Nitrogen vacancy scattering in GaN grown bymetal–organic vapor phase epitaxy

Zhen Chen *, Hairong Yuan, Da-Cheng Lu, Xuehao Sun, Shouke Wan,Xianglin Liu, Peide Han, Xiaohui Wang, Qinsheng Zhu, Zhanguo Wang

Laboratory of Semiconductor Materials Science, Institute of Semiconductors, Beijing 100083, China

Received 24 September 2001; received in revised form 5 April 2002; accepted 22 April 2002

Abstract

Electron mobility limited by nitrogen vacancy scattering was taken into account to evaluate the quality of n-type

GaN grown by metal–organic vapor phase epitaxy. Two assumptions were made for this potential for the nitrogen

vacancy (1) it acts in a short range, and (2) does not diverge at the vacancy core. According to the above assumptions, a

general expression to describe the scattering potential UðrÞ ¼ �U0 exp½�ðr=bÞn�; ðn ¼ 1; 2; . . . ;1Þ was constructed,where b is the potential well width. The mobilities for n ¼ 1; 2; and 1 were calculated based on this equation, cor-

responding to the simple exponential, Gaussian and square well scattering potentials, respectively. In the limiting case

of kb � 1 (where k is the wave vector), all of the mobilities calculated for n ¼ 1; 2; and 1 showed a same result but

different prefactor. Such difference was discussed in terms of the potential tail and was found that all of the calculated

mobilities have T�1=2 temperature and b�6 well width dependences. A mobility taking account of a spatially complicate

scattering potential was studied and the same temperature dependence was also found. A best fit between the calculated

results and experimental data was obtained by taking account of the nitrogen vacancy scattering.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Nitrogen vacancy scattering; GaN; Mobility; MOCVD

1. Introduction

Wide band nitrides are of considerable interest due to

the applications of blue/UV light emitting diodes and

lasers, and of high-temperature electronics [1,2]. Intrin-

sic point defects [3–7] are known to play an important

role in GaN. Nitrogen vacancies were long suspected to

be the source of n-type conductivity. In addition to the

nitrogen vacancies, gallium vacancies with high forma-

tion energy in n-type GaN were regarded as the source

of the ‘‘Yellow luminescence’’ [8]. Other types of point

defects, such as antisites and self-interstitials, have been

also taken into account for explaining some of the ex-

perimental results. Because the formation energies of

antisites, self-interstitials and gallium vacancy are higher

than that of the nitrogen vacancy, the nitrogen vacancy

concentration is thought to be higher than other kinds of

point defects. Therefore the nitrogen vacancy was in-

vestigated in this study.

Hall measurement is a powerful experimental method

to investigate the point defect scattering mechanism. It is

known that two scattering mechanisms are commonly

adopted for characterization in semiconductor materi-

als. One is the impurity scattering that governs the low

temperature mobility, and another is optical phonon

scattering that dominates high temperature mobility.

However, it was found frequently that there was a dis-

crepancy between the theoretical calculations and the

experimental result if taking account only these two

scattering mechanisms. A mobility limited by the ni-

trogen vacancy scattering will be introduced in order to

*Corresponding author. Tel.: +86-10-82304968; fax: +86-10-

82305052.

E-mail address: vibrant@red.semi.ac.cn (Z. Chen).

0038-1101/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0038-1101 (02 )00244-7

Solid-State Electronics 46 (2002) 2069–2074

www.elsevier.com/locate/sse

obtain a better fit between the calculated and experi-

mental results in this work.

Besides the nitrogen vacancy mentioned, GaN epit-

axial layer contains high density threading dislocation

(108–109 cm�2) parallel to the growth direction [3,9],

which also considerably affects the material quality. The

dislocation scattering was proposed by P€ood€oor for thefirst time to study the temperature-dependent mobility in

germanium [10]. Recently, the dislocation scattering in

GaN was studied by Weimann et al. [9]. They showed

that the mobility limited by threading dislocation scat-

tering depended on the dislocation density and carrier

concentration. Since the dislocation scattering affects the

mobility seriously, the dislocation scattering was also

introduced into our study.

