Carrier Dynamics in the Nitride Semiconductors · Carrier Dynamics in the Nitride Semiconductors A...

UNIVERSITY OF CALIFORNIA Santa Barbara

Carrier Dynamics in the Nitride Semiconductors

A Dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in

Electrical and Computer Engineering

Kian-Giap Gan

Committee:

Professor John E. Bowers, Chair Professor Chi-Kuang Sun

Professor Steven P. DenBaars Professor Shuji Nakamura

Professor David D. Awschalom

December, 2006

The dissertation of Kian-Giap Gan is approved

Chi-Kuang Sun

Steven P. DenBaars

Shuji Nakamura

David D. Awschalom

John E. Bowers, Chair

Dec, 2006

Carrier Dynamics in the Nitride Semiconductors

Copyright © 2006 by Kian-Giap Gan All right reserved

Electrical and Computer Engineering Department University of California

Santa Barbara, CA 93106

iii

Acknowledgements

Acknowledgements

iv

I will always remember my time at UCSB, not only because the beautiful

scenery and the gorgeous weather, but also because of the wonderful experience

I have had and the wonderful people I have met.

First of all, I would like to express my deepest appreciation to my

advisor, Professor John Bowers, for provided a stimulating environment with

total academic freedom and allowed me to choose my own research direction.

Along the way, he has been supportive, providing guidance and mentorship and

the funding that the project required. My Ph.D. committee has helped shape and

guides the scope of this work and I thank them for their contribution: Professors

Chi-Kuang Sun, Steve P. DenBaars, Shuji Nakamura, and David D. Awschalom.

I would like to thank Arpan and Rajat for providing me the InGaN thin

films sample that was used in the measurement. I also want to thanks Professor

Claude Weisbush for collaboration and discussion on the study of the carrier

transport dynamics in the InGaN thin films sample.

Finally, I would like to thank my parents for always supporting and

encouraging me throughout the course of my life.

Acknowledgements

v

This dissertation is dedicated

to my family

Curriculum Vitæ

Curriculum Vitæ

vi

Kian-Giap Gan

Employment

1999-2005 Graduate Student Researchers Professor John Bowers, University of California, Santa Barbara, California, USA

1998-1999 Undergraduate Student Research Assistant

Professor Chi-Kuang Sun, National Taiwan University, Taipei, Taiwan

Education

2001-2006 Ph.D. in Electrical and Computer Engineering Professor John Bowers, University of California, Santa Barbara, California, USA.

1999-2001 M.S. in Electrical Engineering

Professor John Bowers, University of California, Santa Barbara, California, USA.

1995-1999 B.S. in Electrical Engineering

National Taiwan University, Taipei, Taiwan.

Awards

Curriculum Vitæ

vii

2004 Walsin Fellowship 1994 First place in the Malaysia National Mathematics Competition 1993 Third place in the Malaysian Mathematical Olympiad

Publications

Publications

viii

First-author journal papers

Kian-Giap Gan, Chi-Kuang Sun, Steven P. DenBaars, and John E. Bowers, “Ultrafast hole intervalence subband relaxation in an InGaN multiple quantum well laser diode,” Applied Physics Letters, 80, pp. 4054-4056, 2002.

Kian-Giap Gan, and John E. Bowers, “Measurements of Gain, Group index, and Linewidth Enhancement Factor of an InGaN Multiple Quantum Well Laser Diode,” IEEE Photonics Technology Letters, 16, pp. 1256-1258, 2004.

Kian-Giap Gan, Jin-Wei Shi, Yen-Hung Chen, Chi-Kuang Sun, Yi-Jen Chiu, and John E. Bowers, “Ultrahigh power-bandwidth-product performance of low-temperature-grown-GaAs based metal-semiconductor-metal traveling-wave photodetectors,” Applied Physics Letters, 80, pp. 4054-4056, May 2002.

First-author conference papers

Kian-Giap Gan, J. E. Bowers, and Chi-Kuang Sun, “Femtosecond carrier dynamics in InGaN multiple-quantum-well laser diodes under high injection levels,” in Proceedings of the 17th Annual Meeting of the IEEE Lasers and Electro-Optics Society, Vol.2, Puerto Rico, 2004, pp. 669-670.

Kian-Giap Gan, Chi-Kuang Sun, John E. Bowers, and Steven P. DenBaars, “Ultrafast carrier dynamics in InGaN MQW laser diode,” in Proceedings of SPIE Vol.4992 Ultrafast Phenomena in Semiconductors VII, 2003, pp. 83-89. (invited)

Publications

ix

Kian-Giap Gan, John E. Bowers, Steven P. DenBaars, and Chi-Kuang Sun, “Ultrafast inter-subband hole relaxation in an InGaN multiple-quantum-well (MQW) laser diode,” in Ultrafast Electronics and Optoelectronics, 2003, pp. 38-41.

Kian-Giap Gan, Jin-Wei Shi, Yi -Jen Chiu, Chi-Kuang Sun, and J. E. Bowers, “Self-aligned MSM low-temperature-grown GaAs traveling wave photodetector for 810 nm and 1230 nm,”. International Topical Meeting on Microwave Photonics 2001, paper Tu-4.15, pp. 153-155, Long Beach, California, January 7-9 (2002).

Kian-Giap Gan, Jin-Wei Shi, Yi-Jen Chiu, Chi-Kuang Sun, and John E. Bowers, “Self-Aligned 0.8ps FWHM MSM Traveling Wave Photodetector Using Low-Temperature-Grown GaAs,” in Ultrafast Electronics and Optoelectronics, 2001, pp. 114-116.

Coauthored papers and presentations

J. Geske, K.-G. Gan, Y. L. Okuno, J. Piprek, and J. E. Bowers, “Vertical-cavity surface-emitting laser active regions for enhanced performance with optical pumping,”, IEEE Journal of Quantum Electronics, 40, pp. 1155-1162, 2004.

J. Geske, K.-G. Gan, Y. L. Okuno, B. Barnes, J. Piprek, and J. E. Bowers, “Vertical-cavity surface-emitting laser active regions for enhanced performance with optical pumping,” in Proceedings of the Conference on Lasers and Electro Optics, 2004, pp. 815-816.

D. Lasaosa, Jin-Wei Shi, D. Pasquariello, Kian-Giap Gan, Ming-Chun Tien, Hsu-Hao Chang, Shi-Wei Chu, Chi-Kuang Sun, Yi-Jen Chiu, and J. E. Bowers, “Traveling-wave photodetectors with high power-bandwidth and gain-bandwidth product performance,” IEEE Journal of Select Topics in Quantum Electronics, 10, pp. 728-741, 2004.

Publications

x

Jin-Wei Shi, Yen-Hung Chen, Kian-Giap Gan, Yi-Jen Chiu, John E. Bowers, Ming-Chun Tien, Tzu-Ming Liu, and Chi-Kuang Sun, “Nonlinear Behaviors of Low-Temperature-Grown GaAs-Based Photodetectors Around 1.3-µm Telecommunication Wavelength,” IEEE Photonics Technology Letters, 16, pp. 242-244, 2004.

Yae Okuno, Jonathan Geske, Kian-Giap Gan, Yi-Jen Chiu, Steven P. DenBaars, and John E. Bowers, “1.3 µm wavelength vertical cavity surface emitting laser fabricated by orientation-mismatched wafer bonding: A prospect for polarization control,” Applied Physics Letters, 82, pp. 2377-2379, 2003.

Jin-Wei Shi, Kian-Giap Gan, Yen-Hung Chen, Chi-Kuang Sun, Yi-Jen Chiu, and J. E. Bowers, “Ultrahigh-power-bandwidth product and nonlinear photoconductance performances of low-temperature-grown GaAs-based metal-semiconductor-metal traveling-wave photodetectors,” IEEE Photonics Technology Letters, 14, pp. 1587-1589, 2002.

Jin-Wei Shi, Yen-Hung Chen, Kian-Giap Gan, Yi-Jen Chiu, Chi-Kuang Sun, and J. E. Bowers, “High-speed and high-power performances of LTG-GaAs based metal-semiconductor-metal traveling-wave-photodetectors in 1.3-µm wavelength regime,” IEEE Photonics Technology Letters, 14, pp. 363-365, 2002.

Jin-Wei Shi, Kian-Giap Gan, Yi-Jen Chiu, Yen-Hung Chen, Chi-Kuang Sun, Ying-Jay Yang, and J. E. Bowers, “Metal-semiconductor-metal traveling-wave photodetectors,” IEEE Photonics Technology Letters, 13, pp. 623-625, 2001.

Jin-Wei Shi, Kian-Giap Gan, Yi-Jen Chiu, J. E. Bowers, and Chi-Kuang Sun, “High power performance of ultrahigh bandwidth MSM TWPDs,” in Proceedings of the 14th Annual Meeting of the IEEE Lasers and Electro-Optics Society, Vol.2, Piscataway, NJ, 2001, pp. 887-888. Jin-Wei Shi, Kian-Giap Gan, Yi-Jen Chiu, Chi-Kuang Sun, Yin-Jay Yang, and J. E. Bowers, “Ultrahigh bandwidth MSM traveling-wave photodetectors,” in Proceedings of the Conference on Lasers and Electro Optics, 2001, pp. 348.

Abstract

Abstract

Carrier Dynamics in the Nitride Semiconductor

by

Kian-Giap Gan

The group-III nitride semiconductor alloys AlN-GaN-InN are recognized as an

important material system for optoelectronic devices in the spectral range from

infrared to ultraviolet. This thesis will investigate the carrier relaxation and

carrier transport dynamics in InGaN, which are important for high speed device

design.

Time-resolved pump-probe measurement is used to study carrier

relaxation dynamics in an InGaN multiple-quantum-well (MQW) laser diode.

Using the optical selection rule in InGaN, different subbands can be selectively

pumped and probed using ultrafast optical pulse with different polarization. An

ultrafast intersubband relaxation process ( <0.35 ps) is found to be important to

carrier dynamics in InGaN.

τ

A novel heterodyne transient grating measurement is developed and used

to study the carrier transport dynamics in InGaN. The measured diffusion

xi

Abstract

xii

constant is very small (~0.2 cm2/s), indicate that carrier localization plays an

important role in the carrier transport in InGaN MQWs. A simple model is

presented to explain the measurement result.

Table of Contents

Table of Contents

xiii

Chapter 1. Introduction......................................1

1.1 Motivation ................................................................1

1.2 Organization of dissertation......................................2

1.3 Reference ..................................................................3

Chapter 2. Gain in nitride semiconductor laser

diodes ............................................................5

2.1 Fourier Transform (FT) method ...............................7

2.1.1 Calculation of net modal gain............................................ 9

2.1.2 Calculation of group index and group velocity

dispersion ........................................................................... 9

2.1.3 Calculation of linewidth enhancement factor.................. 10

2.1.4 Calculation of temperature-induced index change.......... 12

2.2 Measurement results of InGaN MQW laser diode .13

2.2.1 Measurement of gain........................................................ 16

2.2.2 Measurement of group index and group velocity

dispersion ......................................................................... 17

2.2.3 Measurement of linewidth enhancement factor ............... 18

2.2.4 Measurement of thermal-induced index change.............. 21

Table of Contents

xiv

2.3 Summary.................................................................23

2.4 References ..............................................................24

Chapter 3. Intervalence subband carrier

dynamics in InGaN MQW laser diodes ...............26

3.1 Introduction ............................................................26

3.2 Band structure and optical selection rule for

InGaN .....................................................................28

3.3 Bias lead monitoring measurement ........................35

3.4 Time-resolved differential reflection measurement 40

3.4.1 Determination of transparency level ............................... 44

3.4.2 Intersubband relaxation under high carrier density........ 46

3.5 Summary and future work ......................................49

3.6 Reference ................................................................51

Chapter 4. Diffusion constant of InGaN MQW

thin films ..........................................................53

4.1 Conventional transient grating measurement .........54

4.2 New heterdodyne transient grating measurement...57

4.3 Important optical alignment issue...........................63

Table of Contents

xv

4.4 Validation of the new heterodyne transient grating

method ....................................................................65

4.5 Diffusion constant of InGaN MQW .......................67

4.6 Diffusion model......................................................73

4.7 Summary and future work ......................................77

4.8 Reference ................................................................78

Chapter 5. Summary and future work............79

5.1 Gain measurement ..................................................79

5.2 Valence intersubband relaxation.............................79

5.3 Carrier transport in InGaN......................................80

5.4 Reference ................................................................81

Chapter 1. Introduction



important material system for optoelectronic devices in the spectral range from

infrared to ultraviolet. GaN-InN based III-V nitride semiconductors are of

interest in many commercial applications, such as light emitting diodes (LEDs)

and laser diodes (LDs) [1, 2].

