michihiro hosoda department of electrical and electronic ... · department of electrical and...

Carrier transport modeling in diamonds

1

2013 Master’s Thesis


Michihiro Hosoda

Department of Electrical and Electronic Engineering

Tokyo Institute of Technology

Supervisor:

Prof. Hiroshi Iwai and Associate Prof. Kuniyuki Kakushima


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February, 2013 Abstract of Master’s Thesis

Carrier transport modeling in diamonds-

Supervisor: Prof. Hiroshi Iwai

Tokyo Institute of Technology

Department of Electrical and Electronic Engineering

11M36388 Michihiro Hosoda

Due to deep ionization energy of dopants in diamonds, the number of carriers

transporting in either conduction or valence bands becomes less as compared to those in

silicon substrates. While increasing the doping density, hopping conductions become

dominant in diamonds. However, carrier transport modeling from device point of view

considering both components has not been presented so far. In this thesis, we have

attempted to make a current density equation for both homogeneous n- and p-type

diamond, considering both band and hopping conduction. Drift mobility and resistivity

for both n- and p-type diamonds have been modeled based on hopping drift mobility

equations. It has been found that by introducing an effective mobility, which is a weighted

average of each mobility of hopping and band conductions, current density equations can

be successfully expressed. The achievement gives a rough current estimation of diamond

devices. Furthermore, device characteristics of p-channel junction field-effect transistor

are calculated based on a calculation. P-diamond is high breakdown voltage and low

loss, compared to n-Si. When blocking voltage is 5000 V, current of p-diamond is larger

and device length of that is smaller than n-Si. Thus, it is revealed that p-diamond is


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superior to n-Si from various viewpoints.


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Contents Chapter 1 Introduction ............................................................................... 6

1.1 Background of this thesis ................................................................ 7

1.1.1 Characteristics of diamond as a material for power device .................. 7

1.1.2 Research trend on diamond ...................................................................... 9

1.1.3 Transport mechanism in diamonds ........................................................ 10

1.2 Purpose of this thesis ..................................................................... 11

References ............................................................................................... 13

Chapter 2 Electron conduction in n-diamond ........................................ 16

2.1 Nearest-neighbor hopping (NNH) ............................................... 17

2.2 Drift mobility under hopping conduction ................................... 18

2.3 Resistivity of n-type diamonds ..................................................... 21

2.4 Average mobility for current equation ........................................ 24

2.5 Conclusions .................................................................................... 27

References ............................................................................................... 27

Chapter 3 Hole conduction in p-diamond ............................................... 30

3.1 Variable-range hopping (VRH) ...................................................... 31

3.2 Drift mobility under hopping conduction ..................................... 32

3.3 Resistivity of p-type diamonds ........................................................ 35

3.4 Average mobility for current equation........................................... 38

3.5 Conclusion ........................................................................................ 41

References ............................................................................................... 41

Chapter 4 Device characteristics of p-JFET ........................................... 44

4.1 Introduction ...................................................................................... 45

4.2 Derivation of current-voltage characteristics of JFET ................ 47

4.3 Calculation results ........................................................................... 49

4.3.1 Specific on-resistance vs. blocking voltage ................................................ 51

4.3.2 ID-VD characteristics .................................................................................... 52

4.4 Conclusion ........................................................................................ 54

References ............................................................................................... 54

Chapter 5 Conclusion ................................................................................ 56

Acknoledgement ........................................................................................ 58


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Chapter 1 Introduction


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1.1 Background of this thesis

Diamond is a promising material for future devices due to its characteristics such as

wide bandgap, high breakdown voltage, high hole saturation velocity, and high thermal

conductivity.

1.1.1 Characteristics of diamond as a material for power

device

There are several candidates for future power device such as gallium nitride (GaN),

silicon carbide (SiC), and diamond. Although they have many advantages, they are in

the following present conditions [1.1]:

GaN: producing bulk GaN requires reaching extremely high decomposition pressures

and a high melting temperature.

SiC: defect density control and material availability remain issues, despite decades of

research.

Diamond: perhaps the ultimate material for power electronics (PE) due to outstanding

electrical properties; however, power devices may not be feasible for 2-3 decades due

to manufacturing challenges.


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Table 1.1 Physical and electrical properties of Si and the major wide-bandgap

semiconductors [1.2]

Property　(relative to Si) Si GaAs SiC GaN Diamond

Bandgap 1 1.28 2.77 3.08 4.87

Saturation velocity 1 1 2 2.2 >2.5

Electron mobility 1 5.67 0.67 0.83 3

Hole mobility 1 0.67 0.08 0.42 6.3

Breakdown Field 1 1.3 8.3 6.7 33.3

Dielectric constant 1 1.06 0.9 0.9 0.5

Thermal conductivity 1 0.3 3.1 0.9 13.5

Thermal expansion coefficient 1 1.6 1.6 2.2 0.03

There are several indexes of the impact of material parameters on the performance of

semiconductor devices. JFOM is a figure of merit derived by Johnson in 1965 [1.3]:

2JFOM SB vE

, (1.1)

which defines the power-frequency products for a low-voltage transistor. Here, EB is the

critical electric field for breakdown in the semiconductor and vS is the saturated drift

velocity. KFOM is a figure of merit derived by Keyes in 1972 [1.4]:

2

1

4KFOM

Svc

, (1.2)

which provides a thermal limitation to the switching behavior of transistors used in

integrated circuits. Here, c is the velocity of light and is the dielectric constant of the

semiconductor. BFOM is a figure of merit derived by Baliga in 1983 [1.5]:

3BFOM gE , (1.3)

which defines material parameters to minimize the conduction losses in power FETs.

Here, is the mobility and Eg is the bandgap of the semiconductor. The BFOM is based

on the assumption that the power losses are only due to the power dissipation in the


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on-state by current flow through the on-resistance of the power FET. Thus, the BFOM

applies to systems operating at lower frequencies where the conduction losses are

dominant.

