Research Article Barrier Thickness and Hydrostatic

Research ArticleBarrier Thickness and Hydrostatic Pressure Effects onHydrogenic Impurity States in Wurtzite GaNAlxGa

1minusxNStrained Quantum Dots

Guangxin Wang1 Xiuzhi Duan2 and Wei Chen1

1College of Science Hebei United University Tangshan 063000 China2College of Light Industry Hebei United University Tangshan 063000 China

Correspondence should be addressed to Guangxin Wang guangxinwang126com

Received 6 January 2015 Accepted 17 March 2015

Academic Editor J David Carey

Copyright copy 2015 Guangxin Wang et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Within the framework of the effective mass approximation barrier thickness and hydrostatic pressure effects on the ground-statebinding energy of hydrogenic impurity are investigated in wurtzite (WZ) GaNAlxGa1minusxN strained quantum dots (QDs) by meansof a variational approachThe hydrostatic pressure dependence of physical parameters such as electron effective mass energy bandgaps lattice constants and dielectric constants is considered in the calculations Numerical results show that the donor bindingenergy for any impurity position increases when the hydrostatic pressure increases The donor binding energy for the impuritylocated at the central of the QD increases firstly and then begins to drop quickly with the decrease of QD radius (height) in strongbuilt-in electric fields Moreover the influence of barrier thickness along the QD growth direction and Al concentration on donorbinding energy is also investigated In addition we also found that impurity positions have great influence on the donor bindingenergy

1 Introduction

Recently much attention has been paid to wide-band gapwurtzite (WZ) GaNAlGaN quantum heterostructures dueto their promising applications in optoelectronic devicessuch as light-emitting devices (LEDs) and laser diodes(LDs) [1ndash3] Doping impurities in GaN-based confined sys-tems is an effective method for controlling the electronicand optical properties of optoelectronic devices It is wellknown that the hydrostatic pressure applied on a GaN-based semiconductor material can not only modify theparameters such as the band gaps the potential barriers theconduction effective masses the static dielectric constantsand the lattice constants but also change the dimension ofthe low-dimensional systems which is associated with thefractional change in the volume Moreover the strong built-in electric field induced by the spontaneous and piezoelectricpolarizations also affects obviously the distribution of thecarrier wave function in WZ nitride-based quantum het-erostructures According to the above characteristics various

studies concerning impurity states are reported in GaN-based nanostructures such as quantum wells [4ndash8] quantumwell wires [9 10] and (double) quantum dots [11ndash17] Inthese references considering the strong built-in electric fieldandor hydrostatic pressure effects the hydrogenic donorimpurity states are mainly discussed in a radial infiniteconfinement potential barrier Their results demonstrate thatquantum structure size strong built-in electric field andhydrostatic pressure have a significant influence on the donorimpurity binding energies but there are few reports involvedin barrier thickness effects on the donor binding energy inWZ GaNAlxGa1minus119909N strained QD in finite potential barrierto date

To further demonstrate the barrier thickness effect onimpurity states in a WZ GaNAlGaN strained QD wecalculate the donor binding energy of hydrogenic impurity ina WZ GaNAlGaN QD under a strong built-in electric fieldby means of a variational approach in the finite confinementpotential In our calculation effective mass of the electrondielectric constants phonon frequencies energy gaps sizes

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2015 Article ID 937310 9 pageshttpdxdoiorg1011552015937310

2 Journal of Nanomaterials

AlGaN GaN

R

O 120588Lw

Lb

z

Figure 1Thediagramof a cylindrical wurtziteGaNAlxGa1minusxNQD

(radius and height) of QD and piezoelectric polarizations areconsidered as a function of hydrostatic pressureThe paper isorganized as follows In Section 2 we describe the theoreticalframework Then the pressure and the strain coefficients ofGaN and AlxGa1minus119909N are discussed in Section 3 In Section 4the numerical results are discussed Finally conclusions aredrawn from the present study in Section 5

2 Theoretical Framework

In Figure 1 the schematic view of a cylindrical WZGaNAlxGa1minus119909N QD is depicted with a detailed descrip-tion of the different dimensions of the QD (dot radius 119877height 119871

119908 and barrier thickness 119871

119887) Additionally the GaN

quantum dot is embedded in an AlxGa1minus119909N host matrixmaterial and the 119911-axis is defined to be the growth directionWithin the frame of the effective mass approximation theHamiltonian for a hydrogenic donor impurity in cylindricalWZ GaNAlxGa1minus119909N QD under the influence of hydrostaticpressure can be written [4] as

119867 =

119867

0minus

119890

2

4120587120576

0120576 (119901)

1003816

1003816

1003816

1003816

1003816

997888

119903 minus

997888

119903

0

1003816

1003816

1003816

1003816

1003816

(1)

where 997888119903 (9978881199030) denotes the position vector of the electron

(impurity ion) 119890 is the absolute value of the electron charge120576

0is the permittivity of free space and 120576(119901) is the pressure-

dependent effective mean relative dielectric constant of GaNandAlxGa1minus119909NmaterialsTheHamiltonian 119867

0is given by [11]

119867

0= minus

ℏ

2

2119898

perp(119901)

(

1

120588

120597

120597120588

(120588

120597

120597120588

) +

1

120588

2

120597

2

120597120593

2)

minus

ℏ

2

2119898

(119901)