1.1 Motivation

The carrier dynamics, which are important for high speed device design, have

recently been studied by femtosecond time-resolved pump-probe [3] or coherent

spectroscopy [4, 5] with above band gap photons. However, various

contributions such as electron-electron scattering, hole-hole scattering, electron-

hole scattering, electron-phonon interactions, and hole-phonon interactions, mix

together and make it very difficult to extract the fundamental material parameter

for one particular scattering process or single type of carrier. C.-K. Sun et al.

and H. Ye et al. have developed an infrared pump-ultraviolet probe technique to

1


isolate electron and hole dynamics and used it to study the electron relaxation

dynamics in n-type GaN thin films [6, 7] and hole dynamics in p-type GaN thin

films [8]. In this thesis, we will use the optical selection in the InGaN quantum

well to selectively pump and probe different subbands, and this allows us to

study specifically the relaxation of carrier among various subbands in the

valence band.

It is well known that nitrides have strong internal electric fields due to

the spontaneous polarization and piezo-electric effect [9]. K. Omae et al. have

used pump-probe spectroscopy to study the effects of internal electrical field on

the transient absorption in InxGa1-xN quantum well thin films with different

thickness [10]. The effect of the internal electric fields will be included in the

calculation and explanation for the data.

1.2 Organization of dissertation

The dissertation will start by the measurement of the gain spectrum, group

velocity dispersion, and other interesting parameters in an InGaN multiple-

quantum-well (MQW) laser diode with the Fourier transform method in Chapter

2. In Chapter 3, time-resolved pump-probe measurement is used to study the

carrier relaxation dynamic in an InGaN multiple-quantum-well (MQW) laser

diode. Using the optical selection rule in InGaN, difference subbands can be

2


selectively pump and probe using ultrafast optical pulse with difference

polarization. An ultrafast intersubband relaxation process (τ <0.35 ps) is found

to be important to carrier dynamics in InGaN. In Chapter 4, a novel heterodyne

transient grating measurement is developed and used to study the carrier

transport dynamics in InGaN. The measured diffusion constant is very small

(~0.2 cm2/s), indicate that carrier localization plays an important role in the

carrier transport in the InGaN MQW. A simple model is presented to explain

the measurement result.

1.3 Reference [1] S. Nakamura, M. Senoh, and T. Mukai, "High-power InGaN/GaN

double-heterostructure violet light emitting diodes," Applied Physics Letters, vol. 62, pp. 2390-2392, 1993.

[2] S. Nakamura, M. Senoh, S.-i. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, Y. Sugimoto, T. Kozaki, H. Umemoto, M. Sano, and K. Chocho, "InGaN/GaN/AlGaN-based laser diodes with modulation-doped strained-layer superlattices grown on an epitaxially laterally overgrown GaN substrate," Applied Physics Letters, vol. 72, pp. 211-213, 1998.

[3] C. K. Sun, F. Vallee, S. Keller, J. E. Bowers, and S. P. DenBaars, "Femtosecond studies of carrier dynamics in InGaN," Applied Physics Letters, vol. 70, pp. 2004-2006, 1997.

[4] S. Pau, J. Kuhl, F. Scholz, V. Haerle, M. A. Khan, and C. J. Sun, "Femtosecond degenerate four-wave mixing of GaN on sapphire: Measurement of intrinsic exciton dephasing time," Physical Review B, vol. 56, pp. R12718, 1997.

[5] R. Zimmermann, A. Euteneuer, J. Mobius, D. Weber, M. R. Hofmann, W. W. Ruhle, E. O. Gobel, B. K. Meyer, H. Amano, and I. Akasaki, "Transient four-wave-mixing spectroscopy on gallium nitride: Energy splittings of intrinsic excitonic resonances," Physical Review B, vol. 56, pp. R12722, 1997.

3


[6] C. K. Sun, Y. L. Huang, S. Keller, U. K. Mishra, and S. P. DenBaars, "Ultrafast electron dynamics study of GaN," Physical Review B, vol. 59, pp. 13535, 1999.

[7] H. Ye, G. W. Wicks, and P. M. Fauchet, "Hot electron relaxation time in GaN," Applied Physics Letters, vol. 74, pp. 711-713, 1999.

[8] H. Ye, G. W. Wicks, and P. M. Fauchet, "Hot hole relaxation dynamics in p-GaN," Applied Physics Letters, vol. 77, pp. 1185-1187, 2000.

[9] S. Nakamura and S. F. Chichibu, Introduction to Nitride Semiconductor Blue Lasers and Light Emitting Diodes. New York: Taylor & Francis, 2000.

[10] K. Omae, Y. Kawakami, S. Fujita, Y. Narukawa, and T. Mukai, "Effects of internal electrical field on transient absorption in InxGa1-xN thin layers and quantum wells with different thickness by pump and probe spectroscopy," Physical Review B (Condensed Matter and Materials Physics), vol. 68, pp. 085303-5, 2003.

4

Chapter 2. Gain in nitride semiconductor laser diode

Chapter 2. Gain in nitride semiconductor laser diodes

The measurement of gain (absorption) spectra is an important material

characterization tool for the development of semiconductor lasers,

semiconductor optical amplifiers, and other waveguide devices. A number of

different methods have been proposed to determine the net gain spectrum from

transmission spectra or spontaneous emission spectra. Among the many ways of

measuring net gain spectrum, Hakki-Poali (HP) method is well known for it’s

accuracy, versatility and simplicity [1, 2]. Beside the gain spectrum, HP method

can also be use to deduce information on the refractive index such as carrier-

induced index change ( ) by observing the wavelength shift of Fabry-

Perot resonance [3]. In the HP method, a series of peaks and valleys of

individual Fabry-Perot resonances is recorded, and the gain spectrum can be

calculated from the peak-to-valley ratio. In order to accurately measure the

peak-to-valley ratio, the resolution bandwidth of the measurement instrument

has to be high enough to resolve the Fabry-Perot mode in the measured

spectrum. If the instrument’s resolution bandwidth is not enough to resolve the

/dn dN

5


Fabry-Perot mode, then the measured peak (valley) will be lower (higher) than

the actual peak (valley). These will cause the peak-to-valley ratio to be under

estimated and result in an under estimation of the gain. The instrument

resolution limitation is particular acute when the laser diode is biased close to

the threshold. As the biases of the laser diode approached threshold, the peak-

to-valley ratio will approach infinity, and it becomes increasingly difficult to get

a good measure of the peak-to-valley ratio under this situation. There have been

some papers that aim to alleviate the resolution requirement of the HP method

[4-6]. Most of the research in this area involved the assumption of the shape of

the instrument response function that may not be satisfied by the experiment

conditions. Recently, Hofstetter and Thornton proposed a technique that using

Fourier Transform (FT) analysis on the measured spectrum to obtain the gain

spectrum [7]. In contrast with the HP method, FT method allows the instrument

response correction if the instrument response function can be measured [8]. As

a result, the requirement of the high wavelength resolution of the measurement

instrument can be relaxed. This makes the FT method more suitable for the gain

spectrum measurement on nitride waveguide structure than the HP method.

This is because the emission wavelength of the nitride is shorter and thus has a

smaller Fabry-Perot mode spacing and is more difficult to resolve. On top of the

measured instrument response calibration capability, the FT method is also less

sensitive to the noise in the measured spectrum [8]. Just aas with HP method,

6


information about the refractive index of the measured waveguide can also be

obtained using the FT method [9]. In this chapter, we are going to focus on the

FT method. First, we will show how to use the FT method to calculate the

physical parameter of interest; namely, gain, group index ( ), group velocity

dispersion (GVD), carrier-induced index change ( ), linewidth

enhancement factor (α ) and thermal-induced index change ( ). Then we

will apply the equations that we derived in session 2.1 to the measurement of an

InGaN multiple-quantum-well (MQW) laser diode.

gn

/dn dN

/dn dT

2.1 Fourier Transform (FT) method

We start with the calculation of the gain-reflectivity product (b ) and the round

trip phase ( ) from the measured spontaneous emission spectrum [8,

10]. Where is the wavenumber ( is wavelength). Let us assume

that be the measured instrument response function. Define the Fourier

transform pair as

φ ( )ASES β

1/β = λ λ

( )INSF β

2

2

( ) ( )

( ) ( )

j z

j z

H H z e

H z H e d

+∞+ πβ

−∞+∞

− πβ

−∞

β =

= β

∫

∫

dz

β

<2-1>

7


First, take the Fourier transform of the measured spontaneous emission spectrum

and the Fourier transform of the measured instrument response function. ( )ASES β

2( ) ( ) j zASE ASES z S e d

+∞− πβ

−∞

= β∫ β <2-2>

2( ) ( ) j zINS INSF z F e d

+∞− πβ

−∞

= β∫ β <2-3>

The gain reflectivity product and the round trip phase can be calculated by

32

2

( )( )

( )( )( )

nLASE j z

INSnLnL

ASE j z

INSnL

S ze dz

F zb

S ze dz

F z

++ πβ

++

+ πβ

−

β =∫

∫ <2-4>

32

2

( )( )

( )( )( )

nLASE j z

INSnLnL

ASE j z

INSnL

S ze dz

F z

S ze dz

F z

++ πβ

++

+ πβ

−

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟φ β = ⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

∫

∫ <2-5>

The gain-reflectivity product (b ) is related to the net modal gain (g ) by

= ⋅ ⋅ ⋅1 2 exp( )b R R g L <2-6>

and the round trip phase (φ ) is related to the refractive index (n ) as

<2-7> φ π β= ⋅ ⋅ ⋅ ⋅ − 04 n L φ

8


where and are the power reflectivity of the front and back facet, g is the

net modal gain, L is the cavity length, is the wavenumber, n is the

effective refractive index, and φ is an unknown phase constant [10].

1R 2R

1/β = λ

0

2.1.1 Calculation of net modal gain

From the gain-reflectivity product (b ), the net modal gain can be calculated as

= ⋅ − ⋅ ⋅⋅ 1 2

1 1ln( ) ln( )

2g b RL L

R . <2-8>

The net modal gain can be calculated if the values of the facet reflectivity and

cavity length are known.

2.1.2 Calculation of group index and group velocity dispersion

The refractive index n cannot be calculated directly from <2-7>, because the

value of is unknown, unless can be determined by using one known

index within the measured spectra range [10]. Fortunately, without knowing the

value of , some important parameters of interest can still be calculated from

<2-7>. For example, the first derivative of <2-7> can be used to calculated the

group index as

0φ 0φ

0φ

9


φβ

β π= + ⋅ = ⋅

⋅ ⋅g

14

dn dn n

d L βd. <2-9>

And the second derivative of <2-7> can be used to calculate the GVD as

βλ π

−≡ = ⋅

⋅ ⋅

2 2g

2GVD4

dn dd L

φβd

dI

. <2-10>

2.1.3 Calculation of linewidth enhancement factor

If a series of spontaneous emission spectra can be measured under several

different injection current (I ) levels and the step of the current level change are

kept small enough so that the induced Fabry-Perot mode shift is smaller than

one mode spacing, then the linewidth enhancement factor can be calculated with

the following procedure.