2. Theoretical consideration

2.1. Scattering potentials

It is important to find a reasonable nitrogen-vacancy-

induced scattering potential for mobility calculation

firstly. At the point defect, such as the nitrogen vacancy,

the electronic potential differs from the ideal crystal

potential and acts as a scattering center. It is reasonable

to assume that the point-defects-induced potential could

be simplified as: (a) it acts in a short range, that is, the

potential must be distributed within one or several lat-

tice cells (including the potential tail), and (b) does not

diverge at the vacancy core [11]. For example, the

Coulomb potential �A0=r with A0 being the constant isnot only a long range potential, but also diverges at

r ¼ 0. This potential acts as a ‘‘rigid’’ barrier at r ¼ 0,

and any free electron cannot penetrate the barrier. Other

examples are similar potentials ðA0=rÞn for n > 1 and the

Yukawa potential B0 expð�r=aÞ=r with B0 and a beingthe constant potential and well width parameter, re-

spectively. Though these potentials are short range, they

also diverge at r ¼ 0. On the other hand, many types of

the potentials satisfy these two requirements, for exam-

ple, the Gaussian well (GW) potential U0 exp½�ðr=bÞ2�,and the square well (SW) potential described as UðrÞ ¼�U0, for r6 a, and UðrÞ ¼ 0 for r > a, where a is theSW potential well width. In addition to these examples,

a potential UðrÞ ¼ U0 expð�r=bÞ is also a short range

with finite well depth at the core. To sum up the above

examples, we suggest the following equation to describe

all of the cases:

UðrÞ ¼ �U0 exp½�ðr=bÞn�; n ¼ 1; 2; . . . ;1 ð1Þ

This equation satisfies the two assumptions discussed

above. As n ¼ 1, Eq. (1) becomes a single exponential

potential well (EW),

UðrÞ ¼ �U0 exp½�ðr=bÞ� ð2Þ

As n ¼ 2, Eq. (1) reduces to the so-called GW potential

UðrÞ ¼ �U0 exp½�ðr=bÞ2� ð3Þ

When n ! 1, Eq. (1) becomes a SW potential with well

width b

UðrÞ ¼ �U0 exp½�ðr=bÞ1� ¼ �U0 0 < r < b0 b < r < 1

�ð4Þ

The three different potential wells corresponding to Eqs.

(2)–(4) were illustrated in Fig. 1 for comparison.

It is obvious that Eqs. (2) and (4) are the two limiting

cases of Eq. (1). Therefore, we only need to consider the

two limiting cases for our calculation. Eq. (3) is also

used to compare with the mobilities calculated from Eqs.

(2) and (4). In the following, the mobilities due to the

SW, GW and EW scattering potentials will be calculated

and the relations among them will be discussed.

2.2. The case of n ¼ 1

In the case of n ¼ 1, Eq. (1) gives a SW scattering

potential. In comparison with the calculation based on

classical mechanics, the quantum mechanical calculation

of the cross-section based on the SW potential has an

analytical result. The first step is the calculation of the

Hamiltonian matrix element,

Hkk0 ¼Z

Wk0UðrÞWk d3r

¼ 1

V

Zexpð�ik0

!� r!ÞUðrÞ expði k!� r!Þd3r ð5Þ

where Wk and Wk0 are the free electron wave functions

before and after elastic scattering on the SW. V is the

volume. The scattering potential for the point defect can

Fig. 1. Short range scattering potentials for powers n ¼ 1, 2,

and 1000 in Eq. (1). b is the well width parameter, which is closeto the SW width as n ! 1.

2070 Z. Chen et al. / Solid-State Electronics 46 (2002) 2069–2074

be described as a spherical well: UðrÞ ¼ �U0 for r < a,and UðrÞ ¼ 0 for r > a. Thus Eq. (5) becomes

Hkk0 ¼ �U0

V

Z a

0

exp½ið k!� k0!Þ � r!�d3r

¼ � 4pU0

V1

q3½sinðqaÞ � ðqaÞ cosðqaÞ� ð6Þ

where

q ¼ j k!� k0!j ¼ 2k sin

h2

with h being the angle between k!and k0

!. The transition

probability Sð k!; k0!Þ per unit time is given by the Golden

rule,

Sð k!; k0!Þ ¼ 2p

�h2jHkk0 j2d½Eð k

!Þ � Eðk0!Þ� ð7Þ

Thus, the differential cross-section can be obtained as

rðhÞ ¼ ðVm Þ2

ð2p�hÞ31

k

ZSð k!; k0

!Þk0 dE ¼ Vm jHkk0 j

2p�h2

� �2

¼ 4m U0

�h2

� �21

q6½sinðqaÞ � ðqaÞ cosðqaÞ�2 ð8Þ

Above results can be expanded into a series with respect

to qa,

rðhÞ ¼ 4m U0

�h2

� �2

a61

9

�� 1

45q2a2 þ 1

525q4a4 þ � � �

�ð9Þ

Notice that the second and third terms in the paren-

theses are much smaller than the first term, because

qa ¼ 2ka sinh2� 1 ð10Þ

where the wave vector k is deduced from �h2k2=2m ¼KBT , and a is close to the size of GaN lattice constant.

