assuming the parabolic dispersion law with an effective mass m for

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populated with holes, one can find the kinetic energy per hole as

F2D(η) =kBT

½

ηf1

µ

ε0

F

kBT

η

¶

+ (1−η)f1

µ

ε0

F

kBT

(1−η) ¶¾

(2.13)

whereε0

F =π~2n0h/mis the Fermi energy of two-dimensional holes with concentration

n0

h, and

f1(x) = ln (ex−1) +

1

xLi2(1−e

x₎ _(2.14)

with a polylogarithmic function Li2(x) =

R₀

x dtln(1 − t)/t. At low temperature

T ¿ ε0

F/k, equation (2.13) describes the sum of degenerate carrier energies in both

quantum wells. The low temperature assumption is commonly used in the works

on the ferromagnetic ordering in magnetic quantum wells. However, one needs to

account for the arbitrary relation between ε0

F and kBT according to equation (2.14).

The contribution of the energy shift ∆U between the magnetic quantum well and the

nonmagnetic quantum well (Fig. 2.1) is accounted for in the term

U(η) = ∆Uηn0

h. (2.15)

The Coulomb energy is found by approximating the two quantum wells as thin sheets

(36)

limit then takes the form

C(η) = 2πe2

² Lbn 0

h

µ

η−1

2 ¶₂

(2.16)

where e is the electron charge and² is the dielectric constant.

Total Energy

Now one can analyze the total free energy F =FM +Fh with respect to possible

ferromagnetic phase transitions in the magnetic quantum well. Considering that the

dependence TC = TC(nhM) should be taken from experiments, the current analysis

utilizes a model that can be applied to a typical diluted magnetic semiconductor.

Specifically, the following expression is proposed as an approximation that describe

the dependence of TC on η

TC =TC0

³

1−e−αε0Fη/kBTC0 ´

. (2.17)

T0

C is the asymptotic (at a high enoughnhM) value of the critical temperature andα

is the fitting parameter that adjust the dependence [equation (2.17)] to experiments.

Hereinafter, α = 1 is assumed, for it describes the experimental results

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(2.16), the form of the free energy per hole normalized by the energy unitkBTC0 reads

F(η) = −3

8

S S+ 1

ν tc(η)

[tc(η)−t]2θ(tc(η)−t)

+t n ηf1 ³_r tη ´

+ (1−η)f1

³_r

t (1−η)

´o

+uη+w(2η−1)2₊_F

ex (2.18)

whereν =nmLw/n0h,t =T /TC0,tc(η) = TC/TC0,u= ∆U/kBTC0,w=πe2Lbn0h/2²kBTC0,

r =ε0

F/kBTC0, tc(η) = 1−exp(−rη), and θ(x) is the Heaviside step function.

More-over, equation (2.18) includes the exchange energy of free carriers Fex, which can

affect the conditions for bistability formation [60], [61]. The calculation of Fex was

performed in a Hartree-Fock approximation following [62], where Fex is expressed in

the general form as

Fex =−

2 2A

X

k,k0

V(|k−k0|)f(k)f(k0) (2.19)

where

V(|k−k0|) = 2πe2

|k−k0

| (2.20)

is a Fourier transform of the Coulomb interaction in two dimensions and f(k) is the

(38)

approximation in terms of the dimensionless parameter r, given by

Fex ∼= −

25/2_e2_n0

h

3√π²kBT_C0

×

n

η3/2_φ³r tη

´

+ (1−η)3/2_φ³r

t(1−η)

´o

(2.21)

where

φ(x) = 1−e−x/2.4+ x

b

2.4 +xc. (2.22)

b = 0.3 + 0.2θ(x−1) and c = 2.2−0.2θ(x−1). Note that equation (2.18) takes a form similar to the free energy expression used for analyzing the spin/charge

sepa-ration of diluted magnetic semiconductors near a paramagnetic-ferromagnetic phase

transition [63].

2.2.3 Bistability Formation and Conditions

For a numerical evaluation, the following typical values for the parameters of the

double quantum well structure are assumed: m = 0.3m0 (m0 is the free electron

mass), ² = 13, S = 5/2, LwM = LwN = 10 nm, Lb = 5 nm, n0h = 1012cm−2,

nm = 1.3×1021 cm−3, and TC0 = 100 K.