First, the gain (g ) and round trip phase ( ) at different injection current

levels are calculated as shown previously. The next step is to calculate the

derivative of the gain with respect to the current ( ) and the derivative of

the round trip phase with respect to the current ( ) from the calculation

results of the first step. The derivative of the index with respect to the current

( ) is related to by

φ

/dg dI

/d dIφ

/dn dI /dφ

φπ β

=⋅ ⋅ ⋅

14

dn ddI L dI

⋅ . <2-11>

10


Although the facet reflectivity does change as the current injection levels change

due to the carrier-induced refractive index change, is dominated by the

change of the first term in <2-8>. The carrier-induced reflectivity change is

small compared with the first term in <2-8> because the carrier-induced index

change is very small; unless the facet reflectivity is very low (antireflection-

coated facet). So can be calculated by neglecting the carrier-induced

reflectivity change as

/dg dI

/dg dI

= ⋅1 ln(dg d b

dI L dI) . <2-12>

The linewidth enhancement factor (α ) can be calculated with [11]

πα

λ λ⋅ ⋅

= − ⋅ = − ⋅4 4 dn

dIdgdI

dndg

π . <2-13>

From <2-11>, <2-12>, and <2-13>

φ

α = − ln( )

ddId bdI

<2-14>

It is interesting to point out that the linewidth enhancement factor can be

calculated in this way without knowing the cavity length.

Since the carrier density ( ) can be related to the injection current (I )

by [11]

N

2 3i

a life

I NA N B N C N

q V⋅ = ⋅ + ⋅ + ⋅ =

⋅η

τ <2-15>

11


where η is the injection efficiency, q is the charge of an electron, and is the

active volume of the laser diode, A is the non-radioactive recombination rate,

is carrier life time, B is the bi-molecular recombination coefficient, and C

is the Auger recombination coefficient. The derivative is given by

i aV

τlife

/dI dN

2a a

i i life,

( 2 3 )dI q V q V N

A B N C NdN ∆

⋅ ⋅= ⋅ + ⋅ ⋅ + ⋅ ⋅ = ⋅

η η τ <2-16>

Where is the differential carrier life time. life,∆τ

So the differential gain ( ) and carrier-induced index change ( )

can be calculated as

/dg dN /dn dN

1 ln(dg dI dg dI d bdN dN dI dN L dI

= ⋅ = ⋅ ⋅) <2-17>

14

dn dI dn dI ddN dN dI dN L dI

φπ β

= ⋅ = ⋅ ⋅⋅ ⋅ ⋅

<2-18>

2.1.4 Calculation of temperature-induced index change

Similar to the previous analysis of the carrier-induced index change, if a series

of spontaneous emission spectra can be measured under several different

temperatures (T ) at a fixed injection current level, then the thermal-induced

index change can be calculated as show below. Again, the step of the

12


temperature change needs to be kept small enough so that the temperature-

induced Fabry-Perot mode shift is smaller than one mode spacing.

φα

π β+ ⋅ = ⋅

⋅ ⋅ ⋅thermal

14

dn dn

dT L dT <2-19>

The second term on the left-hand-side of <2-19> is due to the thermal expansion

of the cavity length. As will be shown in the next session, this thermal

expansion term is an order of magnitude smaller than and can be

neglected.

/dn dT

2.2 Measurement results of InGaN MQW laser diode

The schematic diagram of the measurement setup is shown in Figure 2-2. The

polarization-resolved spontaneous emission spectrum from an InGaN multiple-

quantum-well (MQW) blue laser diode is collected using a high-resolution

grating spectrometer (SPEX, 0.5-m focal length, 1200 lines/mm grating, 10-µm

slit width) operated at the second order (~ 0.02-nm resolution bandwidth). The

grating installed in our spectrometer has very low diffraction efficiency for the

transverse-magnetic (TM) polarization. Due to this limitation, we limited the

measurement to the transverse-electric (TE) polarization. The laser diode was

mounted on a temperature controlled stage in order to maintain constant

temperature (20 °C). All the spontaneous emission spectra were measured

13


under pulsed operation in order to avoid heating due to current injection. The

properties of the blue laser diode are summarized as following: lasing

wavelength = 403.5 nm, threshold current = 33 mA, and cavity length = 670 µm.

The laser diode we used in this measurement is a commercial device and the

structure of the laser diode is very similar to the one in [12] and is shown in

Figure 2-1. The In0.15Ga0.85N/In0.02Ga0.98N MQW structure consisting of four 3.5

nm Si-doped In0.15Ga0.85N well layers forming a gain medium separated by 10.5

nm Si-doped In0.02Ga0.98N barrier layers. The measured wavelength range of the

spontaneous emission spectrum of 395-415 nm is chosen (an example of

measured spectrum is shown in Figure 2-3) so that the values of the emission

spectrum at both ends of the measured wavelength range are about the same and

small in order to reduce the spectrum leakage when performing the FT method.

No window function was used in the FT method. The instrument response

deconvolutions were performed with a measured response function [8, 10].

14


(0001) Al2O3 substrateGaN buffer layer

n-electrode

SiO2n-GaN

n-In0.1Ga0.9N

n-Al0.14Ga0.86N/GaN MD-SLSn-GaN

In0.02Ga0.98N/In0.15Ga0.85N MQWp-Al0.2Ga0.8N

p-GaNp-Al0.14Ga0.86N/GaN MD-SLS

p-GaN

p-electrode

(0001) Al2O3 substrateGaN buffer layer

n-electrode

SiO2n-GaN

n-In0.1Ga0.9N

n-Al0.14Ga0.86N/GaN MD-SLSn-GaN

In0.02Ga0.98N/In0.15Ga0.85N MQWp-Al0.2Ga0.8N

p-GaNp-Al0.14Ga0.86N/GaN MD-SLS

p-GaN

p-electrode

Figure 2-1 Schematic diagram of the structure of the InGaN MQW laser diode

Lens

Polarizer

Laser diode

Temperature controlled stage

Spectrometer

LensLens

Polarizer

Laser diode

Temperature controlled stage

Spectrometer

Lens

Figure 2-2 Schematic diagram of the experiment setup to measure polarization-resolved spontaneous emission spectrum.

15


395 400 405 410 4150.0

0.2

0.4

0.6

0.8

1.0

403.0 403.20.0

0.2

0.4

0.6

0.8

1.0

Spo

ntan

eous

em

issi

on [A

.U.]

Wavelength [nm]

S

pont

aneo

us e

mis

sion

[A.U

.]

Wavelength [nm]

Figure 2-3 Measured TE-polarization spontaneous emission spectra under bias current of 30 mA at 20 °C. The inset is a zoom-in around 403 nm to show the structure of the Fabry-Perot resonance.

2.2.1 Measurement of gain

Assuming that the facet power reflectivity are , the net modal gain

spectrum can be calculated from the measured gain-reflectivity products (b )

using <2-8>. As can be seen from <2-8>, the value of the power reflectivity

will only introduce an vertical shift in the gain spectrum (if power reflectivity is

insensitive to wavelength). The shape of the gain spectrum will independent of

the value of the facet reflectivity. Because the measurement wavelength range

is small, it is justified that the facet reflectivity remains constant in the

1 2 0.25RR =

16


measurement wavelength range. The measurement results with bias current

from 20mA to 30 mA are shown in Figure 2-4. The wavelength that will reach

transparency first is 405 nm; occur at a bias current of 24 mA. As will be shown

in the later, this number is consistent with the results of the pump-probe

measurement that we performed on this laser diode.

400 402 404 406 408 410-40

-30

-20

-10

0

10

Net

mod

al g

ain

(cm

-1)

Wavelength [nm]

20 mA 22 mA 24 mA 26 mA 28 mA 30 mA

Figure 2-4 Measured net modal gain (g ) spectrum with bias current from 20-30 mA (assuming ). 1 2 0.25RR =

2.2.2 Measurement of group index and group velocity dispersion

The group index and GVD can be calculated using <2-9> and <2-10>

respectively. In order to calculate the derivatives and , the

measured phase ( ) was fitted with a third order polynomial of β . The

/d dφ β 2 /d dφ β2

φ

17


calculation results are shown in Figure 2-5. A higher order polynomial can be

used to improve the measurement accuracy if the signal to noise ratio higher.

The measured results show that the GVD of the blue laser diode is very large

compare with the GVD of infrared laser diode (-0.71 µm-1) [13].

400 402 404 406 408 4103.0

3.2

3.4

3.6

3.8

4.0

Wavelength [nm]

Gro

up in

dex

-50

-40

-30

-20

-10

0

GV

D [µm

-1]

Figure 2-5 Measured group index (dotted line) and group velocity dispersion GVD (solid line)

gn

2.2.3 Measurement of linewidth enhancement factor

A series of spontaneous emission spectra with bias current varying from 25 mA

to 30 mA with 1 mA step were measured. The injection current change was

chosen to be 1 mA so that the induced Fabry-Perot mode shift is smaller than

one mode spacing. This is an important consideration in order to avoid the

18


uncertainty of an integer multiple of 2 when determine the round trip phase

( ). The method described in [10] is used to calculate the gain reflectivity

product (b ) and the round trip phase (φ ) from the measured emission spectrum

under various bias current. The derivatives and were obtained

from the slope of the linear fitting of ln and as a function of bias

current (I ) respectively. The linewidth enhancement factor (α ) can now be

calculated using <2-14>, and the calculated result is shown as a function of

wavelength in Figure 2-6. The α parameter is larger at longer wavelengths as

expected because for the carrier-induced gain change at the long wavelength is

smaller than at the short wavelength due to the band filling effect. Similar

observations have also been reported for infrared laser diode [14].

π

φ

/dg dI /d dIφ

( )/b L φ

19


400 402 404 406 408 4100

5

10

15

20

Line

wid

th e

nhan

cem

ent f

acto

r

Wavelength [nm]

lasing wavelength403.5 nmα = 5.8

Figure 2-6 Measured linewidth enhancement factor of an InGaN MQW laser diode under biases current of 25-30 mA. The open circles are the measured data and the solid line is a 2-nd order polynomial fit to the data.

The differential gain and the carrier-induced index change

can be calculated from and using <2-17> and <2-18>

respectively in combination with <2-16>. In order to get a numerical estimation

for the differential gain and the carrier-induced index change, we neglected the

bi-molecular and Auger recombination terms in <2-16> and get

/dg dN

/dn dN /dg dI /d dIφ

life, life( )∆ =τ τ

a

i life

dI q VdN η τ

⋅=

⋅ <2-20>

20


The values of the injection efficiency η and the carrier life time τ are taken

from [15] to be 0.86 and 2.5 ns. The length and the width of the laser diode is

670 µm and 5 µm respectively and the thickness of a single quantum well is 35

Å (4 QW), so the active volume is 4.69×10

i life

aV-11 cm3. The q is the charge of an

electron. Substituting all the required parameters in <2-20> we get

mA/cm18/ 3.5 10dI dN −= × -3. The calculation results of the differential gain

and the carrier-induced index change is shown in Figure 2-7.

400 402 404 406 408 4100.0

0.5

1.0

1.5

Wavelength [nm]

Diff

rent

ial G

ain

[10-1

7 cm

2 ]

-1.5

-1.0

-0.5

0.0

Carrier-induced index change [10

-22 cm3]

Figure 2-7 Differential gain (dotted line) and carrier-induced index change (solid line) of an InGaN MQW laser diode under biases current of 25-30 mA.

/dg dN/dn dN

2.2.4 Measurement of thermal-induced index change

21


A series of spontaneous emission spectra with temperature varying from 20 °C

to 21.5 °C with 0.5 °C step were measured. Similar to the measurement of the

linewidth enhancement factor, the temperature step needs to be chosen carefully

so that the induced Fabry-Perot mode shift is smaller than one mode spacing to

guarantee the change of round trip phase ( ) is smaller than 2 in order to

avoid the uncertainty of integer multiples of 2 when determining the round

trip phase. The round trip phase (φ ) was calculated from measured emission

spectra under various temperatures (T ). The derivative was obtained

from the slope of the linear fitting of the φ as a function of temperature (T ).