120597

2

120597119911

2+ 119881 (120588 119911 119901) minus 119890119865

119908119887(119901) 119911

(2)

where 119898

and 119898perpare the pressure- and strain-dependent

effective masses of electron along and perpendicular to the

[0001]-direction And 119881(120588 119911 119901) is the electron confinementpotential due to the band offset (119876

119895= 0765) and is given by

[13]

119881 (119911 119901)

=

119876

119895[119864

119892Al119909Ga1minus119909

N minus 119864119892GaN] 119871

119908

2

lt 119911 lt

119871

119908

2

+

119871

119887

2

minus

119871

119908

2

minus

119871

119887

2

lt 119911 lt minus

119871

119908

2

0 |119911| le

119871

119908

2

(3)

119881 (120588 119901) =

0 120588 le 119877

119876

119895[119864

119892Al119909Ga1minus119909

N minus 119864119892GaN] others(4)

In (2) F(p) is the pressure-dependent built-in electricfield (BEF) in the finitely thick barrier layer for WZGaNAlxGa1minus119909NQDThe values of the BEF along the growthdirection in the well (119865GaN) and barrier (119865Al

119909Ga1minus119909

N) that resultfrom the difference in the total electric polarizations in eachregion are given by simple formulas [14]

119865GaN (119901) =

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119871

119887(119875

AlGaNsp + 119875

AlGaNpe minus 119875

GaNsp minus 119875

GaNpe )

120576

0(119871

119887120576

GaN119890+ 119871

119908120576

AlGaN119890

)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119865AlGaN (119901) = minus119871

119908

119871

119887

119865GaN (119901)

(5)

where 119871119908 119871119887and 120576GaN119890

120576AlGaN119890

are the thicknesses and elec-tronic dielectric constants for GaN and AlxGa1minus119909Nmaterials119875

GaNsp 119875AlGaNsp and 119875GaNpe 119875AlGaNpe are the spontaneous and

piezoelectric polarizations forGaN andAlxGa1minus119909Nmaterialsand the bowing parameter 120578 is chosen as minus0019 Cm2 [14]

119875

AlGaNsp = 119909119875

AlNsp + (1 minus 119909) 119875

GaNsp minus 120578119909 (1 minus 119909) (6)

The wave function of the electron confined in the WZGaNAlxGa1minus119909N QD can be written as

120595 (120588 120593 119911) = 119891 (120588) ℎ (119911) 119890

119894119898120593 119898 = 0 plusmn1 plusmn2

119891 (120588) =

119869

0(120573120588) 120588 le 119877

119869

0(120573119877)

119870

0(120573119877)

119870

0(120572120588) 120588 gt 119877

(7)

where the constants (120572 and 120573) are determined by the con-tinuity of the derivative of the radial wave function at theQD boundary and 119898 is the electron 119911-component angularmomentum quantum numberThe radial wave function119891(120588)of the electron can be obtained using the Bessel function 119869

119898

Journal of Nanomaterials 3

and themodified Bessel function119870119898Wave function ℎ(119911) can

be expressed by means of the Airy functions Ai and Bi [16]

ℎ (119911) =

119862

1Ai (1205761) + 119863

1Bi (1205761) minus

119871

119908

2

minus 119871

119887lt 119911 lt minus

119871

119908

2

119862

2Ai (1205762) + 119863

2Bi (1205762) |119911| le

119871

119908

2

119862

3Ai (1205763) + 119863

3Bi (1205763)

119871

119908

2

lt 119911 lt

119871

119908

2

+ 119871

119887

(8)

Here 120576119895= ((2119898

lowast

119895ℏ

2)119890119865)

13(119911 minus 119864

119911minus 119881(119911 119901)119890119865) The coef-

ficients 119862119895and 119863

119895(119895 = 1 2 and 3) can also be obtained by

the transfer matrix methods [16]In order to calculate the donor binding energy the trial

wave function can be chosen as [17]

120601 (120588 120593 119911) = 119873120595 (120588 120593 119911) 119890

minus120572(120588minus1205880)2

minus120573(119911minus1199110)2

(9)

where 119873 denotes the normalization constant with theadiabatic approximation the donor binding energy of ahydrogenic impunity 119864

119887is defined as the difference between

the ground-state energy of the system without impunity andthe ground-state energy of the system with impurity that is

119864

119887= 119864

0120588+ 119864

0119911minusmin120572120573

⟨120595

1003816

1003816

1003816

1003816

119867

1003816

1003816

1003816

1003816

120595⟩

⟨120595

1003816

1003816

1003816

1003816

120595⟩

(10)

3 Pressure and Strain Dependence ofPhysical Parameters

In this model we take the strains induced by the biaxiallattice mismatch into account The components of biaxialstress tensors of GaN and AlGaN materials are given underthe hydrostatic pressure 119875 [18]

120576

119909119909GaN = 120576119910119910GaN =119886eq (119901) minus 119886GaN (119875)

119886GaN (119875)

120576

119909119909AlGaN = 120576119910119910AlGaN =119886eq (119901) minus 119886AlGaN (119875)

119886AlGaN (119875)

(11)

where the equilibrium lattice constant 119886eq for the strainedlayer under hydrostatic pressure 119875 depends on the latticeconstants of the componentmaterials and is weighted by theirrelative thicknesses [19]