Fig. 2.4 depicts F(η) calculated at three different values of the energy shift ∆U

between the minima of the paramagnetic quantum well and nonmagnetic quantum

well (u = ∆U/kBTC0 = 29,5,−9). It is clear that curve 1(u = 29) supports only

(39)

stable state at η = 1. In other words, when the magnetic quantum well lies either

too high (curve 1) or too low (curve 3) compared to the nonmagnetic quantum well,

the holes strongly prefer to be confined in one of the quantum wells at equilibrium.

Even when the holes are transferred to the other quantum well through an external

bias, they will return to the previous preferred state once the applied bias is turned

off. However, one can realize a structure that has free energy minima at or near

both η = 0 (with the magnetic quantum well in the paramagnetic phase) andη = 1

(the magnetic quantum well in the ferromagnetic phase) if ∆U is properly selected

(has a moderate value) (curve 2). Then the two states can be stable with respect

to small fluctuations under the same external conditions. A conclusion is reached

that the relative influence of Fex is small compared to the Coulomb energy of the

inter-quantum well carrier interaction w(2η−1)2_.

The bistability can be achieved in a relatively wide range of ∆U and T as shown

in Fig. 2.5. The range of ∆U, where bistability can be realized, highly depends on

T. The condition for ∆U becomes more flexible with a decreasing T, i.e., the range

of ∆U widens as T decreases. This is due to the fact that the structure can now

operate with a lowerTC, which in turn requires a smaller hole density in the magnetic

quantum well for the paramagnetic-ferromagnetic transition. The highest operating

temperature will be lower than T0

(40)

Write

Erase

V_g V_g

DMS QW (PM)

NM QW NM QW

DMS QW (FM)

EF

Ec

EF

Ec

Eexch

Figure 2.1: Schematic energy diagram (valence band) of the structure in the two coexisting stable states. The first stable state corresponds to a depopulated diluted magnetic semiconductor (DMS) quantum well (QW), which would be paramagnetic (PM). In this state, the holes reside in the nonmagnetic (NM) QW. The second stable state corresponds to a populated DMS QW, which would be ferromagnetic (FM). The mutual alignment of the hole spins (large arrows) and localized spins (small arrows) reduces the energy in the FM QW by Eexch ' 2FM/n0h due to their

(41)

F

_M

M

T > T

_C

T < T

C

Figure 2.2: Free energyF(M) below and above Curie temperatureTC (Landau mean

(42)

M

T TC

Mα (T

C- T)1/2

M =0

(43)

0.0 0.2 0.4 0.6 0.8 1.0 -2

0 2

2

F

(

)

-10 0 10 20

F

(

)

T = T

c 0

/ 2

3 1

Figure 2.4: Free energy trial function F(η) at different values of u (= ∆U/kBTC0).

(44)

0 10 20 30 40 50 0.0

0.2 0.4 0.6

FM

PM

Bistability

Area

T

/

T

c

0

U / k

B

T

c 0

(45)

Chapter 3 Scaling Quantum Wells to

Quantum Dots

In order for the proposed device in Chapter 2, where the active magnetic layer is a

diluted magnetic semiconductor quantum well, to be practical, scaling the magnetic

layer must be feasible. It is crucial to show that a reduction in the magnetic layer

size does not compromise high temperature operability. Hence, a diluted magnetic

semiconductor quantum dot that exchanges itinerant holes under applied bias with a

reservoir seems like a reasonable choice.

The purpose of this chapter is to expand the work discussed in Chapter 2 and

investigate device-relevant issues beyond the bistability effect. Specifically, the effect

of scaling the quantum dot on the bit retention time is analyzed, the dynamic energy

(46)

proposed and examined, and a potential application as a rudimentary device for logic

AND an OR operations is explored.