Substituting the calculated into <2-19>, the can

be calculated, and the results of the calculation are shown in Figure 2-8. Using

the refractive index (n = 2.65) and the thermal expansion coefficient (α =

5.59×10

φ π

π

/d dTφ

/d dTφ

α

α

thermal/dn dT n α+ ⋅

thermal

-6 K-1) of GaN [16], we get n = 1.48×10thermal⋅ -5 K-1. From Fig. 3 the

value of is on the order of 10thermal/dn dT n+ ⋅ -4 K-1, so we conclude that

is much larger than , and the value of is estimated to

be about 1.3×10

/dn dT thermaln α⋅ /dn dT

-4 K-1.

22


400 402 404 406 408 4100.0

0.5

1.0

1.5

2.0

dn/d

T+nα

ther

mal [1

0-4 K

-1]

Wavelength [nm]

Figure 2-8 Thermal-induced index change of an InGaN MQW laser diode under temperature of 20-21.5 °C. The open circles are the measured data and the solid line is a linear fit to the data.

2.3 Summary

In summary, we have used the FT method to measure and calculate the gain

spectrum, linewidth enhancement factor (3 ~ 15), ground index (3.2 ~ 3.55),

group velocity dispersion (-20 µm-1 ~ -40 µm-1), differential gain (

cm

170.2 10−×

2 ~ cm171 10−× 2), carrier-induced index change ( cm220.5 10−− × 3 ~

cm221 10−− × 3) and thermal-induce index change ( K41.3 10−× -1) for an

InGaN MQW laser diode in the wavelength range between 400 nm and 410 nm.

23


2.4 References [1] B. W. Hakki and T. L. Paoli, "cw degradation at 300 °K of GaAs double-

heterostructure junction lasers. II. Electronic gain," Journal of Applied Physics, vol. 44, pp. 4113-4119, 1973.

[2] B. W. Hakki and T. L. Paoli, "Gain spectra in GaAs double-heterostructure injection lasers," Journal of Applied Physics, vol. 46, pp. 1299-1306, 1975.

[3] L. D. Westbrook, "Measurements of dg/dN and dn/dN and their dependence on photon engergy in λ = 1.5µm InGaAsP laser diode," IEE Proceeding J, vol. 133, pp. 135-142, 1986.

[4] T. Tanbun-Ek, N. A. Olsson, R. A. Logan, K. W. Wecht, and A. M. Sergent, "Measurements of the polarization dependence of the gain of strained multiple quantum well InGaAs-InP lasers," IEEE Photonics Technology Letters, vol. 3, pp. 103, 1991.

[5] D. T. Cassidy, "Technique for measurement of the gain spectra of semiconductor diode lasers," Journal of Applied Physics, vol. 56, pp. 3096-3099, 1984.

[6] J. Chen, B. Luo, L. Wu, and Y. Lu, "Instrumental effects on spectrum measurement from a semiconductor diode biased below threshold," IEE Proceeding J, vol. 140, pp. 243-246, 1993.

[7] D. Hofstetter and R. L. Thornton, "Measurement of optical cavity properties in semiconductor lasers by Fourier analysis of the emission spectrum," IEEE Journal of Quantum Electronics, vol. 34, pp. 1914-1923, 1998.

[8] W.-H. Guo, Y.-Z. Huang, C.-L. Han, and L.-J. Yu, "Measurement of gain spectrum for Fabry-Perot semiconductor lasers by the Fourier transform method with a deconvolution process," IEEE Journal of Quantum Electronics, vol. 39, pp. 716-721, 2003.

[9] K.-G. Gan and J. E. Bowers, "Measurement of gain, group index, group velocity dispersion, and linewidth enhancement factor of an InGaN multiple quantum-well laser diode," IEEE Photonics Technology Letters, vol. 16, pp. 1256-1258, 2004.

[10] D. Hofstetter and J. Faist, "Measurement of semiconductor laser gain and dispersion curves utilizing Fourier transforms of the emission spectra," IEEE Photonics Technology Letters, vol. 11, pp. 1372-1374, 1999.

[11] L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits. New York: John Wiley & Sons, 1995.

24



[13] K. L. Hall, G. Lenz, and E. P. Ippen, "Femtosecond time domain measurements of group velocity dispersion in diode lasers at 1.5 µm," Journal of Lightwave Technology, vol. 10, pp. 616-619, 1992.

[14] K. Naganuma and H. Yasaka, "Group delay and α-parameter measurement of 1.3 µm semiconductor traveling-wave optical amplifier using the interferometric method," IEEE Journal of Quantum Electronics, vol. 27, pp. 1280-1287, 1991.

[15] S. Nakamura, M. Senoh, S.-i. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, Y. Sugimoto, and H. Kiyoku, "Optical gain and carrier lifetime of InGaN multi-quantum well structure laser diodes," Applied Physics Letters, vol. 69, pp. 1568, 1996.


25

Chapter 3. Intervalence subband carrier dynamics in InGaN MQW laser diode

Chapter 3. Intervalence subband carrier dynamics in InGaN MQW laser diodes

26

3.1 Introduction


important material system for the optoelectronic devices in the spectral range

from infrared to ultraviolet. GaN-InN based III-V nitride semiconductors are of

interest in many commercial applications, such as light emitting diodes (LEDs)

and laser diodes (LDs) [1, 2]. The carrier dynamics, which are important for

high speed device design, have recently been studied by femtosecond time-

resolved pump-probe [3] or coherent spectroscopy [4, 5] with above band gap

photons. However, various contributions such as electron-electron scattering,

hole-hole scattering, electron-hole scattering, electron-phonon interactions, and

hole-phonon interactions, mix together and make it very difficult to extract the

fundamental material parameter for one particular scattering process or single

type of carrier. C.-K. Sun et al. and H. Ye et al. have developed an infrared


27

pump-ultraviolet probe technique to isolate electron and hole dynamics and used

it to study the electron relaxation dynamics in n-type GaN thin films [6, 7] and

hole dynamics in p-type GaN thin films [8]. In this chapter, we used a time-

resolved bias-lead monitoring pump-probe technique [9] that uses two UV

pulsed of equal amplitude with various polarization configurations (TE-TE, TM-

TE, and TM-TM) to study the carrier dynamics in the InGaN MQW laser diode.

The TM polarization is the direction of electric field parallels the c-axis and TE

polarization is the direction of electric field perpendicular to the c-axis. From

the optical selection rules of TE and TM polarized light, one can selectively

excite and probe different valence subbands to conduction band transitions in

the MQW structure with different polarized pump and probe light. Using this

technique, ultrafast intersubband hole relaxation processes were found to be

important in the observed carrier dynamics.

The measurement of the polarization resolved electroluminescence of an

InGaN MQW laser diode is shown in Figure 3-1. The data in Figure 3-1 clearly

shows that the peak wavelength of the TE-polarized electroluminescence is

longer than the peak wavelength of the TM-polarized electroluminescence. This

observation suggests that one can use pump and probe light of different

polarization to study the carrier interaction between the TM-subband and the

TE-subband.


370 380 390 400 410 420 430 440 450

0

1

2

3

4

6.6nm (50 meV)

λlasing = 403.5 nm

Pow

er [A

U]

TE x 1TM x 10

Wavelength [nm]

Figure 3-1 Electroluminescence spectrum of an InGaN MQW laser diode(the red curve is the lasing spectrum)

3.2 Band structure and optical selection rule for InGaN

In the wurtize crystalline structure, the selection rules for the optical momentum

matrix elements for the transitions between the conduction band and the three

valence bands can be derived from the symmetry properties of the zone center

wave function [10, 11]. In the following section, these acronyms will be used, C:

conduction, HH: heavy hole, LH: light hole and CH: crystal-field splitoff hole.

Let us first describe the band structure and the optical selection rule of bulk

In0.15Ga0.85N. The band structure of bulk In0.15Ga0.85N is shown in Figure 3-1.

28


29

At the zone center (k = 0), the HH-C transition will only occur when the

polarization of the light is perpendicular to the c-axis, i.e., TE polarized. The

CH-C transition will favor TM polarized light, i.e., the light polarized along the

c-axis. For the LH band, LH-C transition will mostly occur when the light is TE

polarized. Away from the zone center, the HH-C transition remains TE

polarized while the CH-C transition and the LH-C transition switch polarization,

i.e., CH-C transition become TE polarized and LH-C become TM polarized.

Because the hole energy of CH band is larger than the hole energy of HH band

and LH band at the zone center, TM polarized light will excite holes with higher

energy compared to the energy of the holes excited by the TE polarized light.

The holes excited by TM polarized light will relax back to the top of the valence

band and thus affect the absorption properties of TE polarized light, but not the

other way around, i.e., TM will affect TE but TE will not affect TM.


-0.10 -0.05 0.00 0.05 0.10-150

-100

-50

0

2800

2850

2900

2950

3000

TE TETE

TM

TM

CHLH

HH

CE

nerg

y [m

eV]

kx [1/A]

Figure 3-2 The band structure of bulk In0.15Ga0.85N

In the quantum well (QW) structure, the valence band turns into different

valence subbands. Because of the valence band mixing effect, the optical

selection rules will need to be modified. We use the finite-difference method to

solve the effective-mass equations [12] for the quantum well structure. The

band structure parameter was taken from reference [13] and a valence band

offset of 33% was used.

Figure 3-3(a) shows the calculation results for valence subbands with

zero electric field in a QW structure of 3 nm of In0.15Ga0.85N well and

In0.02Ga0.98N barrier. The solution of the effective mass equation was used to

calculate the transition matrix element in order to find the optical transition

strength for different valence subbands to conduction band transition. The

30


31

normalized optical transition strengths for the four lowest valence subband to

the first conduction subband transitions are shown in Figure 3-3(b). The results

show that the optical transition from the valence subband HH1 to the conduction

subband C1 (HH1-C1) is TE polarization. LH1-C1 transition is TE polarization

near the zone center and become TM polarization transition away from the zone

center. HH2-C1 transition is mainly TM polarization with some TE polarization.

LH2-C1 transition is TM polarization near the zone center, and is TE

polarization away from the zone center. Note that the first significant TM

polarized transition occurs at a higher energy compared with the TE polarized

transition and the energy separation (~50 meV) is consistence with the electro

luminescence measurement shown in Figure 3-1.

Wurtize crystal is well known to have spontaneous polarization and

piezo-electric effect and will have a strong internal electric (in range of

~100kV/cm) in the quantum well [14]. To understand how the electric field will

affect the subband structure and the optical selection rule, the same calculation

was carried out under electric fields strength of 300kV/cm and the result is

shown in Figure 3-4. With the presence of electric fields, the energy degeneracy

of the valence subband due to the spin at the cross over point (where two or

more subband approach each other) is lifted and the subband is split into two as

can be clearly seen in Figure 3-4(a). The optical selection rule is further

complicated by the presence of electric field, but HH1-C1 transition remains TE


32

polarization and LH1-C1 transition is still TE polarization in the zone center.

Away from the zone center, it becomes a mixed of TE and TM polarization.

The HH2-C1 and LH2-C1 transition is now a mixed of TE and TM polarization.

With the presence of the internal electric field, the TM polarization transition

still occurs at higher energy than the TE-polarization transition.

According to the above calculation, the TM-subband transition occurs at

a higher energy than the TE-subband transition. Imagine that we inject some

photo-generated carrier in the TM-subband (LH1, HH2, LH2) with TM

polarization pump. The photo-generated carrier in the TM-subband is very

energetic and will release its energy with carrier-phonon interaction and transfer

into the lower subband (HH1) which is only sensitive to the TE polarization

light. We can monitor the carrier transfer rate from the TM-subband to the HH1

subband by probing it with a TE-polarized light.