Table 1.2 Figures of merit of some power semiconductors [1.2]

Benchmark Factor　(relative to Si) Si GaAs SiC GaN Diamond

JFOM 1 11 410 790 5800

KFOM 1 0.45 5.1 1.6 31

BFOM 1 28 290 400 12500

Diamond possesses excellent physical and electrical properties such as high thermal

conductivity, low dielectric constant, high carrier mobility, chemical inertness, and

radiation hardness. Then, diamond has outstanding figures of merit amounted based on

eqs. (1.1) – (1.3), as shown in tab. 1.2. These characteristics are suitable for the expected

applications as follows ultraviolet (UV) light-emitting diodes (LED), LD, deep-UV

sensor, high power electronics, high frequency electronics, extreme ambient (such as high

temperature, temperature, space, nuclear facilities and so on) devices.

Crystallinity and resource constraint. However, one of problems of diamond as a power

device is deep doping level due to the low dielectric constant. The deep doping level

causes high resistance and makes it difficult to fabricate high-current devices.

Due to the excellent characteristics such as wide bandgap high saturation velocity, high

breakdown field, high saturation velocity, and high thermal conductivity, as shown in

tab. 1.1, 1.2, diamond is an ideal material as future power devices.

1.1.2 Research trend on diamond

Although it was difficult to synthesize n-type diamond for long time, fabrication of


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high quality p- and n-type diamond has been recently enabled by CVD. It leads to

fabricate several devices such as Schottky-pn diode (SPND), p+in+ diode, pnp transistor,

Schottky barrier diode (SBD), power FET, and bipolar transistor (as shown in fig. 1.1).

Figure 1.1 Diamond bipolar junction transistor device with phosphorus-doped diamond

base layer [1.6]

1.1.3 Transport mechanism in diamonds

Figure 1.2 shows the band diagram of diamond. Bandgap of diamond is 5.47 eV as

mentioned above. P- and n-diamonds have deep impurity levels; acceptor level of boron

(B), EA, located at 0.37 eV above the valence band maximum and donor level of

phosphorus (P), ED, located at 0.58 eV below the conduction band minimum. These

deep levels of dopants make it difficult for carriers to be exited to the conduction or

valence bands as mobile charges. It has been experimentally confirmed that dominant

conduction mechanism in the low-temperature range could be the nearest-neighbor

hopping (NNH) in n+-diamond [1.1] and the variable-range hopping (VRH) in

p+-diamond [1.2]. In NNH, carriers are transported between dopant levels by tunneling.

In VRH, carriers are transported in dopant band formed by wave function overlapping of

dopant levels. Hopping conduction has as much influence as band conduction, even at

room temperature. Then, a new equation for conduction modeling is necessary.


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WH

0.37 eV

0.58 eV

5.4

7 e

V

CB

VB

B

P

Figure 1.2 Band diagram of diamond.

1.2 Purpose of this thesis

Due to the deep ionization energy of dopants in diamonds, the number of carriers

transporting in either conduction or valence bands becomes less as compared to those in

silicon substrates. While increasing the doping density, hopping conductions become

dominant in diamonds. However, carrier transport modeling from device point of view

considering both components has not been presented so far.

Figure 1.3 shows the contents of this thesis. In this thesis, we have attempted to make a

current density equation for both homogeneous n- and p-type diamond, considering both

band and hopping conduction.

In chapter 2 and 3, at first, we estimated drift mobility and resistivity of n+- and

p+-diamond. Here, only the hopping conductions are taken into account. Next, we


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reconstructed the reported experimental data of temperature dependence of the

resistivity taking the both band and NNH and VRH conductions into account, based on

this estimation.

In chapter 4, device characteristics of junction field-effect transistor are calculated

based on a calculation in chapter 3.

Figure 1.3 Contents of this thesis

Chapter 2

Electron conduction in n-diamond

Chapter 3

Hole conduction in p-diamond

Chapter 1

Introduction

Chapter 5

Conclusions

Chapter 4

Device characteristics

of junction field-effect transistor


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References

[1.1] The US Department of Energy, “Power Electronics Research and Development

Program Plan,” April, 2011,

http://energy.gov/sites/prod/files/oeprod/DocumentsandMedia/OE_Power_Electron

ics_Program_Plan_April_2011.pdf.

[1.2] Evince Technology Ltd., “Why Diamond Is Better Than Anything Else,”

http://www.evincetechnology.com/whydiamond.html

[1.3] E. 0. Johnson, “Physical limitations on frequency and power parameters of

transistors,” RCA Rev., pp. 163-177, 1965.

[1.4] R. W. Keyes, “Figure of merit for semiconductors for high-speed switches,” Proc.

IEEE, p. 225, 1972

[1.5] B.J. Baliga, "Power semiconductor device figure of merit for high-frequency

applications", IEEE Electron Device Lett., vol. 10, no.10, pp.455-457, 1989.

[1.6] H. Kato, K. Oyama, T. Makino, M. Ogura, D. Takeuchi, and S.

Yamasaki, ”Diamond bipolar junction transistor device with phosphorus-doped

diamond base layer,” Diamond Relat. Mater. vol. 27-28, pp.19-22, 2012.


14


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Chapter 2 Electron conduction in

n-diamond


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2.1 Nearest-neighbor hopping (NNH)

Diamonds with a wide-band-gap, Eg of 5.47 eV, is one of semiconductors which can be

applied to high-power electronic devices, deep-ultraviolet sensors, and light-emitting

diodes (LED). One of the concerns of diamond semiconductors are that p- and

n-diamonds have deep impurity levels; acceptor level of boron (B), EA, located at 0.37 eV

[2.3] above the valence band maximum and donor level of phosphorus (P), ED, located at

0.58 eV [2.2, 2.3] below the conduction band minimum. These deep levels of dopants

make it difficult for carriers to be exited to the conduction or valence bands as mobile

charges. It has been experimentally confirmed that dominant conduction mechanism in

the low-temperature range could be the nearest-neighbor hopping (NNH) in n+-diamond

[2.1] and the variable-range hopping (VRH) in p+-diamond [2.2]. Figure 2.1 shows a

conceptual diagram of NNH. WH and RH are deference between dopant levels and

distance between dopants, respectively.

Conduction mechanism in diamond has not ever been revealed in detail, different from

that of silicon (Si). It is the key to understand conduction mechanism in diamond in detail.