119886eq (119901) =119886GaN119871119908 + 119886GaAlN119871119887

119871

119908+ 119871

119887

(12)

The lattice constant 119886] of theGaN (AlN)material dependenceof hydrostatic pressure satisfies [20]

119886] (119875) = 1198860] (1 minus119901

3119861

0]) ] = GaNAlN (13)

Here the actual lattice constant 119886AlGaN of theAlxGa1minus119909Nmate-rial can be obtained by the linear interpolation method fromthe corresponding values of GaN and AlN 119871GaN(119871GaAlN) is

the thickness of the dot (barrier) layer The strain inducedby the biaxial lattice mismatch along the z-direction in theheterostructure is [21]

120576

119911119911GaN = 119877119867

GaN120576119909119909GaN

120576

119911119911AlGaN = 119877119867

AlGaN120576119909119909AlGaN(14)

The coefficient 119877119867] of the GaN (AlN) material is given by [21]

119877

119867

] =119862

11] (119875) + 11986212] (119875) minus 211986213] (119875)

119862

33] (119875) minus 11986213] (119875) ] = GaNAlN

(15)

where 119862120582120581] is the pressure-dependent elastic stiffness con-

stant of material ] and is given by [21] 119862120582120581](119875) = 119862120582120581](0) +

120590

120582120581]119875 + 120575120582120581]1198752 and the coefficient 119877119867AlGaN and the pressure-

dependent elastic stiffness constant 119862120582120581AlGaN of AlxGa1minus119909N

material can be obtained by the linear interpolation methodfrom the corresponding values of GaN and AlN The strain-dependent energy gaps of GaN and AlN are [22]

119864

119892GaN = 119864119892GaN (119901) + 2 (1198861GaN + 1198871GaN) 120576119909119909GaN

+ (119886

2GaN + 1198872GaN) 120576119911119911GaN

119864

119892AlN = 119864119892AlN (119901) + 21198861AlN120576119909119909AlN + 1198862AlN120576119911119911AlN

(16)

where 1198861] 1198862] 1198871] and 1198872] (] = GaN and AlN) are the

deformation potentials The energy gap 119864119892](119901) dependence

of hydrostatic pressure 119875 is considered by the followingequation [22]

119864

119892] (119901) = 119864119892] (0) + 120594]119901 (17)

The energy gap of the AlxGa1minus119909N alloy can be calculated bythe following formula [14]

119864

119892AlGaN (119901) = (1 minus 119909) 119864119892GaN + 119909119864119892AlN minus 119887119909 (1 minus 119909) (18)

The biaxial strain and hydrostatic pressure dependences ofthe electron effective masses in the 119909-119910 plane and the 119911-direction can be calculated by [23]

119898

0

119898

perp

] (119901)

= 1 +

119862

119895]

119864

119892] (119901) (19)

where 119862119895] is a fixed value for a given material ] and can be

derived from the values of119898perp] (0) and 119864119892](0) In (5) under

consideration of the biaxial strain in a wurtzite structuremodified by hydrostatic pressure the dielectric constants andthe phonon eigen-frequencies will also change The staticdielectric constant 120576

120585is influenced by biaxial strain and

hydrostatic pressureThe tensor components of 120576]120585for theWZ

structure are derived from the generalized Lyddane-Sachs-Teller relation [24]

120576

]120585(119901) = 120576

]infin120585(119901)(

120596

]LO120585 (119875)

120596

]TO120585 (119875)

)

2

(20)


where the LO- and TO-phonon frequencies influenced bybiaxial strain and hydrostatic pressure can be written as

120596

]119895120585= 120596

]119895120585(119901) + 2119870

]119895120585119909119909

120576

119909119909] (119901) + 119870]119895120585119911119911

120576

119911119911] (119901) (21)

Furthermore the hydrostatic pressure dependence of120596]119895120585

canbe given by [25]

120574

]119895120585= 119861

0]1

120596

]119895120585(0)

(

120597120596

]119895120585(119901)

120597119901

) (22)

where the subscript 119895 represents LO or TO phonon 120585 (120585 =perpor 119911) denotes 119909-119910 plane or 119911-direction 120596]

119895120585(0) is the zone-

center phonon frequency of material ] 120574]119895120585

is Gruneisenparameter of phonon mode given in [25] 119861

0] is bulkmodulus and 119870]

119895120585119909119909and 119870]

119895120585119911119911are the strain coefficients

of zone-center phonon modes Following [14] the influenceof hydrostatic pressure on the high frequency dielectricconstants can be written as

120597120576

]infin120585(119901)

120597119901

= minus

5 (120576

]infin120585minus 1)

3119861

0](09 minus 119891

]ion) (23)

Here 119891]ion (] = GaN and AlN) is Phillips ionicity parameterof material the static bulk modulus 119861

0] under hydrostaticpressure is given by [25]

119861

0] =(119862

11] (119875) + 11986212] (119875)) 11986233] (119875) minus 21198622

13] (119875)

119862

11] (119875) + 11986212] (119875) + 211986233] (119875) minus 411986213] (119875)

(24)

where 11986211] 11986212] 11986213] and 11986233] are the elastic constants of

material ] The effective mean relative dielectric constant in(1) is defined as [26]