0.0 0.2 0.4 0.6 0.8 1.0-0.10

-0.08

-0.06

-0.04

-0.02

0.00

LH2

HH1

LH1

Ener

gy [e

V]

kt [1/nm]

Fi = 0 kV/cm

HH2

0.0

0.1

0.2

0.3

0.4

0.5

Fi = 0 kV/cm

TM

HH1-C1

Nor

mal

ized

Tra

nsiti

on st

reng

th

kt [1/nm]

TE TE

TM

LH1-C1

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

TE

TMHH2-C1

0.0 0.2 0.4 0.6 0.8 1.0

TETM

LH2-C1

(a)

(b)

Figure 3-3 (a) Calculated subband structure of the quantum well structure and (b) normalized transition strength for selected subband transition. TE polarized strength is normalized to

2

xS p X and TM

polarized transition strength is normalized to 2

zS p Z . The transition corresponding to 400 nm light is indicated with vertical lines.

33


0.0 0.2 0.4 0.6 0.8 1.0-0.10

-0.08

-0.06

-0.04

-0.02

0.00

LH2

HH2

HH1

LH1

Fi = 300 kV/cmEn

ergy

[eV

]

kt [1/nm]

0.0

0.1

0.2

0.3

0.4

0.5

Fi = 300 kV/cm

TM

HH1-C1

Nor

mal

ized

Tra

nsiti

on st

reng

th

kt [1/nm]

TE TE

TM

LH1-C1

TM

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

TE

TMHH2-C1

0.0 0.2 0.4 0.6 0.8 1.0

TE

TM

LH2-C1

(a)

(b)

Figure 3-4 (a) Calculated subband structure of the quantum well structure and (b) normalized transition strength for selected subband transition. TE polarized strength is normalized to

2

xS p X and TM

polarized transition strength is normalized to 2

zS p Z . The transition corresponding to 400 nm light is indicated with vertical lines.

34


3.3 Bias lead monitoring measurement

The schematic diagram of the time-resolved bias monitoring setup is shown in

Figure 3-5.

Figure 3-5 Schematic diagram of experiment setup for time-resolved bias monitoring measurement

The pump and probe beam are derived from the second harmonic

generation (SHG) of a tunable 100-fs Ti:Sapphire modelocked laser. The pump

and probe beam are combined collinearly and directed to the laser diode under

test. The pump and probe beam are mechanically chopped at frequencies of 1.7

kHz and 2.0 kHz respectively. The photocurrent collected from the laser diode

was measured by a lock-in amplifier at the sum frequency of 3.7 kHz as a

function of the delay between the pump and probe beam. In order to avoid the

35


interference signal between the pump and probe in the co-polarization

configuration, the frequency of the probe beam was shifted by 40 MHz with an

acousto-optic frequency shifter.

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0

5

10

15

20

Measured Gaussian Fit

tFWHM = 0.37 ps

Pho

to C

urre

nt [n

A]

Delay [ps]

Figure 3-6 Time-resolved photocurrent signal at below-bandgap (425 nm) excitation.

At below-bandgap excitation (425 nm), the time-resolved photocurrent

response signal is enhanced when both pump and probe light overlapped in time.

This positive instantaneous signal is attributed to two-photon absorption (TPA)

and the width of this signal is 0.37 ps and is limited by the autocorrelation width

of the laser pulse.

36


Figure 3-7 Time-resolved photocurrent signal with above bandgap (400 nm) excitation (carrier density ~ 1017 cm-3): (a) TE-TE polarization and TM-TM polarization and (b) TM-TE polarization.

When we tune the laser wavelength to be above the bandgap of InGaN

MQW, different behavior was observed for different pump-probe polarization

configurations. Figure 3-7 shows example traces taken at a wavelength of

400nm. As shown in Figure 3-7(a), when both pump and probe are TE

37


polarized, there is a negative instantaneous signal and a negative double-sided

exponential decay signal. The negative instantaneous signal is attributed to a

phase space filling effect with a fast initial relaxation faster than our system time

resolution. This initial fast relaxation can be attributed to the carrier

thermalization mainly due to carrier-carrier scatterings. The slower negative

exponential decay signal with a time constant ( ) of 2.2 ps is attributed to the

carrier energy relaxation where the carrier-phonon interaction will lead to a new

equilibrium between the carriers and the lattice system. However, when both

pump and probe are TM polarized, only negative instantaneous signal can be

observed. This resolution-limited response suggests an extremely fast (τ < 0.37

ps) intersubband hole relaxation for the TM-generated hole in the LH2 and HH2

subbands into lower HH1 and LH1 subbands, which are only sensitive to the TE

polarized light. In order to study this intersubband hole relaxation process, cross

polarization measurement was performed and the result is shown in Figure

3-7(b).

Rτ

In the cross polarization configuration, positive delay means TM

polarized light (pump) enters the laser diode before the TE polarized (probe) and

negative delay means TE polarized light (pump) enters the laser diode before

TM polarized (probe). At positive delay, there is fast initial decay followed by

another positive single-sided exponential decay signal with the same time

38


constant ( = 2.2 ps) as the one observed in the TE-TE polarization

configuration. The fast initial raise of the observed TE signal supports the

previous suggestion that an extremely fast inter-subband hole relaxation for the

TM generated holes in LH2 and HH2 subbands relaxed into the HH1 and LH1

subbands. These LH2 and HH2 subbands transferred holes in the lower HH1

and LH1 subband will then follow a similar thermalization process as the

directly generated holes. It is interesting to notice that at negative delay, the

signal remains constant suggesting weak HH1 and LH1 subband to LH2 and

HH2 subband transitions as

Rτ

expected. The absence of the signal with time

constant = 2.2 ps in the negative delay further confirm the measure signal is

due to holes and not electrons.

Rτ

In conclusion, the femtosecond carrier dynamics in InGaN MQW laser

diode were studied using a time-resolved bias-lead monitoring technique. Using

the optical selection rules in the wurtize QW structure and various pump-probe

polarization configurations, an ultrafast inter-subband hole relaxation process (τ

< 0.37 ps) can be observed. This inter-subband hole relaxation process is found

to be important in the measurement at high carrier density that will be described

in the later section.

39


40

3.4 Time-resolved differential reflection measurement

Since the sample is a packaged laser diode, there is no access to the transmission

from the back facet of the laser diode. Although there is a monitor photo diode

build in the package that will detect the light form the back facet of the laser

diode, but the pump can not be separated from the probe and background free

measurement can not be possible. In order to separate the pump and probe, we

use cross-polarization time-resolved differential reflection measurement instead

of time-resolved differential reflection measurement. Figure 3-8 shows the

schematic diagram of the time-resolved differential reflection measurement.

The origin of pump and probe are derived from second harmonic generation

(SHG) of a tunable 100-fs Ti:Sapphire modelocked laser. Pump and probe are

combined collinearly with orthogonal polarization and a 10 to 1 ratio before

directed to the laser diode. A beam splitter is used to pick off the reflection

probe form the laser diode. A polarizer is put in front of the photodetector to

block the pump beam so that only the response of pump on the probe is

measured.


Figure 3-8 Schematics of time-resolved differential reflection measurement

The differential reflection measurement will provide the same information as the

differential transmission measurement provided that the contribution due to the

carrier induced absorption change in the waveguide dominates over the

contribution from the carrier induced facet reflectivity change. This condition is

justified when the waveguide is long and absorption is low. If the waveguide is

longer than the coherence length of the light, the effective reflectivity can be

given by

2 2

2 21

L

eff L

T ReR R

R e

− α

− α= +−

<3-1>

41


Where are the power facet reflection and facet transmission (assume back

and front facet is the same), α is the absorption coefficient, and L is the cavity

length. R and T are related to the index of the waveguide (n ) by

,R T

22

1( ) ; 1

1 (n

R T Rn n−

= = − =+ +

41)n <3-2>

The ratio of the index-induced reflection change ( ) to the absorption-

induced reflection change ( ) is given by

,eff nR∆

,effR α∆

2 2 2 4

, 22 3

,

1 (1 4 ) ( 1)2 ( 1)

L Leff n L

Lineeff

R R R e R e n ne

R RT n

− α − α+ α

α

∆ + − + + − λ= − α

∆ + Lπ<3-3>

Where is the linewidth enhancement factor given by Lineα

4Line

nπ ∆α = −

λ ∆α <3-4>

Note that there is a factor in the expression of in <3-3>,

which mean that the index induced reflection change will dominate when the

loss is large. However, when the loss is small and cavity is long, this ratio can

be much smaller than 1 and the absorption-induced reflection change will

dominate. The magnitude of the ratio as function of the loss is

plot in Figure 3-9.

2 Le+ α, /eff n effR R α∆ ∆ ,

,, /eff n effR R α∆ ∆

42


0 10 20 30 40 50 6010-4

10-3

10-2

10-1

100

Rat

io |∆

Ref

f,n/∆

Ref

f,α|

Modal loss α [cm-1]

α=42.5cm-1

Figure 3-9 as a function of modal loss α ,| /eff n effR R α∆ ∆ , |

The following parameter have been used , ,

(index of GaN), (use the measured linewidth enhancement

factor in Figure 2-5). From Figure 3-9, when the modal loss is less than 42.5

cm

0.4 mλ = µ 670L m= µ

2.65n = 5Lineα =

-1 the index-induced reflection change is less than the 10 % of the absorption-

induced reflection change. And according to the gain measurement in Chapter 2

(Figure 2-3 have been reproduced as Figure 3-10 in this chapter for

convenience), the modal loss is less than 40 cm-1 for wavelength from 400nm to

410 nm with injection current larger than 20 mA. For this wavelength range, the

43


differential reflection measurement can be used to study the gain dynamics in

the InGaN laser.

400 402 404 406 408 410-40

-30

-20

-10

0

10

Net

mod

al g

ain

(cm

-1)

Wavelength [nm]

20 mA 22 mA 24 mA 26 mA 28 mA 30 mA

Figure 3-10 Measured net modal gain (g ) spectrum with bias current from 20-30 mA. (Reproduced from Chapter 2, Figure 2-3)

3.4.1 Determination of transparency level

The transparency level of the laser diode can be determined by using TE-

polarized light as the pump light and used TM-polarized light to probe the

carrier density change due to TE pump. Figure 3-11 show a TE-pump TM-

probe measurements taken with 402 nm excitation at temperature of 20 °C and

the electrical injection current is increase from 10 mA to 30 mA with a 5 mA

44


step. The curves in the figure have been normalized to the peak at the time zero

in order to clearly show the sign of the reflection change at large delay (t = 6 ps).

-2 -1 0 1 2 3 4 5 6-1.2-1.0-0.8-0.6-0.4-0.20.00.20.4

Injection current increase from10 mA to 30 mA with 5 mA stepT = 20 oCN

orm

aliz

ed ∆

R

Time delay [ps]

Figure 3-11 Normalized TE-pump TM-probe time-resolved differential reflection signal at 402nm excitation and temperature = 20 °C, the injection current increase from 10 mA to 30 mA with 5 mA step, temperature.

Figure 3-11 shows that as the injection current increases, the sign of the

differential reflection change from positive to negative. At low injection current,

there is no population inversion in the laser, so the TE pump light will be

absorbed and increase the carrier density in the laser and thus reduce the

absorption for the TM probe light and as a result, more TM probe light will be

reflected back (positive differential reflection signal at long delay). At the high

injection current, there is population inversion and the TE pump light will

45


46

induce the stimulated emission and decrease the carrier density in the laser and

thus increase the absorption for the TM probe light and result in the negative

differential reflection signal at long delay. For a certain injection current level

(~25 mA), the differential reflection signal at long delay go to zero, this is the

point where the laser reaches transparency. Comparing this with the measure

net modal gain in Figure 3-10, the internal modal loss at 402 nm can be

estimated to be about 10 cm-1.

3.4.2 Intersubband relaxation under high carrier density

Figure 3-12 and Figure 3-13 showed the time-resolved differential reflection

measurement taken at below bandgap excitation (415 nm) for TM-pump TE-

probe and TE-pump TM-probe respectively. The negative instantaneous signal

is due to two photon adsorption.