The integration overall angles gives momentum-transfer

cross-section [12]

r ¼ 2pZ p

0

rðhÞð1� cos hÞ sin hdh

¼ 8pa6m U0

�h2

� �22

9

�� 16

135k2a2 þ 16

525k4a4

�

� 16

9pa6

m U0

�h2

� �2

; for ka � 1

ð11Þ

Finally we obtain the mobility:

lS ¼em

1

Nvvr

¼ 9

16ffiffiffi3

pp

e�h4

ðm Þ5=21

Nv

1

U 20 a6

ðKBT Þ�1=2 ð12Þ

where Nv, KB, and T are the vacancy concentration, the

Boltzmann’s constant, and the temperature, respec-

tively, and v is the electron velocity given by v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3KBT=m

p.

2.3. The case of n ¼ 2

When n ¼ 2, Eq. (1) becomes a GW scattering po-

tential. To compare with the SW potential scattering, we

will calculate the GW potential scattering in this section.

Using the potential given by Eq. (3), the Hamiltonian

matrix element is

Hkk0 ¼Z 1

0

Wk0 UðrÞWk d3r

¼ 1

V

Z 1

0

�U0 expð�r2=b2Þ exp½�ið k!� k0!Þ � r!�d3r

¼ �1V

1

j k!� k0!j4pU0

�Z 1

0

exp

�� r2

b2

�sin j k!

�� k0!jr�rdr ð13Þ

For integrating Eq. (13), we use a relationZ 1

0

expð�nr2Þ sinðgrÞrdr

¼ gffiffiffip

p

4nffiffiffin

p exp�g2

4n

� �for n > 0 ð14Þ

Substituting n ¼ 1=b2 and g ¼ q ¼ 2k sinðh=2Þ in Eq.

(14), the Hkk0 then becomes

Hkk0 ¼1

Vp3=2b3U0 exp

�� 1

4q2b2

�ð15Þ

The momentum-transfer cross-section is then given by

r ¼Z p

0

Vm 1V p3=2b3U0 exp � 1

4q2b2

�� ��2p�h2

" #2

� 2pð1� cos hÞ sin hdh ð16Þ

since qb ¼ 2kb sinðh=2Þ � 1, the exponential term in Eq.

(16) can be approximated as

exp

�� 2k2b2 sin2

h2

� ��� 1� 2k2b2 sin2

h2

� �ð17Þ

Then, the momentum cross-section is

r ¼ p2m b6U 20

2�h42

�� 16

5k2b2

�� p2m b6U 2

0

�h4ð18Þ

Finally, we obtain the mobility as follows

lG ¼ 1ffiffiffi3

pp2

e�h4

ðm Þ5=21

Nv

1

U 20b

6ðKBT Þ�1=2 ð19Þ

The result is good consistent with lS given by Eq. (12).The difference in prefactor between lG and lS will bediscussed in Section 2.5.

Z. Chen et al. / Solid-State Electronics 46 (2002) 2069–2074 2071

2.4. The case of n ¼ 1

In the case of n ¼ 1, Eq. (1) gives a single exponential

potential �U0 expð�r=bÞ. The Hamiltonian matrix ele-ment is,

Hkk0 ¼1

V

Z 1

0

�U0 exp

�� r

b

�exp½�ið k!� k0

!Þ � r!�d3r

¼ �U0

V4pq

Z 1

0

exp

�� r

b

�sinðqrÞrdr

¼ �U0

V4pq

b2

1þ q2b2sin½2 arctanðqbÞ�

¼ �8pU0

Vb3

ð1þ q2b2Þ2

ð20Þ

The differential scattering cross-section is

rðhÞ ¼ 64p2Vm

2p�h2

� �2 U 20

V 2

b6

ð1þ q2b2Þ4ð21Þ

Substitute rðhÞ into following equation, we can obtainthe momentum cross-section,

r ¼ 2pZ p

0

rðhÞð1� cos hÞ sin hdh

¼ 128p3m

2p�h2

� �2

U 20b

6R ð22Þ

where

R ¼Z p

0

1

½1þ 4k2b2 sin2ðh=2Þ�4ð1� cos hÞ sin hdh

Because k2b2 � 1, the following approximation is

valid, that is:

1

1þ 4K2b2 sin2 h2

� 1� 4k2b2 sin2h2

� �þ � � �

then,

R ¼ 2� 64

3k2b2 � 2

Substituting this result into Eq. (22) leads to the mobility

lE ¼ 1

64ffiffiffi3

pp

e�h4

ðm Þ5=21

Nv

1

U 20b

6ðKBT Þ�1=2 ð23Þ

This result is consistent with lG and lS given by Eqs.(19) and (12), respectively. They all have the same T, U0

and b dependences, but the different prefactors.