3.1 Theoretical Model

The nanostructure under investigation consists of a single diluted magnetic

semi-conductor quantum dot separated from a reservoir of itinerant holes that controls the

chemical potential µ0 of the system (see Fig. 3.1). Thus, the system under

consid-eration can be viewed as a small subsystem (the quantum dot) in equilibrium with

a much larger reservoir (quantum well - the hole reservoir) with respect to both hole

and energy exchange. Hence, it is appropriate to use the grand canonical ensemble

to describe the system. Then the population of the eigenstates in the quantum dot is

governed by the grand canonical distribution. The free energy (also called the grand

potential) of the system reads

F =−kBT ln(Z) (3.1)

where Z is the grand partition function, and it takes the form

Z =X

j

eµ·j/kBT X

i

(47)

where j, µ, and Ei are the number of holes in the quantum dot, the electrochemical

potential, and the energy of a microstate, respectively. It is important to note that

the sum is carried over all the microstates not the energy levels. The mean number

of particles j is determined by the ensemble average

j =−∂F

∂µ =kBT ∂lnZ

∂µ . (3.3)

Note that equation (3.3) gives the relation between the hole occupancyjand chemical

potential µ.

In order to solve for the equilibrium hole population at given external conditions,

one needs to know the energy eigenvalues of the system. Typically, this is done by

deriving the complete Hamiltonian of the system including the magnetic Hamiltonian

Hm that describes the effective ferromagnetic inter-ion spin-spin interaction in the

presence of free holes. Then the free energy F of the total system consisting of

magnetic ions and free carriers can be calculated. Nonetheless, this approach faces

the difficulty of specifying the mechanisms responsible for ferromagnetic ordering in

the magnetic quantum dot. Therefore, an approach that utilizes experimental data

extracted from routine experimental measurements is considered.

A general property of the partition function [equation (3.2)] can be useful in

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energy states Ei’s are shifted by δE, the partition function is modified as

Ei →Ei+δE ⇒Z →e−δE/kBT. (3.4)

Then the free energy [equation (3.1)] is shifted by the same amount δE

F →F +δE. (3.5)

Therefore, this property is utilized to write the free energy of the system as a sum of

the free energy of the holes in the quantum dot (FN) and the magnetic energy (FM)

F =FN +FM. (3.6)

Note that the quantum dot dimension in the vertical (i.e., growth) direction is

assumed to be much smaller than that in the lateral directions, forming a pancake-like

shape. Hence, the confinement in the vertical direction is very strong with only the

ground states considered for hole occupation. As for the lateral directions, however,

one must take into account the details of the hole states. A parabolic potential in

these directions is adopted. The parabolic potential reflects the effect of gate bias

on the hole electrostatic confinement in the quantum dot [64], [65]. For simplicity,

it is assumed that µ0 À kBT, and the possible temperature dependence of µ0 in

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example, through modulation doping, etc.) can be used as the desired reservoir.

3.1.1 Magnetic Energy - Landau Expansion

If the diluted magnetic semiconductor quantum dot is near the

paramagnetic-ferromagnetic transition, the Landau expansion over the magnetization M can be

applied for FM. Following the same steps discussed in Chapter 2, one can derive

FM = −TCM

2 0 C0

µ 1− T

TC

¶₂

, T < TC

FM = 0, T ≥TC. (3.7)

It is important to emphasize that the localized spins S of the magnetic ions provide

the major contribution to the diluted magnetic semiconductor magnetization, whereas

the spin of the free holes add a minor role. The parameters M0 and C0 can be easily

obtained for Nm localized spin moments leading to the estimation

M2 0 C0

= 3SNm

8(S+ 1) (3.8)

which is independent of the hole population. In the presented approximation, the

only dependence of FM on j comes from the term TC = TC(j). This term will be

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3.1.2 Hole Energy

To obtain the total free energy, the nonmagnetic part FN for j particles located

in the quantum dot must be included as well. Unfortunately, the calculation of FN

requires very specific details such as the material composition, size and shape of

the quantum dot, presence of dopants and external fields, etc. Consequently, this

problem cannot be solved in a general manner. As aforementioned, a quantum dot

with a parabolic potential profile that merges to a flat barrier at the boundary (see

Fig. 3.1) is considered. Note that in the case of a quantum well, the energy spectrum

calculated for such a truncated potential profile is well approximated by that of infinite

parabola [66]. Assuming that a similar argument holds for quantum dots, the edge

effect for the reasonably deep energy levels is ignored. Then the energy spectrum of

the quantum dot can be solved analytically [67]. With the confining potential in the

lateral x−y plane represented as 1

2mωr2 (m is the in-plane hole effective mass and

r2 ₌_x2₊_y2_{), the energy} _ε

n,l of a localized particle is given by the equation

εn,l =~ω(2n+|l|+ 1). (3.9)

Here, ~ω denotes the confinement energy, which is assumed to be controlled by the electrostatic potential, and n (= 0,1,2, . . .) and l (= 0,±1, . . . ,±n) are the radial and the angular momentum quantum numbers, respectively.