In the present work, we have attempted to make a current density equation for

homogeneous n+-diamonds.

RH

WH

Figure. 2.1 Conceptual diagram of NNH


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2.2 Drift mobility under hopping conduction

At first, we estimated drift mobility and resistivity of n+-diamond applying the

following Eqs. (2.1) and (2.2). Here, only the hopping conduction is taken into account.

Next, we reconstructed the reported experimental data of temperature dependence of the

resistivity taking both band and nearest-neighbor hopping conductions into account,

based on this estimation.

In the following parts, the ways to calculate drift mobility and resistivity of n-diamond

are showed in detail.

The hopping drift mobility of NNH at temperature T is given by [2.6]

kT

WR

kT

qR HphH

HH exp2exp

6

12

, (2.1)

where q is the elementary charge, RH is the hop length, k is Boltzmann’s constant. ph is

the phonon scattering probability, WH is the hopping activation energy, and is an

electronic wavefunction overlap parameter. The term exp(-2αRH) describes the overlap of

the wavefunctions on neighboring hopping sites, with the parameter α-1 denoting the

spatial extension of a localized donor wavefuntion; νph exp(-WH/kT) represents the

probability per unit time that a localized electron hops to a new site at an energy WH above

the original one.

As the tunneling probability is sensitive to the tunneling distance RH, doping

concentrations strongly affect the H. Figure 2.2 shows calculated H with different

doping concentrations. A steep decrease in H can be observed with lower doping

concentration. As NNH is based on activation type carrier conduction, H is further


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reduced at low temperature.

Figure 2.2 shows a relationship between RH and doping concentration. The larger

doping concentration becomes, the smaller RH becomes. Although this fact is natural,

doping concentration is then related to RH. 1E+16 1E+17 1E+18 1E+19 1E+20

1.E-07

1.E-06

1.E-05

1.E-07

1.E-06

1.E-05

1E+16 1E+17 1E+18 1E+19 1E+20

RD

(cm

)

Doping density (/cm3)

1016 1018 1020

Doping concentration (cm-3)1017 1019

10-5

RH

(cm

)

10-7

10-6

Figure 2.2 Relationship between RH and doping concentration

Figure 2.3 shows temperature dependence of H of n-diamonds with two different

doping concentrations. In the low-temperature range, the activation energy term, WH, in

Eq. (2.1) becomes dominant, so that H decreases along with temperature. In the

high-temperature range, the influence of the term on WH2 in Eq. (2.1) becomes dominant,

so that H starts to deviate from Arrhenius plot as shown in fig. 2.4.


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1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-031E+16 1E+17 1E+18 1E+19 1E+20

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1E+16 1E+17 1E+18 1E+19 1E+2010-9

10-5

1016 1018 1020

Doping concentration (cm-3)

H

(cm

2 /V

s)

100 K

300 K

500 K

10-3

10-7

1017 1019

10-4

10-6

10-8

-1 = 1.8 nm

ph = 2.0×1011 s-1

WH = 51 meV

500 K

Figure 2.3 Doping concentration dependence of hopping drift mobility

1.E-07

1.E-06

1.E-05

1.E-04

1.E-030 4 8 12 16

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

0 4 8 12

1×1018 cm-3

8×1019 cm-3

10-7

10-5

10-3

10-4

10-6

0 4 8 12

1000/T (K-1)

-1 = 1.8 nm

ph = 2.0×1011 s-1

WH = 51 meV

H

(cm

2 /V

s)

WH

WH

Figure 2.4 Temperature dependence of hopping drift mobility


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2.3 Resistivity of n-type diamonds

Resistivity of a homogeneously doped semiconductor is given by

qn

1 , (2.2)

where n is the carrier density and is the mobility. As there are two conduction

mechanisms in n-diamonds, the resistivity of n-diamond, n, can then be given by

HHBBn

qnqn

1, (2.3)

where B is the mobility of band conduction. Here, the concentration of mobile charge

carriers related to band conduction, nB, can be calculated by the following equation:

kT

E

h

kTm

nNN

nNn De

BAD

BAB exp2

122

1)( 2

3

2

*, (2.4)

where me* is the effective mass of the electron, and h is Planck’s constant. Assuming that

there is a simple proportional relationship between ND and the concentration of mobile

charge carriers related to NNH, nH can be written as

BDH nNn . (2.5)

Figure 2.5 shows the temperature dependence of resistivity obtained for n- and

n+-diamonds with doping concentration of 11018 (n-diamond) and 81019 (n+-diamond)

cm-3, respectively. The dashed and chained lines are the contribution of hopping and band

conductions, respectively. And the solid line represents the total resistivity of both

conductions. Triangle and circle symbols are the experimental data reported in Ref. 2.7.

The electron Hall mobility reported in Ref. 2.8 was adopted as B, with exponential

functions approximation as shown in fig. 2.6. Table 2.1 shows exponential approximated

Hall electron mobility of deferent doping concentration. We assumed that the


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compensation rates NA/ND of n- and n+-diamonds are 22 and 52 %, and activation rate in

Eq. (2.5) is 90 %. In the cases of both n- and n+-diamonds, the values of the experimental

data can be reconstructed by the calculation as shown in fig. 2.5. Although there is no

experimental data in the low-temperature range with n-diamond, the resistivity can be

obtained with Eq. (2.3).

For n- and n+-diamonds in the higher-temperature range over 300 and 500 K,

respectively, band conduction becomes the main conduction owing to enough thermal

energy to excite electrons from ED to the conduction band. On the other hand, the

n+-diamond in a low-temperature range and the n-diamond in the lower-temperature

range than 300 K have a weaker temperature dependence, which comes from hopping

conduction. The hopping conduction is the dominant transport mechanism, which

reduces the resistivity of the n+-diamond at 300 K to ~110 -cm. At 300 K, when the

doping concentration exceeds 1019 cm-3, the resistivity can be largely reduced since

hopping conduction dominates the electron transport [2.5, 2.9].