120576] (119901) =2

3

120576

]120585perp(119901) +

1

3

120576

]120585119911(119901)

(25)

Then the hydrostatic-pressure-modified biaxial and uniaxialstrain dependence of the static dielectric constant is fullyconsidered whereas the dielectric constant of AlxGa1minus119909N canbe calculated by the SCPA [13]The piezoelectric polarizationalong the [0001] oriented WZ GaNAlxGa1minus119909N QD can becalculated as [27]

119875

]pe = 119890

]33(119901) 120576

]119911119911+ 2119890

]31(119901) 120576

]119909119909 (26)

where 119890]31

and 119890]33

are the pressure-dependent piezoelectricconstants of material ]

4 Results and Discussions

Under the strong built-in electric field induced by the sponta-neous and piezoelectric polarizations barrier thickness andhydrostatic pressure effects on the donor binding energy ofhydrogenic impurity are investigated inWZGaNAlxGa1minus119909NstrainedQD All material parameters used in our calculationsare listed in Tables 1ndash4Material parameters of AlxGa1minus119909Nareestimated using a linear interpolation between the values ofthe corresponding parameters of GaN and AlN

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

03

06

09

12

15

18

F (m

Vc

m)

FGaN

Lb (nm)

FAlGaN

P = 0GPaP = 4GPaP = 8GPa

Lw = 3nmx = 03

Figure 2The built-in electric field119865GaN (119865AlGaN) in thewell (barrier)layer along the QD growth direction as a function of the barrierthickness 119871

119887in WZ GaNAlxGa1minus119909N strained QD with the QD

height 119871119908= 3 nm 119909 = 03 for different hydrostatic pressures 119875

Figure 2 presents that the built-in electric field (BEF)119865GaN (119865AlGaN) in the well (barrier) layer along the growthdirection 119911-axis as a function of the barrier thickness 119871

119887in

WZ GaNAlxGa1minus119909N QD for different hydrostatic pressures119875 Numerical results show that the BEF 119865GaN (119865AlGaN) is anincreasing (a decreasing) function of the barrier thickness 119871

119887

This is because the fact that the equilibrium lattice constant119886eq of the strained layer and the components of the straintensor 120576

119909119909119908of the dot layer decrease with the increase of

the barrier thickness 119871119887according to (11) which induces

the decrement of the piezoelectric polarization in the dotlayer and the increment of the piezoelectric polarization inthe barrier layer along the QD growth direction Thereforethe BEF 119865GaN (119865AlGaN) gradually increases (decreases) (see(5)) Moreover Figure 2 also shows that the built-in electricfields (119865GaN and 119865AlGaN) increase correspondingly as thehydrostatic pressure 119875 increases this is caused by the changeof the pressure-dependent piezoelectric constants the biaxialstrains and the dielectric constants of WZ GaNAlxGa1minus119909NQD when the hydrostatic pressure increases

In Figure 3 the built-in electric field (BEF) 119865GaN (119865AlGaN)in the well (barrier) layer along the QD growth direction 119911-axis is displayed as a function of the barrier thickness 119871

119887

for different Al compositions 119909 in WZ GaNAlxGa1minus119909N QDNumerical results show that the BEF 119865GaN (119865AlGaN) increases(decreases) graduallywhen the barrier thickness119871

119887increases

This is caused by the change of the pressure-dependentpiezoelectric constants the biaxial strains and the dielectricconstants of WZ GaNAlxGa1minus119909N strained QD In additionFigure 3 also shows that the BEF 119865GaN (119865AlGaN) increasesas Al composition 119909 increases but the BEF 119865AlGaN remainsinsensitive to the bigger barrier thickness 119871

119887 The reason


Table 1 Lattice constant 119886 (in units of nm) effective mass 119898119890(in units of a free-electron mass 119898

0) piezoelectric constants 119890

31and 119890

33(in

units of Cm2) and deformation potentials 1198861 1198862 1198871 and 119887

2(in units of eV) for GaN and AlN

119886 119898

perp119898

119890

31119890

33119886

1119887

1119886

2119887

2

GaN 03189a 018a 02a minus044b 067b minus409c minus887c minus702c 365c

AlN 03112a 025a 033a minus053b 150b minus339c minus1181c minus942c 402caReference [14] bReference [28] and cReference [18]

Table 2 Band gap 119864119892(in units of eV) spontaneous polarization (in units of Cm2) elastic constants 119862

11 11986212 11986213 and 119862

33(in units of GPa)

Phillips iconicity parameter 119891ion and the high frequency dielectric constant 120576infinfor GaN and AlN

119864

119892119901

sp119862

11119862

12119862

13119862

33119891ion 120576

infinperp120576

infin119911

GaN 3507a minus0034a 365b 139b 101b 405b 05a 520a 539a

AlN 6230a minus0090a 397b 143b 112b 371b 0499a 430a 452aaReference [7] bReference [28]

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

03

06

09

12

15

F (m

Vc

m)

FGaN

Lb (nm)

FAlGaN

x = 03

x = 02

x = 01

P = 0GPaLw = 3nm



thickness 119871119908= 3 nm 119875 = 0GPa for different Al compositions 119909

can be given as follows When Al composition 119909 increasesthe equilibrium lattice constant 119886eq of the strained layerdecreases according to (11) therefore the absolute valuesof strain tensor 120576