2.5. Discussion

The calculated mobilities given by Eqs. (12), (19) and

(23) are in very good agreement. There is only a differ-

ence among the prefactors of them. For lE, lG and lS,the prefactors are 1=64

ffiffiffi3

pp, 1=

ffiffiffi3

pp2, and 9=16ð1=

ffiffiffi3

ppÞ,

respectively. It is apparent that lE < lG < lS. Suchdifference is believed to be due to the spatial distribution

of scattering potential with a small tail exceeding well

width b (or a) (see Fig. 1). From Eq. (1), for n ¼ 1, the

simple exponential potential UðrÞ has the longest tail,for n ¼ 2, it has the shorter tail, and for n ¼ 1, it has no

tail. It can be shown that lE, lG and lS are all pro-portional to b�6 or a�6 from Eqs. (12), (19) and (23),

thus the potential tail for n ¼ 1 in Eq. (1) can ‘‘kill’’ the

mobility significantly.

It is also noteworthy that lE, lG and lS are all pro-portional to T�1=2. This is a main feature for all of the

short-range scattering potentials. The nitrogen vacancy

concentration is less than 1017 cm�3. The point defect

width a closes to the GaN lattice constant if U0 is several

eVs deep.

2.6. Complicate short-range scattering potential

So for, we have described the scattering potentials

based on the typical cases, i.e., SW, GW and EW scat-

tering potentials. However, the real nitrogen-vacancy-

induced potential in GaN might be complicate, which

may differ from the ideal cases as discussed earlier.

Usually there is no analytical function to describe these

potentials. However, we can decompose such potential

into several rectangular wells with an equal well width

aiþl � ai but different well depth Ui for ith well. Such

total rectangular well width is given by

A ¼Xn

i¼0ðaiþ1 � aiÞ ð24Þ

Fig. 2 shows a typical potential distribution with respect

to the spatial position r. In this case, the total Hamil-

tonian Hkk0 is a summation over all of Hikk0 , given by

Hkk0 ¼P

i Hikk0 . Thus, the differential cross-section for

this system has a complicate form

rðhÞ ¼ 4m 2

�h41

q6Xn�1i¼0

Ui½sinðqaiÞ(

� sinðqaiþ1Þ

� ðqaiÞ cosðqaiÞ þ ðqaiþ1Þ cosðqaiþ1Þ�)2

ð25Þ

In the limiting case, when ai ¼ 0 for i ¼ 0, Eq. (25) re-

duces to Eq. (8) for the case of single SW potential.

It is impossible to integrate Eq. (25) in order to obtain

an analytic result for further discussion. However, Eq.

(25) gives us an important information that the rðhÞ areindependent of temperature for the limiting case ka � 1.

As a result, the mobility is expected to be proportional

to T�1=2. It is apparent that all of the short-range scat-

2072 Z. Chen et al. / Solid-State Electronics 46 (2002) 2069–2074

tering potentials (they do not diverge at the core) show

the same temperature dependence.

3. Experiment

3.1. Crystal growth and measurement

The samples used in this study were grown by

MOVPE method [13]. In brief, trimethylgallium and

NH3 are reacted in a chamber at atmospheric pressure

with SiH4 used for the Si source gas. The wurtzite

(0 0 0 1) GaN is deposited on (1 1 20) sapphire at 1150 �Cusing an AlN buffer layer. By this method, the donor

concentration has been shown to be controllable from

2� 1016 cm�3 to higher concentration by means of the

SiH4 flow rate. The GaN layers used for Hall measure-

ments were about 2 lm thick. Hall measurements were

performed using the Van der Pauw technique at tem-

perature ranging from 4 K to room temperature in a

closed loop helium cryostat. Indium dot was applied to

the crystal face to form the ohmic contacts. Temperature

dependent Hall measurements of sample were per-

formed for a numerical fitting.