(51)

contributions, FN =Ej+F1(T, j). The first term

Ej =jU +

1

2j(j −1)C (3.10)

accounts for the energy acquired byj particles due to their localization in the quantum

dot with the ground state energy U (related to ε0,0; see Fig. 3.1) as well as their

Coulomb repulsion energy; C = e2_/²√_A

0, where e is the electron charge, ² is the

dielectric constant, and A0 reflects the area occupied by holes in the quantum dot.

When estimated classically, A0 is given as

A0 = 2π~2(

√

j m~ω +

kBT

m~2_ω2). (3.11)

The remaining term of the nonmagnetic free energy

F1(T, j) = Ω(T, µ1) +jµ1(j) (3.12)

is similar to the free electron gas contribution with the thermodynamic potential

Ω(T, µ1) =−kBT

X

s

ln[1 +e(µ1−εs)/kBT_]_. _(3.13)

Here, εs(s = n, l) and µ1 represent the energy spectrum [equation (3.9)] and the

(52)

is excluded (i.e., the nonmagnetic version of the quantum dot), respectively, and j

is treated as a mean value hji over the grand canonical Gibbs ensemble. The sum in equation (3.13) can be approximated by an integral over εs with the density of

states 2εs/~2ω2 + 1/~ω. Taking into account the relation j = −∂Ω(T, µ1)/∂µ1, the

dependence µ1 =µ1(j) can be found numerically from the transcendental equation

j =−2 µ

kBT

~ω

¶₂

Li2(−eµ1/kBT_{) +} kBT

~ω ln(1 +e

µ1/kBT₎ _(3.14)

where Li2(x) is a polylogarithm function. Since the dependence µ1 = µ1(j) is fixed

by equation (3.14), the thermodynamic potential [equation (3.13)] can be expressed

in terms of the quantum dot population

Ω[T, µ1(j)] =−2(kBT)

3

(~ω)2

∞

Z

0

xln(1 +e(µ1−x)/kBT₎_dx₋ (kBT)

2

~ω

∞

Z

0

ln(1 +e(µ1−x)/kBT₎_dx.

(3.15)

Equations (3.10) and (3.12) along with Ω[T, µ1(j)] from equation (3.15) determine the

nonmagnetic part of the free energy for the holes localized in the parabolic potential,

while the total free energy of the diluted magnetic semiconductor quantum dot is the

sum F =FM +FN.

Notice that the quantum dot is in contact with a large reservoir providing two-way

exchange of carriers through the potential barrier (see Fig. 3.1). This results in the

(53)

Thus, the equation that determines the population of the quantum dot takes the form

µQD(j) = µ0. (3.16)

Note that µQD(j) 6= µ1(j) since both the nonmagnetic and magnetic interactions

contribute to µQD(j). As the chemical potential of the quantum dot µQD(j) can be

obtained from µQD(j) = dF/dj in general, the stable solutions of equation (3.16)

must correspond to the local minima ofF =F(j) or equivalentlydµQD(j)/dj >0.

Finally, the desired solutions require a specific expression for the dependenceTC =

TC(j) in equation (3.7). In the case of a rectangular quantum dot potential, one can

take advantage of the following semiphenomenological expression

TC(j) = TC0

¡

1−e−αjξc¢ _(3.17)

that approximates the experimental data in [41] provided the fitting parameterα = 1;

here, T0

C is the asymptotic value of the critical temperature (at a sufficiently large

hole concentration) andξc=π~2/mA0kBTC0. Although this expression was originally

developed based on the results of diluted magnetic semiconductor quantum wells, a

similar behavior is expected for the case under consideration. Indeed, when the

di-luted magnetic semiconductor quantum dot contains a large number of magnetic ions,

the statistics of the localized spins nearly reproduce the property of the bulk

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the magnetic ions by means of their concentration rather than the hole statistics.