119909119909119887and 120576

119911119911119887of the barrier layer along

the QD growth direction increase which induces that thepiezoelectric polarization in the barrier layer increases andthen the BEF 119865GaN (119865AlGaN) increases correspondingly in (4)

In Figure 4 the ground-state donor binding energy isshown as a function of theQD radius inWZGaNAlxGa1minus119909NQD with the parameters (119871

119908= 2 nm 119871

119887= 5 nm and

119909 = 03) for different hydrostatic pressures (119901 = 0Gpa4Gpa and 8Gpa) The impurity ion is placed at the centerof the QD As shown in Figure 4 the donor binding energyincreases with decreasing the radius 119877 in all cases reaches

a maximum value and then decreases rapidly The behavioris related to the variation of the electron confinement inQD the electron wave function is firmly localized inside theQD with decreasing the QD radius Therefore the Coulombinteraction between the electron and the impurity ion isenhanced and the donor binding energy increases corre-spondingly Moreover when the radial thickness decreasescontinuously to a certain value the kinetic energy of theconfined electron rises greatly which increases greatly theprobability of the electron penetrating into the potentialbarrier by the uncertainty principle and therefore the donorbinding energy starts decreasing quickly Moreover Figure 4also displays that the larger the hydrostatic pressure is thebigger the donor binding energy is The main reasons can begiven as followsWith the increase of the hydrostatic pressurethe dielectric constants the electron effective mass and thefinite confinement potential barrier will increase which willresult in bigger donor binding energy

Figure 5 demonstrates that the ground-state donor bind-ing energy in cylindricalWZGaNAlxGa1minus119909NstrainedQDasa function of the height 119871

119908with the parameters (119877 = 10 nm

119871

119887= 20 nm and 119909 = 03) for different hydrostatic pressures

(119875 = 0Gpa 4Gpa and 8Gpa) The impurity ion is placed atthe center of the QD setting 120588

0= 0 and 119911

0= 0 As expected

in all cases of hydrostatic pressures the donor bindingenergy increases with a decrease of the QD height reachesa maximum value and then decreases quickly in finitepotential barrier For a fixed value of the barrier thickness119871

119887 the size quantization confinement of the electron wave

function goes strongerwith the decrease of theQDheight theelectron-impurity Coulomb interaction becomes larger andwhen the QD height becomes small enough the probabilityof the electron leaking into the potential barrier increasesgreatly by the uncertainly principle Therefore the donorbinding energy decreases correspondingly In addition thecurves in Figure 5 also show that the stronger the appliedhydrostatic pressure is the bigger the donor binding energyis Take the QD height 119871

119908= 4 nm for example a change

of the hydrostatic pressure from 0 to 8Gpa results in anincrease of the impurity binding energy119864

119887from 3403meV to

4835meV As expected with the increase of the hydrostatic


Table 3 Strain coefficients of the zone-center phonon modes 119870]119895120585119909119909

and 119870]119895120585119911119911

(in units of cmminus1) for GaN and AlN

119870Toperp119909119909 119870To119911119909119909 119870Toperp119911119911 119870Toperp119911119911 119870Loperp119909119909 119870Lo119911119909119909 119870Loperp119911119911 119870Loperp119911119911

GaN minus1139a minus931a minus300a minus443a minus1198a minus885a minus389a minus618a

AlN minus1208a minus1330a minus391a minus70a minus1233a minus1038a minus442a minus434aaReference [14]

Table 4 Band gap pressure coefficient 120594 (meVGPa) phonon frequencies 120596LO and 120596TO (cmminus1) for GaN and AlN and Gruneisen parameterof phonon mode 120574]

119895120585

120594 120596Loperp 120596Lo119911 120596Toperp 120596To119911 120574Loperp 120574Lo119911 120574Toperp 120574To119911

GaN 39a 757b 748b 568b 540b 091b 082b 118b 102b

AlN 40a 924b 898b 677b 318b 099b 098b 119b 121baReference [29] bReference [28]

0 2 4 6 8 10 1220

40

60

80

100

120

140

R (nm)

Eb

(meV

)

Lw = 2nmLb = 5nmx = 03


Figure 4 The ground-state donor binding energy in cylindricalWZ GaNAlxGa1minus119909N strained QD as a function of the radius 119877 for119871

119908= 2 nm 119871

119887= 5 nm 119909 = 03 and several values of the hydrostatic

pressure 119875

pressure 119901 electron effective masses and dielectric constantsof GaN and AlxGa1minus119909N materials become lager and agrowth of the hydrostatic pressure leads to the increase ofthe finite potential barrier which will lead to the electronwave function being firmly squeezed around the impurityion and consequently the donor binding energy increasescorrespondingly

The ground-state donor binding energy as a function ofAl composition 119909 in WZ GaNAlxGa1minus119909N QD is displayedin Figure 6 for different hydrostatic pressures 119875 Numericalresults show that the donor binding energy for the centralimpurity inWZGaNAlxGa1minus119909N strained QD increases withthe increase of Al composition This is because that thecompetition effects between the built-in electric field andthe potential barrier confinement will change the strength ofthe electron-impurity interaction For the small QD height119871