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Figure 2.5 Temperature dependence of resistivity

1.E+01

1.E+02

1.E+03

1.E+04100 1000

1.E+01

1.E+02

1.E+03

1.E+04

100 10001002

1000

Temperature (K)

3 4 5 6 7 8 9100

102

Ha

ll m

ob

ility

(cm

2 /V

s)

101

103

7×1016 cm-3

2×1017 cm-3

1×1018 cm-3

3×1018 cm-3

. Figure 2.6 Electron mobility adopted as B by Hall effect measurement [2.8]


24

Table 2.1 Exponential approximated Hall electron mobility of deferent doping

concentration

Doping concentration

N D (cm-3

)7 10

162 10

171 10

183 10

18

Hall electron mobility

B = a exp(-bT ) (cm2/Vs)

a = 2030

b = 0.004

a = 1279

b = 0.0035

a = 592

b = 0.0027

a = 203

b = 0.0017

2.4 Average mobility for current equation

Eq. (2.3) should be renewed as the following equation:

navgtotHHBB

nqnqnqn ,

11

, (2.6)

where avg is the average mobility defined as

H

tot

H

B

tot

B

navgn

n

n

n , . (2.7)

Since nH is comparable to ND, Eq. (2.7) can be approximated as

HB

tot

Bnavg

n

n , . (2.8)

Figure 2.6 shows the temperature dependence of the avg,n, calculated based on Eq.

(2.8). B was obtained in the same way as the resistivity calculation. avg,n has a tendency

to increase with an increase with temperature for all doping concentrations. For lower

doping concentrations, namely 71016 and 21017 cm-3, avg,n becomes as small as ~10-13

cm2/Vs in the low-temperature range, in which hopping conduction dominates the

electron transports. For higher doping concentrations of 11018 and 81019 cm-3, namely

in the low-temperature range, avg,n shows larger value with higher doping concentrations.


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On the other hand, while increasing the temperature, in which band conduction dominates

the electron transports, a steep increase in avg,n can be observed for samples with lower

doping concentrations, and exceeds the avg,n of higher doping concentrations.

1.0E-141.0E-121.0E-101.0E-081.0E-061.0E-041.0E-021.0E+001.0E+02

100 1000

1.0E-141.0E-121.0E-101.0E-081.0E-061.0E-041.0E-021.0E+001.0E+02

100 1000

Mob

ility

(cm

2/V

s)

Temperature (K)

10-8

10-6

10-4

10-2

100

102

1002

1000

Temperature (K)

3 4 5 6 7 8 9

10-10

10-12

10-14

7×1016 cm-3

2×1017 cm-3

1×1018 cm-3

8×1019 cm-3

300K

avg

,n(c

m2

/Vs)

Figure 2.7 Temperature dependence of the average mobility

Figure 2.7 shows the doping concentration dependence of the avg,n. As shown in fig.

2.7, in hopping conduction regime, the higher the doping concentrations are, the higher

the avg,n becomes. On the other hand, in band conduction regime, the lower the doping

concentration is, the higher the avg,n becomes. At 100 K, since n-diamonds with all the

doping concentrations are in hopping conduction regime, avg,n increases with an increase

in the doping concentration. At 300 K, there is only a low correlation between average

mobility and doping concentration since n-diamonds with the doping concentrations of


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71016, 21017, and 11018 cm-3 are in transition from hopping to band conduction

regimes, and n+-diamond with the doping concentration of 81019 cm-3 is still in hopping

regime toward low temperature. At 500 K, since n-diamonds with almost all of the doping

concentrations are in band conduction regime, avg,n decreases with an increase in the

doping concentration.

1.0E-141.0E-121.0E-101.0E-081.0E-061.0E-041.0E-021.0E+001.0E+02

1.E+161.E+171.E+181.E+191.E+20

1.0E-141.0E-121.0E-101.0E-081.0E-061.0E-041.0E-021.0E+001.0E+02

1E+16 1E+17 1E+18 1E+19 1E+20

Mobili

ty (

cm

2/V

s)

Doping density (/cm3)

1018 1019 1020


10-8

10-6

10-4

10-2

100

10-10

10-12

10-14

102

500 K

300 K

100 K

10171016

avg

,n(c

m2

/Vs)

Figure 2.7 Doping concentration dependence of the average mobility

From Einstein relation given by

ng,avnq

kTD , (2.9)

avg and diffusion coefficient (Dn) can be expressed. Therefore, the current density

equation for n-diamond can be then obtained as the following equation:

dx

dnqDEqnJ nnavgn , . (2.10)


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2.5 Conclusions

We have succeeded in fitting the resistivity calculated by taking both band and

nearest-neighbor hopping conductions into account to the experimental data, especially in

the case of n+-diamond (81019 cm-3). Then, we revealed the equation of the average

mobility of n-diamond. It is suggested that the current density for n-diamond can be

calculated based on it.

References

[2.1] A. Collins and A. Williams, “The nature of the acceptor centre in semiconducting

diamond,” J. Phys. C: Solid State Phys., vol. 4, no. 13, pp. 1789–1800, 1971.

[2.2] S. Koizumi, M. Kamo, Y. Sato, H. Ozaki, and T. Inuzuka, “Growth and

characterization of phosphorous doped {111} homoepitaxial diamond thin films,”

Appl. Phys. Lett., vol. 71, no. 8, pp. 1065–1067, 1997.

[2.3] H. Kato, S. Yamasaki, and H. Okushi, “n-type doping of (001)-oriented

single-crystalline diamond by phosphorus,” Appl. Phys. Lett., vol. 86, no. 22, p.

222111, 2005.

[2.4] J. Prins, “Using ion implantation to dope diamond—an update on selected issues,”

Diamond Relat. Mater., vol. 10, no. 9–10, pp. 1756–1764, 2001.

[2.5] T. Sato, K. Ohashi, H. Sugai, T. Sumi, K. Haruna, H. Maeta, N. Matsumoto, and H.

Otsuka, “Transport of heavily boron-doped synthetic semiconductor diamond in the

hopping regime,” Phys. Rev. B, vol. 61, no. 19, pp. 12970–12976, 2000.


28

[2.6] L. Comber, W. Spear, and D. Allan, “Transport studies in doped amorphous

silicon,” J. Non-Cryst. Solids, vol. 32, no. 1–3, pp. 1–15, 1979.