3.2. Fitting results

The mobility ldis due to the dislocation line scatteringis given by [10].

ldis ¼30

ffiffiffiffiffiffi2p

pee0a2d

e3f 2LDm 1=2ðKBT Þ3=2

Ndis

ð26Þ

where LD is the ‘‘Debye length’’ given by

LD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiee0KBT=e2nðT Þ

pwith the temperature-dependent net carrier concentra-

tion nðT Þ. In Eq. (26), kB is the Boltzmann’s constant, adis the distance between the imperfection centers along

the dislocation line, f is their occupation probability and

Ndis is the dislocation density. f, Ndis and ad are employedas fitting parameters. The dislocation density Ndis is de-termined by accounting the etch pit density in the sam-

ple surface, which is about 1� 109 cm�2. Eq. (26) shows

temperature-dependent factor of TffiffiffiffiffiffiffiffiffiffinðT Þ

p, which domi-

nates the total mobility in very low temperature region

(several tens Kelvins).

In Fig. 3, lim, the mobility limited by the impurity

scattering, which dominates the total mobility in the low

temperature region, is given by the Brooks–Herring

formula [14]; lop, the mobility limited by the longitudi-nal optical-phonon scattering, which dominates the

mobility in the high temperature region, was derived by

Homarth and Sondheimer [15].

To obtain best fitting between calculated and mea-

sured data, the mobility lS due to SW potential

Fig. 3. Experiment plots and theory with full account of the

scattering. In this figure the calculated mobility components

lim, ldis, lop and lv are, respectively, due to ionized impurity,dislocation, optical phonon, and nitrogen vacancy scatterings.

The measured data are also plotted in the same figure by closed

squares for a comparison with the calculated total mobility ltot.The free carrier concentration for this sample is about 1� 1017

cm�3, and the dislocation density is about 1� 109 cm�2. In this

figure, we also show a total mobility l0tot calculated with full

account of scatterings but only without the mobility lv due tonitrogen vacancy scattering.

Fig. 2. An example of complicate short-range scattering po-

tential spatial distribution. This potential can be decomposed

into several rectangular wells with an equal well width but

different well height Ui for ith well.

Z. Chen et al. / Solid-State Electronics 46 (2002) 2069–2074 2073

scattering (here lS is replaced by the notation lv in Fig.3) was adopted for fitting. Using the Matthiesen’s rule,

the reciprocal total mobility, l�1tot is given by

1=ltot ¼ 1=lim þ 1=lop þ 1=ldis þ 1=lv ð27Þ

Fig. 3 shows the temperature-dependent total mobility

ltot given by above equation for a sample. As shown inthis figure, the calculated ltot can be well fitted to themeasured data. In order to know how the lv contributesto ltot, in the same figure we show a total mobility l0

tot in

which the lv is excluded from Eq. (27). It is obvious that

l0tot deviates from the measured data at high tempera-

ture. Thus we conclude that the point defect scattering

should be introduced to analyze the total mobility of

GaN. In Fig. 3, two measured mobility data appearing

at lower temperature deviate apparently from the cal-

culated total mobility. The reason cannot be well ex-

plained at the present time.

4. Conclusion

In conclusion, we studied the nitrogen vacancy scat-

tering for high quality n-GaN films grown by MOVPE

technique. A scattering potential, which is in a short

range with a finite potential depth at the core, can

be expressed by a formula UðrÞ ¼ �U0 exp½�ðr=bÞn�;ðn ¼ 1; 2; . . . ;1Þ and utilized as the model for point

defect scattering potential. The calculated results exhibit

that for n ¼ 1; 2 and1, the mobilities lE, lG and lS arethe same for the limiting case of kb � 1 (where b is thepotential well width and k the wave vector). It is also

found that the amplitude of prefactor in such mobility

increases with the power n in UðrÞ increasing. The longerthe scattering potential tail is, the lower the mobility will

be. Those calculated mobilities are found to be pro-

portional to T�1=2, being a dominant one in the high

temperature range, like lop. The mobility calculated oncomplicate scattering potential also shows the T�1=2

temperature dependence. The suggested SW potential

used for fitting to the experimental data seems too rough

in describing a real nitrogen-vacancy-induced scattering

potential, but in fact there are no significantly different

results between an assumed short range potential and a

real one for GaN.

Acknowledgements

This work is financially supported by the National

Natural Science Foundation of China (No. 60086001),

and by the Special Funds for Major State Basic Re-

search program no. G20000683 of China.

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