119908= 2 nm as the Al concentration increases conductor

1 2 3 4 5 6 7 830

35

40

45

50

55

60

65

Eb

(meV

)


Lw (nm)

R = 10nmLb = 20nmx = 03

Figure 5The ground-state donor binding energy in cylindricalWZGaNAlxGa1minus119909NstrainedQDas a function of theQDheight (119871

119908) for

119877 = 10 nm119871119887= 20 nm119909 = 03 and several values of the hydrostatic

pressure 119875

band offset of WZ GaNAlxGa1minus119909N QD increases and thepotential barriers on the surfaces of QD play the mainrole in the distribution of the electron wave function Inaddition the bigger the Al composition 119909 is the largerthe potential barrier is which results in the fact that theprobability of the electron leaking into the barrier regionbecomes small Accordingly increasing the Coulomb effectbetween the electron and the impurity ion leads to theenhancement of the binding energy Figure 6 also shows thatthe donor binding energy increases with the increment ofthe hydrostatic pressure 119875 This is due to the fact that theelectron wave function is strongly compressed in the QDas hydrostatic pressure 119875 increases and the strength of theelectron-impurity interaction becomes larger leading to theenhancement of the binding energy correspondingly

To clarify the effect of the hydrostatic pressure on theground-state donor binding energy we investigated the


010 015 020 025 030 035 04035

40

45

50

55

60

x

Eb

(meV

)

Lw = 2nmLb = 5nm

6nmR =


Figure 6The ground-state donor binding energy in cylindricalWZGaNAlxGa1minus119909N strained QD as a function of Al composition 119909for 119877 = 6 nm 119871

119908= 2 nm 119871

119887= 5 nm and several values of the

hydrostatic pressure 119875

0 2 4P (GPa)

6 8 1035

40

45

50

55

60

65

70

75

Eb

(meV

)

x = 01 Lw = 4nm Lb = 6nmx = 01 Lw = 3nm Lb = 5nmx = 03 Lw = 4nm Lb = 6nmx = 03 Lw = 3nm Lb = 5nm

Figure 7 The ground-state donor binding energy in cylindricalWZ GaNAlxGa1minus119909N strained QD as a function of the hydrostaticpressure (P) with 119877 = 5 nm and 119909 = 01(03) and for different QDheights and barrier thicknesses

donor binding energy in cylindrical WZ GaNAlxGa1minus119909Nstrained QD with the parameters (119877 = 5 nm 119909 = 01(03)120588

0= 0 nm and 119911

0= 0 nm) for several values of

QD height and barrier thickness From Figure 7 one canobserve that the donor binding energy increases almostlinearly as the hydrostatic pressure 119875 increases The pressurebehavior in WZ GaNAlxGa1minus119909N QD can be explained by

4 6 8 10 12 14 16 18 2030

35

40

45

50

55

Eb

(meV

)

Lb (nm)

R = 10nmLw = 3nm

x = 02


Figure 8The ground-state donor binding energy in cylindricalWZGaNAlxGa1minus119909N strained QD as a function of the barrier thickness119871

119887along the QD growth direction for 119877 = 10 nm 119871

119908= 3 nm 119909 =

02 and several values of the hydrostatic pressure P

the modification of the polarization in the dot layer by thepressure induced strain which leads to a significant increaseof the BEF 119865GaN in the QD This behavior is in agreementwith the result of [13] In addition the stronger the appliedhydrostatic pressures is the bigger the electron effectivemasses and dielectric constants of GaN and AlxGa1minus119909Nmaterials are and the finite confinement potential at theboundary of GaN QD also becomes large under the biggerhydrostatic pressure hence the expected value of the distancebetween the electron and the impurity ion reduces and thestrength of the electron-impurity interaction becomes largerwhich will lead to the increase of the donor binding energycorrespondingly Taking the solid curve for example thedonor binding energy increases by 2115meV approximatelyif the hydrostatic pressure 119875 increases from 0 to 8GPa Thusthe hydrostatic pressure has an important influence on thedonor binding energy

Figure 8 displays the ground-state donor binding energyas a function of the barrier thickness 119871

119887along the QD growth

direction with the parameters (119877 = 10 nm 119871119908= 3 nm

120588

0= 0 nm 119909 = 02 and 119911

0= 0 nm) and different values

of the hydrostatic pressure 119875 The impurity ion is locatedat the centre of the QD We can see from Figure 8 thatthe donor binding energy increases reaching a maximumvalue and then reduces gradually with the increase of thebarrier thickness 119871

119887in all cases As expected the cures in

Figure 8 also show that the donor binding energy has amaximum value This is because the enhancement of thebarrier thickness leads to the change of the finite confinementpotential along the QD growth direction As the barrierthickness 119871

119887increases the finite confinement potential at the

bottom of the QD becomes small and the one at the top ofthe QD becomes big due to the strong built-in electric field


minus05 minus04 minus03 minus02 minus01 00 01 02 03 04 05400

425

450

475

500

525

550

z0 (nm)

Eb

(meV

)

R = 6nm

Lw = 3nmLb = 5nm

x = 03


Figure 9 The ground-state donor binding energy in a cylindricalWZGaNAlxGa1minus119909N strainedQD as a function of the axial impurityposition 119911