[2.7] K. Oyama, S.-G. Ri, H. Kato, M. Ogura, T. Makino, D. Takeuchi, N. Tokuda, H.

Okushi, and S. Yamasaki, “High performance of diamond p+-i-n+ junction diode

fabricated using heavily doped p+ and n+ layers,” Appl. Phys. Lett., vol. 94, no. 15, p.

152109, 2009.

[2.8] J. Pernot, C. Tavares, E. Gheeraert, E. Bustarret, M. Katagiri, and S. Koizumi, “Hall

electron mobility in diamond,” Appl. Phys. Lett., vol. 89, no. 12, p. 122111, 2006.

[2.9] H. Kato, H. Umezawa, N. Tokuda, D. Takeuchi, H. Okushi, and S. Yamasaki, “Low

specific contact resistance of heavily phosphorus-doped diamond film,” Appl. Phys.

Lett., vol. 93, no. 20, p. 202103, 2008.


29


30

Chapter 3 Hole conduction in

p-diamond


31

3.1 Variable-range hopping (VRH)

Diamonds with a wide-band-gap, Eg of 5.47 eV, is one of semiconductors which can be

applied to high-power electronic devices, deep-ultraviolet sensors, and light-emitting

diodes (LED). One of the concerns of diamond semiconductors are that p- and

n-diamonds have deep impurity levels; acceptor level of boron (B), EA, located at 0.37 eV

[3.2] above the valence band maximum and donor level of phosphorus (P), ED, located at

0.58 eV [3.2, 3.3] below the conduction band minimum. These deep levels of dopants

make it difficult for carriers to be exited to the conduction or valence bands as mobile

charges. It has been experimentally confirmed that dominant conduction mechanism in

the low-temperature range could be the nearest-neighbor hopping (NNH) in n+-diamond

[3.5] and the variable-range hopping (VRH) in p+-diamond [3.6].

Conduction mechanism in diamond has not ever been revealed in detail, different from

that of silicon (Si). It is the key to understand conduction mechanism in diamond in detail.

In the present work, we have attempted to make a current density equation for

homogeneous n+-diamonds.

Figure 3.1 shows the conceptual diagram of VRH. In VRH, carrier is transported to

smaller dopant level instead of nearer physical distance.


32

Figure 3.1 Conceptual diagram of VRH

3.2 Drift mobility under hopping conduction

At first, we estimated drift mobility and resistivity of n+-diamond applying the

following Eqs. (3.1) and (3.2). Here, only the hopping conduction is taken into account.

Next, we reconstructed the reported experimental data of temperature dependence of the

resistivity taking both band and nearest-neighbor hopping conductions into account,

based on this estimation.

In the following parts, the ways to calculate drift mobility and resistivity of n-diamond

are showed in detail.

The hopping drift mobility of VRH at temperature T is given by [3.7]

kT

WR

kT

qR m

mph

m

m 2exp

2

m

mph

m

T

T

kT

qRexp

2

, (3.1)

where Rm is the hop length, Wm is the hopping activation energy, Tm is a constant, and m is

unity corresponding to simple activated behavior. m are 1/4 and 1/2 for Mott and Efros

VRH, respectively. In the case of Mott VRH,

W(R0)

R

W

R0


33

4

1

4/1

1

8

9

kTNR

F and

4

1

3

4/14/13

4

FNRW , (3.2)

where NF is the density of states near the Fermi energy:

3

1

2

2 4

3

AF

N

qN . (3.3)

T1/4 is then as follows

kNT

F

9

8 33

4/1 . (3.4)

On the other hand, in the case of Efros VRH,

4

1

22/1))((8

9

kTEN

ER

V

A

and

4

1

3

2/12/1 )(3

4

VENRW , (3.5)

T1/2 is then as follows

2

1

2

33

2/1)(9

8

kEN

ET

V

A

. (3.6)

As the tunneling probability is sensitive to the tunneling distance RH, doping

concentrations strongly affect the H. Figure 3.2 shows calculated H with different

doping concentrations. A steep decrease in H can be observed with lower doping

concentration. As NNH is based on activation type carrier conduction, H is further

reduced at low temperature.


34

Figure 3.2 Doping concentration dependence of Mott VRH drift mobility

Figure 3.3 shows temperature dependence of H of n-diamonds with two different

doping concentrations. In the low-temperature range, the activation energy term, WH, in

Eq. (3.1) becomes dominant, so that H decreases along with temperature. In the

high-temperature range, the influence of the term on WH2 in Eq. (3.1) becomes dominant,

so that H starts to deviate from Arrhenius plot as shown in fig. 3.5.

1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00

1.E+16 1.E+17 1.E+18 1.E+19 1.E+20

1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00

1.E+16 1.E+17 1.E+18 1.E+19 1.E+20

10-3

1016 1018 1020


(cm

2 /V

s)

100 K

300 K

500 K101

10-7

1017 1019

10-5

10-9

-1 = 1.0 nm

ph = 2.0×1015 s-1


35

Figure 3.3 Temperature dependence of Mott VRH drift mobility

3.3 Resistivity of p-type diamonds

As there are two conduction mechanisms in p-diamonds, the resistivity of p-diamond,

p, can be given by

2/12/14/14/1

1

qpqpqp BBp

, (3.7)

where B is the mobility of band conduction. Here, the concentration of mobile charge

carriers related to band conduction, pB, can be calculated by the following equation:

kT

E

h

kTm

pNN

pNp Ah

BDA

BDB exp2)( 2

3

2

*, (3.8)

where mh* is the effective mass of the hole. Assuming that there is a simple proportional

relationship between NA and the concentration of mobile charge carriers related to Mott

and Efros VRH, p1/4 and p1/2, respectively can be written as

1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00

0 4 8 12

1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00

0 4 8 120 4 8 12

1000/T (K-1)

10-3

101

10-7

10-5

10-9

-1 = 1.0 nm

ph = 2.0×1015 s-1

1×1018 cm-3

1×1020 cm-3

(cm

2 /V

s)


36

BA pNpp 2

1, 2/14/1

. (3.9)

Figure 3.4 shows the temperature dependence of resistivity obtained for n- and

n+-diamonds with doping concentration of 11018 (n-diamond) and 81019 (n+-diamond)

cm-3, respectively. The dashed and chained lines are the contribution of hopping and band

conductions, respectively. And the solid line represents the total resistivity of both

conductions. Triangle and circle symbols are the experimental data reported in Ref. 4.7.