Even in the extreme case of a single hole, the long range ferromagnetic correlations

are inversely proportional to the hole localization area that forms a bound magnetic

polaron at a sufficiently low temperature. Thus, equation (3.17) can be used to

de-scribe the increase of TC with the hole concentration j/A0 in the quantum dot; the

parameter α is adjusted to quantitatively fit a particular system.

Then, by adopting equation (3.17), equation (3.16) can be written explicitly as

µ0 = ∂Ej

∂j +µ1(j) + ∂FM

∂j . (3.18)

This expression along with equations (3.7), (3.10), and (3.14) provides the theoretical

framework of the rest of the discussion.

3.2 Bistability Conditions

For numerical evaluation, a typical carrier mediated diluted magnetic

semiconduc-tor such as Ga0.95Mn0.05As is assumed. The parameters used are: ² = 13, S = 5/2,

nm = 1.3×1021 cm−3, TC0 = 110 K, and the quantum dot width Lw = 5 nm. An

additional parameter that needs to be specified is the quantum dot dimension in the

lateral directions (i.e., on the x− y plane). Practical realization of the proposed device assumes a reasonable restriction of the magnetic layer sizes and, consequently,

(55)

cannot be less than maximal area A0 occupied by holes during the device

opera-tion. This important parameter A0 is sensitive to the confinement energy ~ω, i.e.,

the stronger ~ω is, the smaller A0 becomes. In addition, A0 tends to increase with the operating temperature. Figs. 3.2 and 3.3 present the quantitative analysis for

the radius of A0 as a function of ~ω and T, respectively. While the confinement

potential has a strong influence on A0 as expected, the temperature does not show

a pronounced effect. For the cases considered throughout this work, the parabolic

confinement energy ~ω of kBTC0 (corresponding toA0 = 750 nm2) is assumed unless

specified otherwise. A diluted magnetic semiconductor quantum dot of this dimension

typically contains a large number of magnetic ions (&1000) justifying the theoretical

model [e.g., equation (3.17)] as aforementioned.

The dependence µQD(j) vs. j calculated according to equation (3.18) is shown

in Fig. 3.4. Three characteristic levels of µ0 −U distinguish the monostable and

bistable states. The results indicate that a sole solution for hole population j exists

at sufficiently high or low energies U in reference to µ0 (e.g., dashed line 1 or 3 with

µ0−U =−0.5kBTC0 or 5kBTC0). However, moderate values of U (e.g., dashed line 2

with µ0 −U = 2.3kBTC0) can support multiple solutions of equation (3.16). Two of

them (with the smallest and largest j) are stable considering the positive derivative

(dµQD/dj > 0), while the intermediate solution is not (dµQD/dj < 0). Note also

that the stable solutions with the larger (smaller)j are realized in the ferromagnetic

Nonvolatile Spin Memory based on Diluted Magnetic Semiconductor and Hybrid Semiconductor Ferromagnetic Nanostructures

DEDICATION

BIOGRAPHY

ACKNOWLEDGMENTS

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

Chapter 1

Introduction

Chapter 2

Memory Effect Based on Electric

Tuning of Ferromagnetism in a

Diluted Magnetic Semiconductor

Quantum Well

2.1

Diluted Magnetic Semiconductors

2.1.1

Progress on GaMnAs

2.1.2

Theoretical Picture

2.1.3

Electric Control of Ferromagnetism

2.1.4

Room Temperature Ferromagnetism

2.2

Nonvolatile Memory Based on Diluted

Mag-netic Semiconductor Quantum Wells

2.2.1

Device Concept

2.2.2

Theoretical Model

2.2.3

Bistability Formation and Conditions

F

M

T > T

T < T

Chapter 3

Scaling Quantum Wells to

Quantum Dots

3.1

Theoretical Model

3.1.1

Magnetic Energy - Landau Expansion

3.1.2

Hole Energy

3.2

Bistability Conditions