0for119877 = 6 nm 119871

119908= 3 nm 119871

119887= 5 nm 119909 = 03 and several

values of the hydrostatic pressure P

effects When the QD barrier thickness 119871119887increases to about

11 nm the finite confinement potentials at two sides of theQDalong the growth direction are approximately equal whichleads to the fact that the electron wave function is stronglycompressed around the central impurity ion Therefore theCoulomb action between the electron and the impurity ionreaches a maximum

In Figure 9 the ground-state donor binding energy isinvestigated as a function of the impurity position 119911

0along

the QD growth direction with the parameters (119877 = 6 nm119871

119908= 3 nm 119909 = 03 and 120588

0= 0 nm) and different values of

the hydrostatic pressure 119875 As shown in Figure 9 the curvesin all cases are absolutely asymmetry and the donor bindingenergy demonstrates a maximum value when the impurityis located from the plane 119911 = minus119871

1199082 to the symmetry

plane 1198711199082 along the growth direction of the QD and the

maximum value of the donor binding energy is not located atthe point [0 0] This is because the fact that the strong BEFmodifies the spread of the electron wave function in the QDand the direction of the built-in electric field 119865

119908in the dot

layer is opposite to the growth direction of the QDThus thebuilt-in electric field 119865

119908pushes the electron toward the right

side of the QD This behavior is in agreement with the resultof [17] In addition the curves also show that the strongerthe hydrostatic pressure is the larger the peak value of thedonor binding energy is with the same parameters (119871

119908 119877

and 1205880) Moreover the position of the peak value of the donor

binding energy is also shifted to positive 119911-direction This isbecause the fact that the electronic wave function is obviouslymodified and the bigger concentration of the electron wavefunction is squeezed strongly around the impurity ion Inaddition the stronger the hydrostatic pressure is the bigger

the localization effect of the electron wave function is so thatthe peak value of the binding energy increases accordinglyTherefore the distribution of the electron wave function isnot central symmetrical about the QD in the presence of thestrong BEF

5 Conclusions

With the framework of the effective mass approximationthe ground-state donor binding energy in a cylindrical WZGaNAlxGa1minus119909N strained QD is investigated theoretically inthe presence of built-in electric field and hydrostatic pressureby using a variational approach The ground-state donorbinding energy depends strongly on dot radius hydrostaticpressure impurity position and barrier thickness in thefinite confinement potential Numerical results show that thedonor binding energy increases firstly reaches a maximumvalue and then drops slowly as the QD radius (height)decreases And the donor binding energy is an increasingfunction of Al composition 119909 andor hydrostatic pressure Inaddition the donor binding energy has a maximum valuewhen the impurity position moves along the symmetry axisof the QD from the bottom of the QD to the top and theposition of the peak value of the donor binding energy is alsoshifted towards positive 119911-direction Moreover the strongerthe hydrostatic pressure is the larger the peak value of thedonor binding energy is with the same spatial confinementThe electronic wave function distribution in the QD is alsoobviously modified by the hydrostatic pressure We hope thatour results would stimulate further researches and lead tosome potential applications on group-III nitride materials

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific and TechnologicalDepartment Foundation of Hebei Province (no 12210617)and the Natural Science Foundation of Hebei Province (noA201420308)

References

[1] M Gladysiewicz R Kudrawiec J Misiewicz et al ldquoThesurface boundary conditions in GaNAlGaNGaN transistorheterostructuresrdquo Applied Physics Letters vol 98 no 1 ArticleID 231902 2011

[2] L Duggen and M Willatzen ldquoCrystal orientation effectson wurtzite quantum well electromechanical fieldsrdquo PhysicalReview BampCondensed Matter and Materials Physics vol 82 no20 Article ID 205303 2010

[3] A Armstrong A A Allerman T A Henry and M H Craw-ford ldquoInfluence of growth temperature on AlGaN multiquan-tum well point defect incorporation and photoluminescenceefficiencyrdquo Applied Physics Letters vol 98 no 16 Article ID162110 2011


[4] M Zhang and S L Ban ldquoPressure influence on the Starkeffect of impurity states in a strained wurtzite GaNAl

119909Ga1minus119909

Nheterojunctionrdquo Chinese Physics B vol 18 no 10 pp 4449ndash4455 2009

[5] M Pattammal and A J Peter ldquoElectronic states of a hydrogenicimpurity in a zinc-blende GaNAlGaN quantum wellrdquo AppliedSurface Science vol 256 no 22 pp 6748ndash6752 2010

[6] C X Xia Z P Zeng S Y Wei and J B Wei ldquoShallow-donorimpurity in vertical-stacked InGaNGaN multiple-quantumwells electric field effectrdquo Physica E vol 43 no 1 pp 458ndash4612010

[7] Z Y Feng S L Ban and J Zhu ldquoBinding energies of impuritystates in strained wurtzite GaNAl

119909Ga1minus119909

N heterojunctionswith finitely thick potential barriersrdquo Chinese Physics B vol 23no 6 Article ID 066801 2014

[8] Y N Wei Y Ji Q Sun C X Xia and Y Jia ldquoBarrier width andbuilt-in electric field effects on hydrogenic impurity in wurtziteGaNAlGaN quantum wellrdquo Physica E Low-Dimensional Sys-tems and Nanostructures vol 44 no 2 pp 511ndash514 2011