The electron Hall mobility reported in Ref. 4.8 was adopted as B, with exponential

functions approximation as shown in fig. 3.5. Table 3.1 shows exponential approximated

Hall hole mobility of deferent doping concentration. We assumed that the compensation

rates NA/ND of n- and n+-diamonds are 22 and 52 %, and activation rate in Eq. (3.5) is

90 %. In the cases of both n- and n+-diamonds, the values of the experimental data can be

reconstructed by the calculation as shown in the fig. 4.3. Although there is no

experimental data in the low-temperature range with n-diamond, the resistivity can be

obtained with Eq. (3.3).

For n- and n+-diamonds in the higher-temperature range over 300 and 500 K,

respectively, band conduction becomes the main conduction owing to enough thermal

energy to excite electrons from ED to the conduction band. On the other hand, the

n+-diamond in a low-temperature range and the n-diamond in the lower-temperature

range than 300 K have a weaker temperature dependence, which comes from hopping

conduction. The hopping conduction is the dominant transport mechanism, which

reduces the resistivity of the n+-diamond at 300 K to ~110 -cm. At 300 K, when the

doping concentration exceeds 1019 cm-3, the resistivity can be largely reduced since

hopping conduction dominates the electron transport [3.5, 3.9].


37

Figure 3.4 Temperature dependence or resistivity

Figure 3.5 Hole mobility adopted as B by Hall effect measurement [4.8]

1.E+01

1.E+02

1.E+03

1.E+04100 1000

1.E+01

1.E+02

1.E+03

1.E+04

100 1000

1×1016 cm-32.5×1017 cm-3

2×1018 cm-3

1×1020 cm-3

101

103

Hall

mob

ility

(cm

2 /V

s)

102

104

1002

1000

Temperature (K)

3 4 5 6 7 8 9

1.E-031.E-021.E-011.E+001.E+011.E+021.E+031.E+041.E+051.E+061.E+071.E+08

0 4 8 12

1.E-031.E-021.E-011.E+001.E+011.E+021.E+031.E+041.E+051.E+061.E+071.E+08

0 4 8 12

Res

istivi

ty (ohm

-cm

)

1000/T (/K)

105

107

Resis

tivity

(-c

m)

103

101

10-1

10-3

Exp.

Mott VRH

Band

Total

Efros VRH

Exp.

Mott VRH

Band

Total

Efros VRH

0 4 8 12

1000/T (K-1)

1×1018 cm-3

1×1020 cm-3


38

Table 3.1 Exponential approximated Hall hole mobility of deferent doping

concentration

Doping concentration

N A (cm-3

)1 10

162.5 10

172 10

181 10

20

Hall hole mobility

B = a exp(-bT ) (cm2/Vs)

a = 7892

b = 0.0061

a = 4260

b = 0.0053

a = 2922

b = 0.0052

a = 411

b = 0.0029

3.4 Average mobility for current equation

Eq. (3.7) should be renewed as the following equation:

pavgtotMMEEBB

pqpqpqpqp ,

11

, (3.10)

where avg,p is the average mobility defined as

M

tot

M

E

tot

E

B

tot

B

pavgp

p

p

p

p

p , . (3.11)

Since pH is comparable to NA, Eq. (3.11) can be approximated as

MEB

tot

B

pavgp

p

2

1

2

1, . (3.12)

Figure 3.6 shows the temperature dependence of the avg,p, calculated based on Eq. (3.12).

B was obtained in the same way as the resistivity calculation. avg,p has a tendency to

increase with an increase with temperature for all doping concentrations. For lower

doping concentrations, namely 71016 and 21017 cm-3, avg,p becomes as small as ~10-13

cm2/Vs in the low-temperature range, in which hopping conduction dominates the

electron transports. For higher doping concentrations of 11018 and 81019 cm-3, namely

in the low-temperature range, avg,p shows larger value with higher doping concentrations.


39

On the other hand, while increasing the temperature, in which band conduction dominates

the electron transports, a steep increase in avg,p can be observed for samples with lower

doping concentrations, and exceeds the avg,p of higher doping concentrations.

1.E-121.E-101.E-081.E-061.E-041.E-021.E+001.E+02

100

1.E-121.E-101.E-081.E-061.E-041.E-021.E+001.E+02

100 1000

μef

f (c

m2/

Vs)

Temperature (K)

10-8

10-6

10-4

10-2

100

102

10-10

10-12

a

vg

,p(c

m2

/Vs)

1002

1000

Temperature (K)

3 4 5 6 7 8 9

1×1016 cm-3

2.5×1017 cm-3

1×1018 cm-3

2×1020 cm-3

Figure 3.6 Temperature dependence of the average mobility

Figure 3.7 shows the doping concentration dependence of the avg,n. As shown in fig.

3.6, in hopping conduction regime, the higher the doping concentrations are, the higher

the avg,n becomes. On the other hand, in band conduction regime, the lower the doping

concentration is, the higher the avg,n becomes. At 100 K, since n-diamonds with all the

doping concentrations are in hopping conduction regime, avg,n increases with an increase

in the doping concentration. At 300 K, there is only a low correlation between average

mobility and doping concentration since n-diamonds with the doping concentrations of

71016, 21017, and 11018 cm-3 are in transition from hopping to band conduction

regimes, and n+-diamond with the doping concentration of 81019 cm-3 is still in hopping


40

regime toward low temperature. At 500 K, since n-diamonds with almost all of the doping

concentrations are in band conduction regime, avg,p decreases with an increase in the

doping concentration.