-2 -1 0 1 2 3 4 5 6

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Nor

mal

ized

Ref

lect

ion

Cha

nge

Time delay [ps]

TM pump, TE probe, 415nm 5 mA 10 mA 15 mA 20 mA 25 mA 30 mA

Figure 3-12 Normalized TM-pump TE-probe time-resolved differential reflection signal at 415 nm

-2 -1 0 1 2 3 4 5 6

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Nor

mal

ized

Ref

lect

ion

Cha

nge

Time delay [ps]

TE pump, TM probe, 415nm 5 mA 10 mA 15 mA 20 mA 25 mA 30 mA

Figure 3-13 Normalized TE-pump TM-probe time-resolved differential reflection signal at 415 nm

47


A very different behavior is observed when the excitation wavelength is

tuned to above bandgap. We will focus on the TM-pump TE-probe result in this

section. The general behavior of TE-pump TM-probe signal has been discussed

and shown in Figure 3-11 in section 3.4.1.

-2 -1 0 1 2 3 4 5 6-1.2-1.0-0.8-0.6-0.4-0.20.00.20.4

Injection current increase from15 mA to 30 mA with 5 mA steppump power = 300 µWtemperature = 20 oC

Nor

mal

ized

∆R

Time delay [ps]

Figure 3-14 Normalized TM-pump TE-probe time-resolved differential reflection signal at 402 nm at temperature of 20 °C.

Shown in Figure 3-14 is the TM-pump TE-probe measurement taken

with 402 nm excitation. The same negative instantaneous signal due to two

photon absorption is present in the data. The positive signal at long delay

consistent with the fact that TM-pump light does not experience stimulated

emission. Near the time zero, there is fast positive going signal. The strength of

this signal reduces as the bias level increases. The behavior of this positive fast

48


49

transient signal is consistent with the ultrafast inter-subband hole transfer from

the TM-sensitive subbands (LH2, HH2) to the TE-sensitive subbands (HH1,

LH1) that was observed in the bias-lead monitoring measurement (see section

3.3). As the bias current increases, more and more TE-sensitive states in HH1

and LH1 are occupied by the injected hole, as a result the photo-generated hole

in the TM-sensitive states in LH2 and HH2 is less likely to transfer to TE-

sensitive states in HH1 and LH1. Once the holes transfer to the HH1 and LH1

subbands, they will increase the hole temperature in the TE-sensitive subbands

(HH1, LH1) through carrier-carrier scattering, the resulting hot hole distribution

will cool down to the lattice temperature at time constant of 0.8 ps through

carrier phonon interaction.

It is interesting to note that, the inter-subband hole relaxation and hole

heating processes are found to dominate the carrier dynamic response in InGaN

laser diodes and this is very different from the previous GaAs [15] and InP [16]

based devices. Pevious studies on GaAs and InP based device found that carrier

dynamics are dominated by electron heating through free carrier absorption or

interband transitions.

3.5 Summary and future work


50

In summary, the femtosecond carrier dynamics in InGaN MQW laser diode

were studied using time-resolved bias-lead monitoring technique at low carrier

density region and the dynamics at high carrier density region were studied

using time-resolved differential reflection measurement. In both regions, the

carrier dynamics were found to be consistent with the presence of an ultrafast

inter-subband hole relaxation process between the TM-sensitive subbands (LH2,

HH2) and TE-sensitive subbands (LH1, HH1). The time constant of the transfer

process is within the time resolution of the experiment instrument (τ < 0.37 ps).

The differential transmission measurement is a preferred method to study

the carrier dynamics. In this thesis, due to the limitation of the sample,

differential transmission measurement is not possible. In the future, if the back

facet of the laser diode is accessible, measurement described in [17-19] should

be performed to make a more detailed analysis of the valence intersubband

relaxation dynamics.

Measurement on the a-plane (or m-plane) InGaN thin film sample can

also shed more light on the valence intersubband relaxation because a-plane

corresponds to the mirror facet of the InGaN laser diode. Measurement on the

a-plane sample had been tried, but repeatable measurement can not be obtained

due to the surface roughness and uniformity of the sample. In the future, when

the better quality a-plane (m-plane) sample is available, the measurement should


51

be performed to gain more insight to the valence intersubband relaxation

dynamics.

3.6 Reference [1] S. Nakamura, M. Senoh, and T. Mukai, "High-power InGaN/GaN

double-heterostructure violet light emitting diodes," Applied Physics Letters, vol. 62, pp. 2390-2392, 1993.


[3] C. K. Sun, F. Vallee, S. Keller, J. E. Bowers, and S. P. DenBaars, "Femtosecond studies of carrier dynamics in InGaN," Applied Physics Letters, vol. 70, pp. 2004-2006, 1997.

[4] S. Pau, J. Kuhl, F. Scholz, V. Haerle, M. A. Khan, and C. J. Sun, "Femtosecond degenerate four-wave mixing of GaN on sapphire: Measurement of intrinsic exciton dephasing time," Physical Review B, vol. 56, pp. R12718, 1997.

[5] R. Zimmermann, A. Euteneuer, J. Mobius, D. Weber, M. R. Hofmann, W. W. Ruhle, E. O. Gobel, B. K. Meyer, H. Amano, and I. Akasaki, "Transient four-wave-mixing spectroscopy on gallium nitride: Energy splittings of intrinsic excitonic resonances," Physical Review B, vol. 56, pp. R12722, 1997.

[6] C. K. Sun, Y. L. Huang, S. Keller, U. K. Mishra, and S. P. DenBaars, "Ultrafast electron dynamics study of GaN," Physical Review B, vol. 59, pp. 13535, 1999.

[7] H. Ye, G. W. Wicks, and P. M. Fauchet, "Hot electron relaxation time in GaN," Applied Physics Letters, vol. 74, pp. 711-713, 1999.

[8] H. Ye, G. W. Wicks, and P. M. Fauchet, "Hot hole relaxation dynamics in p-GaN," Applied Physics Letters, vol. 77, pp. 1185-1187, 2000.

[9] K. L. Hall, E. P. Ippen, and G. Eisenstein, "Bias-lead monitoring of ultrafast nonlinearities in InGaAsP diode laser amplifiers," Applied Physics Letters, vol. 57, pp. 129-131, 1990.


52

[10] G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductor. New York: Wiley, 1974.

[11] S. L. Chuang and C. S. Chang, "Effective-mass Hamiltonian for strained wurtzite GaN and analytical solutions," Applied Physics Letters, vol. 68, pp. 1657-1659, 1996.

[12] S. L. Chuang and C. S. Chang, "A band-structure model of strained quantum-well wurtzite semiconductors," Semiconductor Science and Technology, vol. 12, pp. 252-263, 1997.

[13] Y. C. Yeo, T. C. Chong, and M. F. Li, "Electronic band structures and effective-mass parameters of wurtzite GaN and InN," Journal of Applied Physics, vol. 83, pp. 1429-1436, 1998.


[15] C. T. Hultgren, D. J. Dougherty, and E. P. Ippen, "Above- and below-band femtosecond nonlinearities in active AlGaAs waveguides," Applied Physics Letters, vol. 61, pp. 2767-2769, 1992.

[16] K. L. Hall, G. Lenz, E. P. Ippen, U. Koren, and G. Raybon, "Carrier heating and spectral hole burning in strained-layer quantum-well laser amplifiers at 1.5 mu m," Applied Physics Letters, vol. 61, pp. 2512-2514, 1992.

[17] K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon, "Femtosecond index nonlinearities in InGaAsP optical amplifiers," Applied Physics Letters, vol. 62, pp. 1320-1322, 1993.

[18] C. T. Hultgren and E. P. Ippen, "Ultrafast refractive index dynamics in AlGaAs diode laser amplifiers," Applied Physics Letters, vol. 59, pp. 635-637, 1991.

[19] C. K. Sun, H. K. Choi, C. A. Wang, and J. G. Fujimoto, "Femtosecond gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers," Applied Physics Letters, vol. 63, pp. 96-98, 1993.

Chapter 4. Diffusion constant of InGaN MQW thin films


53

In this chapter, we will study the lateral carrier diffusion in InGaN multiple

quantum wells. The method that will be employed for the study of the carrier

diffusion is an optical measurement and is called the transient grating

measurement [1]. In this measurement, two pump pulses are arranged to

interfere and generate a carrier grating on the sample. The time evolution of the

generated grating is recorded by monitoring the diffraction of a third probe pulse

that arrives at some delay time after the pump pulses had reached the sample.

The physical process that is involve in the transient grating measurement is a

third order nonlinear process and the resulting signal is usually very weak and

required amplified laser system to perform the measurement. In the following

session, a novel experiment configuration will be described. This new

experiment configuration has the advantage of higher sensitivity and it allowed

the use of an unamplified laser system to perform the experiment. The

problematic parasitic interference between the diffracted probe and the scattered

probe can also be eliminated using this new experimental configuration.


54

4.1 Conventional transient grating measurement

The schematic diagram of the conventional transient grating measurement is

shown in Figure 4-1. The pump and probe beams are derived from second

harmonic generation (SHG) of a tunable 100-fs Ti:Sapphire modelocked laser.

The power ratio of the pump and probe beam are 10 to 1. The transient grating

is formed on the sample by the interference of the pump1 and pump2, the path

difference between pump1 and pump2 has to be carefully adjusted so that the

pulses from pump1 and pump2 are overlapped in time. A probe beam is counter

propagating against pump2 and in this configuration the diffracted probe beam

will be counter propagating against pump1. A beam splitter is inserted in the

path of pump1 to pick off the diffracted probe beam. The diffracted probe light

is then measured with a photomultiplier tube (PMT). To prevent the scattered

pump light from over exposing the PMT, the polarization of pump and probe are

perpendicular to each other and a polarizer is used to block out the scatted pump

beam. By monitoring the power of the diffraction of the probe beam as a

function of the delay between the probe beam and the pump beams, the time

evolution of the transient grating can be recorded.


MQW

Substrate

Probe

Pump1

Pump2

Diffracted probe

Beam splitter

Polarizer (blocked pump1 and pump2)

PMT

Lock-in@ 80 kHz

AOM

160 kHzAO

M24

0 kH

z

Figure 4-1 Schematic diagram of the conventional transient grating measurement setup

The decay time of the signal ( ) is related to the carrier life time ( )

and the period of the grating (Λ ). The formula is given by [1]

sτ cτ

2

2

1 2 8

s c

Dπ

τ τ⋅

= + ⋅Λ

. <4-1>

where D is the ambipolar diffusion constant. We repeat the experiment with

several grating periods (adjust the grating period by varying the angle between

the pump1 and pump2) and record the corresponding decaying time and the

ambipolar diffusion constant D can be obtained.

55


56

In practice, the diffracted probe beam is very weak and can be very

difficult to measure directly. One particularly serious problem is due to the

unwanted interference between the diffracted probe beam and the scattered

probe beam in the direction of the diffracted probe beam. The strength of this

scattered probe may actually be stronger than the diffracted probe, so the

interference signal actually dominates over the diffracted probe signal. One may

think that we can then just measured the interference term and we can still get

the decaying time of the grating. Unfortunately this is very difficult, because the

phase difference between the diffracted probe and the scattered probe is related

to the phase difference between pump1 and pump2. The path length difference

between pump1 and pump2 will be randomly varying on the order of

wavelength due the mechanical vibration of the experiment setup. This effect is

shown in Figure 4-2, the inset is a zoom in view of 0.3 second and it shows that

the signal can undergo a very large change in a very short time span. As a result,

without the path length stabilization control, the interference signal behaves like

noise and obscures the measurement of the power of the diffracted probe beam.


0 20 40 60 80 100

-6

-4

-2

0

2

4

6

54.5 54.6 54.7 54.8-5

-4

-3

-2

Sign

al [A

U]

Time [s]

B

Sign

al [A

U]

Time [s]

Figure 4-2 The signal variation due to the parasitic interference between the scattered probe and diffracted probe. In this 100 seconds time span, the experiment conditions remain nominally the same except the uncontrollable path length variation between pump1 and pump2.