Figure 3.7 Doping concentration dependence of the average mobility

From Einstein relation given by

pavpq

kTD g, , (3.13)

avg,p and diffusion coefficient (Dp) can be expressed. Therefore, the current density

equation for p-diamond can be then obtained as the following equation:

dx

dpqDEqnJ ppavgp , . (3.14)

1.E-12

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

1.E+16 1.E+17 1.E+18 1.E+19 1.E+20

1.E-12

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

1.E+16 1.E+17 1.E+18 1.E+19 1.E+20

μef

f (c

m2/

Vs)


10-8

10-6

10-4

10-2

100

102

10-10

10-12

1018 1019 1020


10171016

500 K

300 K

100 K

avg,p

(cm

2 /V

s)


41

3.5 Conclusion

We have succeeded in fitting the resistivity calculated by taking both band and

nearest-neighbor hopping conductions into account to the experimental data, especially in

the case of p+-diamond (11020 cm-3). Then, we revealed the equation of the average

mobility of p-diamond. It is suggested that the current density for p-diamond can be

calculated based on it.

References

[3.1] J. Pernot,* P. N. Volpe, F. Omnès, and P. Muret, “Hall hole mobility in boron-doped

homoepitaxial diamond,” Phys. Rev. B, vol. 81, no. 20, p205203, 2010.

[3.2] A. T. Collins and A. W. S. Williams, “The nature of the acceptor centre in

semiconducting diamond,” J. Phys. C: Solid St. Phys., vol. 4, no. 13, pp. 1789-1800,

1971.


42


44

Chapter 4 Device characteristics of

p-JFET


45

4.1 Introduction

A junction field-effect transistor (JFET) is one of the representative power switching

devices along with metal-oxide-semiconductor FETs. Figure 4.1 shows a conceptual

structure of p-type channel JFET. There is a depletion layer between gate (n-type) and

p-type in JFET as shown in shadow areas of fig. 4.1. An area except of the depletion layer

is a channel. Applying voltage between source and drain, moving electrons in the channel

make drain current. Figure 4.2 shows symbols of JFET.

G

n

n

pS D

Depletion layer

Channel

Figure 4.1 Conceptual diagram of JFET.

Figure 4.2 Symbols of JFETs

G

D

S G

D

S

n-JFET p-JFET


46

Figure 4.3 shows bias voltage of JFET. JFET is used, applying reverse bias voltage to

the gate.

When VD is kept constant and the absolute value of VG increases, the bias voltage

applied to p-n junction increases and the depletion layer extends. Then, the thickness of

the channel decreases and ID shrinks. Thus, VG can control ID. Note that current does not

flow into the gate since VG is the reverse bias voltage.

Next, we consider when keeping VG constant and increasing VD. When VD is still small,

ID increases as VD becomes bigger. As VD becomes bigger further, the reverse bias voltage

between gates and drain increases, and the depletion layers extend toward the drain

electrode. When VD reaches pinch-off voltage VP, the upper and lower depletion layers

overlap each other in the vicinity of the drain, and the channel disappears, as shown in fig.

4.4 (a). As VD becomes lager than VP, the depletion layers extend further, as shown in fig.

4.4 (b) but ID does not change because of no carrier in the depletion layers.

VDDVG

G

S

D

VD

ID

Figure 4.3 bias voltage of p-JFET


47

G

n

n

pS D

Depletion layerG

G

n

n

pS D

Depletion layerG

(a) (b)

Figure 4.4 (a) pinch-off just occurs and (b) a state after it, in p-JFET

4.2 Derivation of current-voltage characteristics of JFET

Here, we approximately derive current-voltage characteristics of JFET. VD is applied to

both sides of n-type semiconductor (thickness a, length L, width W, and mobility n) as

shown in fig. 4.5 (a).

p

VG

VD

L

pVD

L

n

0

a a

x

l (x)

(a) (b)

Figure 4.5 Quantitative analysis model of JFET (a) if there is no depletion layer and (b) if

there is a depletion layer.

Conductivity, is given by


48

DnNq . (4.1)

Since cross-sectional area is 2aW and channel length is L, Conductance, g0 is given by

L

aWNq

L

aWg Dn 2

20 . (4.2)

Therefore, drain current, ID is given by

DDnDD VL

aWNqVgI 20 . (4.3)

Next, as shown in fig. 4.4 (b), p-type semiconductor is contacted to n-type semiconductor

in fig. 4.4 (a). We assume that width of depletion layer in p-type semiconductor can be

neglected since NA ≫ ND. Width of depletion layer, l (x) is treated as a function of position

x

D

Gbi

qN

xVVVxl

)(2)(

, (4.4)

where is the dielectric constant, Vbi is the built-in potential between p-n junction, and

V(x) is the potential difference from both sides of the depletion layer. g0 reduces to g0’ as

there is no carrier in the depletion layer. g0’ is given by

L

dxxlaL

gg0

00 )(1

1 , (4.5)

V(x) is treated as a primary function of x:

DVL

xxV )( . (4.6)

Then,

2

3

2

3

003

21 GbiDGbi

aD

VVVVVVV

gg , (4.7)

where Va is the voltage to deplete fully at VD = 0, including Vbi, and is defined as

2

2

A

a

qNaV . (4.8)


49

When VD does not reach VP, current-voltage characteristics of JFET is as follows

2

3

2

3

003

2GbiDGbi

a

DDD VVVVVV

VgVgI . (4.9)

VP is given by

abiGP VVVV . (4.10)

When VDS reaches VP, current-voltage characteristics of p-JFET is as follows

2

3

0 2313 a

Gbi

a

Gbia

DsatV

VV

V

VVVgI . (4.11)

4.3 Calculation results

First of all, we explain why p-JFET was selected instead of n-JFET. Generally

speaking, electron mobility is higher than hole mobility. However, the average hole

mobility is higher than the average electron mobility in 300 K. That is why a calculation

of p-JFET was decided.

For a calculation of device characteristics of diamond p-JFET, we assumed as

following:

The doping concentrations of a boron-doped p-type layer and of the highly

phosphorus-doped n+ diamonds were 1 1016, 8 1019 cm-3, respectively.

The channel width (2a) was 2 m, the channel length (L) was 7 m, and the channel

thickness (W) was 0.7 m.

avg,p calculated in Chapter 3was adopted to the hole mobility of the p-type layer.


50

After calculating the device characteristics at 300 K, based on the equations derived in

the previous section, the results were compared with those of Si.