4.2 New heterdodyne transient grating measurement

In this section, we describe a novel experiment configuration that will eliminate

the above interference problem and also boost the sensitivity of the

measurement by using heterodyne measurement. The schematic diagram of the

new experiment configuration is shown in Figure 4-3. In the new configuration,

the transmitted probe beam is used as the reference probe beam for the

57


heterodyne measurement. The same beam splitter that was used to split the

pump beam into pump1 and pump2 is also used to combine the diffracted probe

beam and reference probe beam. In this configuration, the reference probes

beam and the diffracted probe beam going trough the same path as pump1 and

pump2 respectively. The detail phase analysis given as following will show that

the resulting interference signal between diffracted probe beam and the

reference probe beam is independent of the path length difference between

pump1 and pump2. So the heterodyne measurement can be used without the

stabilization of the path length difference.

58

)

)

) )

r

For the purpose of phase analysis, we assume that the polarization of the

pump is TE (s) polarization and the probe is TM (p) polarization. We define the

transmission coefficient and reflection coefficient for TM (TE) polarization light

incidence from the front side of the beam splitter as and

respectively and the transmission coefficient and reflection

coefficient for TM (TE) polarization light incidence from the back side of the

beam splitter as and respectively. The phase of the

reference probe (transmitted probe) at the detector is given by

, ,(bs TM bs TEt t

, ,(bs TM bs TEr r

, ,(bs TM bs TEt t′ ′, ,(bs TM bs TEr r′ ′

<4-2> ( )1, , 1,

iref TM bs TM TM

i

E r ′= φ + +∑


Where is the phase of the optical path length of arm number 1 for TM

polarization light, is the phase of reflection coefficient of the TM light

from the back side of the beam splitter, is the phase of the reflection

coefficient of the i-th mirror for TM light in the path number 1.

1,TMφ

,bs TMr ′

( )1,iTMr

pumpprobe

detection

pump1

pump2

sample

Mirror

diffracted probe

reference probe

Mirror

TE (s)TM (p)

,bs Pr ′

,bs Pt ′,bs Pr

,bs PtBeam splitter

pumpprobe

detection

pump1

pump2

sample

Mirror

diffracted probe

reference probe

Mirror

TE (s)TM (p)

,bs Pr ′

,bs Pt ′,bs Pr

,bs Pt,bs Pr ′

,bs Pt ′,bs Pr

,bs PtBeam splitter

Figure 4-3 Schematic diagram of new heterodyne transient grating measurement setup.

The phase of the diffracted probe is given by

( )2, , 2,

jdif TM bs TM TM grating

j

E t r= φ + + +φ∑ <4-3>

Where is the phase of the optical path length of arm number 2 for TM

polarization light, is the phase of transmission coefficient of the TM

2,TMφ

,bs TMt

59


light from the front side of the beam splitter, is the phase of the reflection

coefficient of the i-th mirror for TM light in the path number 2. is the

phase difference between the diffracted and transmitted light of the grating and

is given by

( )2,iTMr

gratingφ

0, 1 2grating grating pump pumpE Eφ = φ + − <4-4>

Where is the phase of the pump 1(2) at the sample. is a

constant that relate to the shape of the grating.

1(2)pumpE 0,gratingφ

The phase of pump1 and pump2 at the sample is given by

( )1 1, , 1,

( )2 2, , 2,

ipump TE bs TE TE

i

jpump TE bs TE TE

j

E t

E r

= + +

= + +

∑

∑

φ

φ

r

r

)

)

<4-5>

Where is the phase of the optical path length of arm number 1 (2)

for TE polarization light. is the phase of the transmission

(reflection) coefficient of the beam splitter for TE light from the front side.

( ) is the phase of the reflection coefficient of the i-th mirror in the

path number 1 (2) for TE light.

1, 2,(TE TEφ φ

, ,(bs TE bs TEt r

( )1,iTEr

( )2,iTEr

From <4-2>, <4-3>, <4-4> and <4-5>, the phase difference between the

diffracted probe and the reference probe is given by

<4-6> 2, 2, 1, 1, 0( ) ( )

dif ref

TM TE TM TE

E Eθ = −

= φ −φ − φ −φ + θ

60


Where is a constant that is not related the optical path length of arm 1 and

arm 2 and is given by

0θ

0 0, , , , ,

( ) ( ) ( ) ( )2, 1, 1, 2,

grating bs TM bs TM bs TE bs TE

j i iTM TM TE TE

j

j i i j

t r t r

r r r

′θ = φ + − + −

+ − + −∑ ∑ ∑ ∑ r

Γ

<4-7>

Since the air is an isotropic medium and plug this

into <4-6>

1, 1, 2, 2,,TM TE TM TEφ = φ φ = φ

<4-8> 0θ = θ

This complete the proof that the phase difference between the diffracted probe

and the reference probe is independence of the optical path length of the arm 1

and arm 2.

Under a very special experiment condition, it is possible that and

when this happen, the interference signals between the transmitted probe and

diffracted probe (proportional to cos( ) is exactly zeros and the heterodyne

measurement will fail. To account for this situation, a variable compensator can

be inserted into arm 2 and align the optical axis of the compensator with the

polarization of the light. When the compensator is inserted,

/2θ = π

)θ

<4-9> 1, 1,

2, 2,

0TM TE

TM TE

φ −φ =

φ −φ =

And the phase difference θ is given by

<4-10> 0θ = θ + Γ

61


The retardance ( ) of the compensator can be adjusted to maximum the

interference signal.

Γ

Although the measurement using this configuration increases the

detection sensitivity of the diffracted probe beam by using the heterodyne

technique, the signal of measurement is not background free. On top of the

signal due to the transient grating diffracted probe beam (TG), there is also

another signal due the pump induced differential transmission change of the

reference probe beam (DT). These two signals can be separated by doing one

more measurement with the same exact experiment conditions except with a

small mismatch of the path length difference between pump1 and pump2 so that

they are not overlap in time and thus not interfere with each other to form the

grating. In this mismatch case, the TG signal will cease to exist and only DT

signal is left in the measurement. By subtracted the results of this two

measurements, the TG signal can be separated out.

The decay time of the TG signal ( ) is related to the carrier life time

( ) and the period of the grating ( ). The formula is given by

TGτ

cτ Λ

2

2

1 1 4

TG c

Dπ

τ τ⋅

= + ⋅Λ

<4-11>

Where D is the ambipolar diffusion constant. As oppose to the conventional

method, there is no need to change the grating period for one to obtain the

62


diffusion constant, because the carrier life time ( ) can be obtained from the

DT signal.

cτ

To further increase the sensitivity of the measurement, lock-in detection

is used and the pump beam is modulated with an acousto-optic modulator at 1.6

MHz (this frequency is choose to avoid the low frequency noise in the

Ti:Sapphire laser). To prevent the scattered pump beam from reaching the

detector, the polarization of the pump beam and the probe beam are

perpendicular to each other and a polarizer is place in front of the detector to

block the pump beam.

4.3 Important optical alignment issue

When doing the experiment setup of the heterodyne transient grating

measurement, it is very important that the total number of mirror in the arm 1

and arm 2 to be differed by an odd number. The alignment will become difficult

if the above rule is not follow. For the purpose to demonstrate that why this is

the case we consider a special case where there is a single mirror in the arm 1

and arm 2 as show in Figure 4-4. The solid line represents the perfect alignment

in which the probe and pump are overlap with each other. The dash line

represents the case where there is a small angle misalignment of δ in the

direction of the probe and pump 1. As can be seen from the figure, the

63


diffracted probe and transmitted probe will displace toward different direction

and causing the angle deviation ( ) between the transmitted probe and the

diffracted probe to be increase from 0 to 2 . The angle deviation ϕ has to be

keep below a certain value in order for the detector to be able to measure the

heterodyne signal. The condition is given by

ϕ

δ

40 4 10 radDETr

−λϕ < = × <4-12>

Where (405 nm) is wavelength of the light and (1 mm) is the radius of

the detector.

0λ DETr

pumpprobe

detectionpump1

pump2

sample

Mirror

Mirror

Beam splitterδ

δ

2ϕ = δ

pumpprobe

detectionpump1

pump2

sample

Mirror

Mirror

Beam splitterδ

δ

2δϕ =

Figure 4-4 Experiment configuration with same number of mirrors in both arm.

64


pumpprobe

detectionpump1

pump2

sample

Mirror

Mirror

Beam splitterδ

δ

0ϕ =

Mirror

pumpprobe

detectionpump1

pump2

sample

Mirror

Mirror

Beam splitterδ

δ

0ϕ =

Mirror

Figure 4-5 Experiment configuration with mirror number differ by 1 in arm 1 and arm 2.

For comparison, Figure 4-5 shown that the case with 2 mirrors in arm 1 and 1

mirror in arm2. As can be seen from the figure, when there is a small angle

misalignment, the diffracted probe and transmitted probe will displace toward

the same direction and the angle deviation ϕ remain to be zero.

4.4 Validation of the new heterodyne transient grating method

To validate the new method, the heterodyne transient grating measurement was

performed on an InGaN sample with the same experiment conditions

(wavelength of the laser = 405 nm) except the grating period was varying from

65


1.0 µm to 1.6 µm. The measurement results was found to satisfy the formula

<4-11>.

-5 0 5 10 15 20 25 30 35 400.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1.84 µJ/cm2 (1.25x1010 cm-2/well)

D = 0.147 cm2/s, τc = 1.58 ns

6.77 µJ/cm2 (4.63x1010 cm-2/well)

D = 0.169 cm2/s, τc = 1.49 ns

Gra

ting

deca

y ra

te 1

/τG [1

09 s-1]

q2 [1012 m-2]

10 % total absorption in 30 QW

Figure 4-6 Grating decaying rate as a function of grating period

According the detail phase relation analysis, adding a half-waveplate ( )

into one of the arm will change the sign of the heterodyne signal. And this is

confirmed by the experiment as well. The asymmetry of the data shown in

Figure 4-7 is because the wavelength of the lignt used in the experiment is not

the nominal wavelength of the half-waveplate used in the experiment.

Γ = π

66


-200 0 200 400 600 800 1000 1200 1400

0.0

0.2

0.4

0.6

0.8

1.0 P1&P2 overlap not overlap overlap with HWP

∆P/P

[10-4

]

Delay [ps]

(a)

(b)

(c)

Figure 4-7 Heterodyne signal (a) pump1 and pump2 overlap and no half-waveplate (HWP), (b) pump1 and pump2 overlap and with HWP in one arm, (c) pump1 and pump2 not overlap

4.5 Diffusion constant of InGaN MQW

The measurement results in this session is measured on a multiple-quantum-well

(MQW) InGaN thin film sample with 30 period of 2.5 nm well, 8 nm n-doped

barrier (The schematic of the sample structure is shown in Figure 4-8). The

photoluminescence (PL) of the sample is shown in Figure 4-9. From the PL

spectrum, it is clear that the sample is not uniform across the sample. We will

use this fact to study the dependence of the diffusion constant on the

composition of InGaN. Unless otherwise states, all the results present in this

67


chapter are measured with 405 nm excitation. From the peak of the PL

spectrum in Figure 4-9, assuming that the non-uniform of the sample is all due

to the composition variation, the composition is found to be vary between

In0.1Ga0.9N and In0.15Ga0.85N. If we assume the fixed composition In0.13Ga0.87N,

the well width is vary between 1.5 nm and 4 nm.

(0001) Al2O3 Substrate

GaN buffer layer

In0.12Ga0.88N (2.5 nm)

In0.02Ga0.98N (8.0 nm n-doped)30 period

(0001) Al2O3 Substrate

GaN buffer layer

In0.12Ga0.88N (2.5 nm)

In0.02Ga0.98N (8.0 nm n-doped)30 period

Figure 4-8 Sample structure of InGaN MQW thin film

68


375 400 425 4500

5

10

15

20Position

-1mm 1mm 3mm 5mm

PL [A

.U.]

Wavelength [nm]

405nm

Figure 4-9 The photoluminescence spectrum at various position on the InGaN MQW sample.

In order to choose which spot on the sample to do the measurement, we do a

series of differential transmission measurements a various position. The

differential transmission signal ( ) at long delay at various position is

plot in Figure 4-10.

/T T∆

69


-4 -3 -2 -1 0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

3.0

∆T/T

[10-4

]

Position [mm]

Figure 4-10 Differential transmission signal at long delay across the sample (405 nm excitation wavelength)

The diffusion constant at various position on the sample is shown in Figure 4-11.

As can be seen from the figure, the diffusion constant is fairly constant across

the sample ~ 0.18 cm2/s. Using the Einstein relationship, the ambipolar mobility

can be found to be 7.2 cm2/Vs and the hole mobility is 3.6 cm2/Vs. This number

is consistence with the numbers of hole mobility of that have been report in the

literature [2].

70


0.00

0.04

0.08

0.12

0.16

0.20

-4 -3 -2 -1 0 1 2 3 4 5 60.00

0.04

0.08

0.12

0.16

0.20

Diffusion constant

Diff

usio

n C

onst

ant [

cm2 /s

]

Position [mm]

Diffusion length

Diff

usio

n Le

ngth

[µm

]

Figure 4-11 Diffusion constant and diffusion length across the sample (405nm excitation wavelength).

Figure 4-12 shown that the diffusion constant increase as the temperature

increase. Figure 4-13 shown that the diffusion constant increase as the photo-

generated carrier density increase. Both this behavior suggests that the localized

state play an important role in the carrier transport. At elevated temperature, the

localized carrier can be thermally excited out of the localized state and thus the

increase of the diffusion constant. As the number of photo-generated carrier

density increase, most of the localized state will be filled up and more carriers

will occupied the de-localized state and lead to an increase in the diffusion

constant.

71


50 100 150 200 250 300 350 400 450 500

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a)

(d)(b)

Position -1mm 1mm 3mm 5mm

GaN:Mg 1 GaN:Mg 2 InGaN: Mg

Am

bipo

lar d

iffus

ion

cons

tant

[cm

2 /s]

Temperature [K]

(c)

Figure 4-12 The temperature dependent of the diffusion constant. (a) 30 QW 2.5nm In0.13Ga0.87N is my measurement data, (b) and (c) are calculated from the measured mobility data of p-GaN taken from[3] using Einstein relation, (d) are calculated from the measured mobility data of p-In0.09Ga0.91N taken from [2] using Einstein relation.

1016 1017 1018 1019 1020

0.01

0.1

1

10

(a) (b) (c) Calculated

Diff

usio

n co

nsta

nt [c

m2 /s

]

Photo Generated Carrier Density [1/cm3]

72


Figure 4-13 The power dependent of the diffusion constant, (a) 30 QW 2.5nm In0.13Ga0.87N is my measurement data, (b) 6 QW 3.4nm In0.3Ga0.7N and (c) 10 QW 4nm In0.3Ga0.7N are taken from [4].

4.6 Diffusion model

It is important to recognize that the diffusion constant that measure with the

transient grating method is the ambipolar diffusion constant. The ambipolar

diffusion constant is given by (page 87 in [5])

p n n pa

n p

D n D pD

n p

µ + µ=

µ + µ <4-13>

Where is the electron (hole) diffusion constant, is the electron

(hole) mobility, is the electron (hole) carrier density.

(n pD D ) ( )n pµ µ

( )n p

Here we introduce a simple model shown in Figure 4-14 we consider only the

heave-hole subband in the valence band and also we assume that there is a

localized state with energy level and the density of the localized state is

given by . We assume that the relaxation between the localized state and the

extended state is fast that they share a common quasi Fermi level ( ).

loc,pE

locP

FpE

73


locP

FpE E=

loc,pE E=

0hhE E= =h2

( )m

EL

ρ =π

locP

FpE E=

loc,pE E=

0hhE E= =h2

( )m

EL

ρ =π

Figure 4-14 Schematic of the diffusion model

The hole occupy in the extended state is given by

Fph Bext 2

B

ln[1 exp( )]Em k T

pL k

−= +

π T <4-14>

Where T is the temperature, is the Plank’s constant, is the Boltzmann’s

constant, L is the width of the quantum well, is the effective mass of the

hole.

Bk

hm

The hole occupy in the localized state is given by

locloc

Fp loc,p

B

1 exp( )

Pp E E

k T

= −+

<4-15>

The total number of hole is given by

<4-16> ext locp p p= +

Using the Boltzmann transport equation and under the relaxation time

approximation, the mobility and diffusion constant of hole is given by [6]

74


ext

Fpext

p

ph

pFpp p

h

qt pm p

dE dEp tD p

q dp dp m

µ =

= − µ = − <4-17>

Where is the electron charge, is the hole effective mass, q hm pt is the

relaxation time of hole.

A similarly model can be applied to the conduction band. The hole occupy in

the extended state is given by

c B Fnext 2

B

ln[1 exp( )]m k T E

nL k

= +π T

<4-18>

Where is the electron effective mass, is the electron quasi Fermi level. cm FnE

The electron occupy in the localized state is given by

locloc

loc,n Fn

B

1 exp( )

Nn E E

k T

= −+

<4-19>

Where is the localized energy level near the conduction band, is the

density of the localized state near the conduction band.

loc,nE locN

The total number of electron is given by

<4-20> ext locn n n= +

The mobility and diffusion constant of hole is given by

75


ext

Fn Fnext

n

nc

n

n nc

qt nm n

n dE dE tD n

q dp dn m

µ =

= µ = <4-21>

Where is the electron charge, is the electron effective mass, q cm nt is the

relaxation time of electron.

Under the optical excitation, same number of electron and hole is generated

<4-22> back

back

n N s

p P s s

= +

= + ≈

Where is the background electron (hole) carrier density, s is the

photo-generated carrier density. The background electron carrier density

is not negligible because the barrier is n-doped.

(back backN P )

backN

For an optical excitation level (s ), the quasi Fermi level of electron and hole can

be found by solving <4-16> and <4-20> combine with <4-22>. From there the

mobility and diffusion constant of electron and hole can be obtained from

<4-17> and <4-21>. The ambipolar diffusion constant is then calculate with

<4-13>.

In the numerical calculation, the following value have been used

0 0

loc,n loc,p

18 3 18 3loc loc

17 3

0.2 ; 1.2

100 ; 150

10 ; 5.5 10

0.1 ; 10

c h

n p back

m m m m

E meV E meV

N cm P c

t t ps N cm

− −

−

= =

= − =

= =

= = =

m× <4-23>

76


77

The calculation results are plot in Figure 4-12 and Figure 4-13. The calculation

results agree with the measurement data reasonably well. The proposed model

is able to reproduce the qualitative behaviors of the diffusion constant increase

as the temperature increase and the diffusion constant increase as the photo-

generated carrier density increase.

4.7 Summary and future work

In summary, a novel heterodyne transient grating measurement without the need

of electronic feedback stabilization has been developed and used to measure the

diffusion constant of InGaN MQW thin film. The measure diffusion constant is

very small (~0.2 cm2/s), indicated that the carrier localization play an important

role in the carrier transport. A simple model has been developed to explain the

temperature dependence and the carrier density dependence of the diffusion

constant.

The heterodyne transient grating developed in this chapter is a very

powerful method to study the carrier transport in thin film sample. Due to the

system limitation, only temperature around the room temperature is accessible to

our experiment. Measurement at lower and higher temperature should give us

more information about the carrier transport properties. Performing the


78

measurement with higher photo-generated carrier density will also shed more

light on the carrier transport behavior.

4.8 Reference [1] H. J. Eichler, P. Gunter, and S. W. Pohl, Laser-Induced Dynamics

Gratings. Berlin: Springer-Verlag, 1986. [2] S. Yamasaki, S. Asami, N. Shibata, M. Koike, K. Manabe, T. Tanaka, H.

Amano, and I. Akasaki, "p-type conduction in Mg-doped Ga0.91In0.09N grown by metalorganic vapor-phase epitaxy," Applied Physics Letters, vol. 66, pp. 1112-1113, 1995.

[3] T. Tanaka, A. Watanabe, H. Amano, Y. Kobayashi, I. Akasaki, S. Yamazaki, and M. Koike, "p-type conduction in Mg-doped GaN and Al0.08Ga0.92N grown by metalorganic vapor phase epitaxy," Applied Physics Letters, vol. 65, pp. 593-594, 1994.

[4] A. Vertikov, I. Ozden, and A. V. Nurmikko, "Diffusion and relaxation of excess carriers in InGaN quantum wells in localized versus extended states," Journal of Applied Physics, vol. 86, pp. 4697-4699, 1999.

[5] S. M. Sze, Physics of Semiconductor Devices, 2-nd ed. New York: John Wiley & Sons, 1981.

[6] J. R. Christman, Fundamentals of Solid State Physics. New York: John Wiley & Sons, 1988.

Chapter 5. Summary and future work


5.1 Gain measurement

In Chapter 2, we have used the FT method to measure and calculate the gain

spectrum, linewidth enhancement factor (3 ~ 15), ground index (3.2 ~ 3.55),

group velocity dispersion (-20 µm-1 ~ -40 µm-1), differential gain (

cm

170.2 10−×

2 ~ cm171 10−× 2), carrier-induced index change ( cm220.5 10−− × 3 ~

cm221 10−− × 3) and thermal-induce index change ( K41.3 10−× -1) for an

InGaN MQW laser diode in the wavelength range between 400 nm and 410 nm.

5.2 Valence intersubband relaxation

In Chapter 3, the femtosecond carrier dynamics in InGaN MQW laser diode

were studied using time-resolved bias-lead monitoring technique at low carrier

density region and the dynamics at high carrier density region were studied

using time-resolved differential reflection measurement. In both regions, the

carrier dynamics were found to be consistent with the presence of an ultrafast

79


inter-subband hole relaxation process between the TM-sensitive subbands (LH2,

HH2) and TE-sensitive subbands (LH1, HH1). The time constant of the transfer

process is within the time resolution of the experiment instrument (τ < 0.37 ps).

The differential transmission measurement is a preferred method to study

the carrier dynamics. In this thesis, due to the limitation of the sample,

differential transmission measurement is not possible. In the future, if the back

facet of the laser diode is accessible, measurement described in [1-3] should be

performed to make a more detailed analysis of the valence intersubband

relaxation dynamics.

Measurement on the a-plane (or m-plane) InGaN thin film sample can

also shed more light on the valence intersubband relaxation because a-plane

corresponds to the mirror facet of the InGaN laser diode. Measurement on the

a-plane sample had been tried, but repeatable measurement can not be obtained

due to the surface roughness and uniformity of the sample. In the future, when

the better quality a-plane (m-plane) sample is available, the measurement should

be performed to gain more insight to the valence intersubband relaxation

dynamics.

5.3 Carrier transport in InGaN

80


The heterodyne transient grating developed in chapter 4 is a very powerful

method to study the carrier transport in thin film sample. This novel heterodyne

transient grating measurement eliminated need of electronic feedback

stabilization. We have used this method to measure the diffusion constant of

InGaN MQW thin film. The measure diffusion constant is very small (~0.2

cm2/s), indicated that the carrier localization play an important role in the carrier

transport. A simple model has been developed to explain the temperature

dependence and the carrier density dependence of the diffusion constant.

Due to the system limitation, only temperature around the room

temperature is accessible to our experiment. Measurement at lower and higher

temperature should give us more information about physical mechanism that

governs the carrier transport properties. Performing the measurement with

higher photo-generated carrier density will also shed more light on the carrier

transport behavior.

5.4 Reference [1] K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon,

"Femtosecond index nonlinearities in InGaAsP optical amplifiers," Applied Physics Letters, vol. 62, pp. 1320-1322, 1993.

[2] C. T. Hultgren and E. P. Ippen, "Ultrafast refractive index dynamics in AlGaAs diode laser amplifiers," Applied Physics Letters, vol. 59, pp. 635-637, 1991.

81


[3] C. K. Sun, H. K. Choi, C. A. Wang, and J. G. Fujimoto, "Femtosecond gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers," Applied Physics Letters, vol. 63, pp. 96-98, 1993.

82

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