Depletion layer

Depletion layer

8×1019cm-3

1×1016cm-3

Channel

W = 0.7m

LVG

VD

Gate

Gate

Source Drain

8×1019cm-3

a = 1m

Figure 4.6 Adopted device architecture of JFET

Figure 4.7 shows how to decide the drift length Ldrift. Ldrift is decided as the region

between gate end and drain electrode can bear breakdown field. Then, each breakdown

field is adopted to 0.3 and 10 MV/cm in the case of Si and diamond, respectively.


51

VG

pVD

Depletion layer

LdriftLch = 0.7m

L

Figure 4.7 How to decide drift length Ldrift

4.3.1 Specific on-resistance vs. blocking voltage

The built-in potential, Vbi was calculated by an equation as follows:

2ln

i

ADbi

n

NN

q

kTV , (4.12)

where ni2 of diamond and Si are 2.2 10-27 and 9.6 109 cm-3, respectively. Based on eqs.

(4.8) and (4.12), Va and Vbi were calculated and the values as shown in tab. 4.1.

Table 4.1 Values of diamond and Si

Material V bi (V) V a (V) V TH = V a - V bi (V)

Diamond 5.3 15.6 10.3

Si 1.0 7.7 6.7


52

Figure 4.6 shows the specific on-resistance RonA vs. blocking voltage. In more than

400 V, it is confirmed that p-diamond has higher blocking voltage and lower loss than

n-Si.

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+02 1.E+03 1.E+04 1.E+05

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+02 1.E+03 1.E+04 1.E+05

100 100k10k

Blocking voltage (V)

1k

Sp

ecific

On

-Re

sis

tan

ce

Ro

nA

(-c

m2)

0.01

0.1

1

10

100

n-Si

p-diamond

Figure 4.6 Specific on-resistance vs. blocking voltage with p-diamond and n-Si

4.3.2 ID-VD characteristics

Figure 4.7 show the ID-VD curves of JFETs of n-Si and p-diamond when blocking

voltage is 5000 V. Then, the device lengths of n-Si and p-diamond are 170 and 12 m,

respectively calculated as mentioned above. When blocking voltage is 5000 V, current


53

of p-diamond is larger and device length of that is smaller than n-Si. Thus, it is revealed

that p-diamond is superior to n-Si from various viewpoints.

-4

-3

-2

-1

0

-10-8-6-4-20

-4

-3

-2

-1

0

-10-8-6-4-200 -6-4 -10

VD (V)

-2 -8

VG =

-5 V

7 V

V

G=

2 V

I D(

A)

0

-1

-2

p-diamond

L = 12 m-3

-4

0

1

2

3

4

0 2 4 6 8 10

0

1

2

3

4

0 2 4 6 8 10

0 -6-4 -10

VD (V)

-2 -8

VG =

0 V

7 V V

G=

1 V

n-Si

L = 170 m

I D(

A)

0

-1

-2

-3

-4

Figure 4.7 ID-VD curves of JFETs of n-Si and p-diamond


54

4.4 Conclusion

Device characteristics of p-channel junction field-effect transistor are calculated

based on a calculation. P-diamond is high breakdown voltage and low loss, compared to

n-Si. When blocking voltage is 5000 V, current of p-diamond is larger and device length

of that is smaller than n-Si. Thus, it is revealed that p-diamond is superior to n-Si from

various viewpoints.

References

[5.1] T. Iwasaki, Y. Hoshino, K. Tsuzuki, H. Kato, T. Makino, M. Ogura, D. Takeuchi,

T. Matsumoto, H. Okushi, S. Yamasaki, and M. Hatano, “Diamond Junction

Field-Effect Transistors with Selectively Grown n+-Side Gates,” Appl. Phys. Exp.,

vol. 5, p. 091301, 2012.


55


56

Chapter 5 Conclusion


57

The conclusions in this thesis are summarized as following:

1) In the case of n-diamonds, it is confirmed that the resistivity consists of those of

the band and nearest-neighbor hopping (NNH) conductions. In the case of

p-diamonds, on the other hand, it is confirmed that the resistivity consists of

those of the band and Mott and Efros variable-range hopping (VRH). We have

succeeded in fitting the resistivity, calculated by taking both band and

nearest-neighbor hopping conductions into account, to the experimental data,

especially in the case of n+- (8 1019 cm-3) and p+- diamonds (1 1020 cm-3).

Then, we revealed the equation of the average mobility of n- and p-diamond. It

is suggested that the current density for n- and p-diamond can be calculated

based on it.

2) It has been found that by introducing an average mobility, which is a weighted

average of each mobility of the hopping and band conductions, current density

equations can be successfully expressed.

The achievement gives a rough current estimation of diamond devices.

3) Device characteristics of p-channel junction field-effect transistor are calculated

based on a calculation. P-diamond is high breakdown voltage and low loss,

compared to n-Si. When blocking voltage is 5000 V, current of p-diamond is

larger and device length of that is smaller than n-Si. Thus, it is revealed that

p-diamond is superior to n-Si from various viewpoints.


58

Acknoledgement

First of all, I would like to express my gratitude to my supervisor Prof. Hiroshi Iwai

and Associate Prof. Kuniyuki Kakushima for his continuous encouragement and advices

for this thesis. He also gave me many chances to attend conferences. The experiences are

precious for my present and future life.

I deeply thank to Prof. Takeo Hattori, Prof. Kenji Natori, Prof. Nobuyuki Sugii, Prof,

Akira Nishiyama, Prof. Kazuo Tsutsui, Prof. Yoshinori Kataoka, Associate Prof. Parhat

Ahmet, and Associate Prof. Kuniyuki Kakushima for useful advice and great help

whenever I met difficult problems.

I also thank research colleagues of Iwai Lab. for their friendship, active many

discussions and many of encouraging words.

I would like to appreciate the support of secretaries, Ms. Nishizawa and Ms.

Matsumoto.

I would like to thank Mr. Yeonghun Lee.

Finally, I would like to thank my grandparents Kohji and Sono, parents Yasuo and

Toshiko, and my brother Kensuke for their endless support and encouragement.

Michihiro Hosoda

February, 2013


59