[9] H ElGhazi A Jorio and I Zorkani ldquoPressure-dependentshallow donor binding energy in InGaNGaN square QWWsrdquoPhysica B Condensed Matter vol 410 no 1 pp 49ndash52 2013

[10] P Baser S Elagoz and N Baraz ldquoHydrogenic impurity statesin zinc-blende In

119909Ga1minus119909

NGaN in cylindrical quantum wellwires under hydrostatic pressurerdquo Physica E Low-DimensionalSystems and Nanostructures vol 44 no 2 pp 356ndash360 2011

[11] M Zhang and J-J Shi ldquoExciton states and interband opticaltransitions in wurtzite InGaNGaN quantum dot nanowireheterostructuresrdquo Superlattices and Microstructures vol 50 no5 pp 529ndash538 2011

[12] M Kırak S Yılmaz M Sahin and M Gencaslan ldquoThe electricfield effects on the binding energies and the nonlinear opticalproperties of a donor impurity in a spherical quantum dotrdquoJournal of Applied Physics vol 109 no 9 Article ID094309 2011

[13] M Zhang and J J Shi ldquoInfluence of pressure on exciton statesand interband optical transitions in wurtzite InGaNGaN cou-pled quantum dot nanowire heterostructures with polarizationand dielectric mismatchrdquo Journal of Applied Physics vol 111 no11 Article ID 113516 6 pages 2012

[14] DM Zheng Z CWang and B Q Xiao ldquoEffects of hydrostaticpressure on ionized donor bound exciton states in strainedwurtzite GaNAl

119909Ga1minus119909

N cylindrical quantum dotsrdquo Physica BCondensed Matter vol 407 no 21 pp 4160ndash4167 2012

[15] M G Barseghyan A A Kirakosyan and C A Duque ldquoHydro-static pressureelectric and magnetic field effects on shallowdonor impurity states and photoionization cross section incylindrical GaAs-Ga

1minus119909Al119909As quantum dotsrdquo Physica Status

Solidi (B) Basic Research vol 246 no 3 pp 626ndash629 2009[16] C X Xia Z P Zeng and S Y Wei ldquoBarrier width dependence

of the donor binding energy of hydrogenic impurity in wurtziteInGaNGaN quantum dotrdquo Journal of Applied Physics vol 106no 9 Article ID 094301 2009

[17] C X Xia S Y Wei and X Zhao ldquoBuilt-in electric field effecton hydrogenic impurity in wurtzite GaNAlGaN quantum dotrdquoApplied Surface Science vol 253 no 12 pp 5345ndash5348 2007

[18] J-M Wagner and F Bechstedt ldquoPressure dependence of thedielectric and lattice-dynamical properties of GaN and AlNrdquoPhysical Review BmdashCondensed Matter and Materials Physicsvol 62 no 7 pp 4526ndash4534 2000

[19] J A Tuchman and I P Herman ldquoGeneral trends in changingepilayer strains through the application of hydrostatic pressurerdquoPhysical Review B vol 45 no 20 pp 11929ndash11935 1992

[20] P Perlin LMattosNA Shapiro et al ldquoReduction of the energygap pressure coefficient ofGaNdue to the constraining presenceof the sapphire substraterdquo Journal of Applied Physics vol 85 no4 pp 2385ndash2389 1999

[21] S P Łepkowski ldquoNonlinear elasticity effect in group III-nitridequantum heterostructures Ab initio calculationsrdquo PhysicalReview B vol 75 no 19 Article ID 195303 11 pages 2007

[22] W Shan R J Hauenstein A J Fischer et al ldquoStrain effects onexcitonic transitions in GaN deformation potentialsrdquo PhysicalReview BmdashCondensedMatter andMaterials Physics vol 54 no19 pp 13460ndash13463 1996

[23] N E Christensen and I Gorczyca ldquoOptical and structuralproperties of III-V nitrides under pressurerdquo Physical Review Bvol 50 no 7 pp 4397ndash4415 1994

[24] M Holtz M Seon O Brafman R Manor and D Fekete ldquoPres-sure dependence of the optic phonon energies in Al

119909Ga1minus119909

AsrdquoPhysical Review BmdashCondensed Matter and Materials Physicsvol 54 no 3 pp 8714ndash8720 1996

[25] S P Łepkowski ldquoNonlinear elasticity effect in group III-nitridequantum heterostructures Ab initio calculationsrdquo PhysicalReview BmdashCondensed Matter and Materials Physics vol 75 no19 Article ID 195303 11 pages 2007

[26] J M Wagner and F Bechstedt ldquoProperties of strained wurtziteGaN andAlN Ab initio studiesrdquo Physical Review BmdashCondensedMatter and Materials Physics vol 66 no 11 20 pages 2002

[27] S H Ha and S L Ban ldquoBinding energies of excitons in astrained wurtzite GaNAlGaN quantum well influenced byscreening and hydrostatic pressurerdquo Journal of Physics Con-densed Matter vol 20 no 8 Article ID 085218 2008

[28] S P Łepkowski J A Majewski and G Jurczak ldquoNonlinearelasticity in III-N compounds ab initio calculationsrdquo PhysicalReview BmdashCondensed Matter and Materials Physics vol 72 no24 Article ID 245201 12 pages 2005


Submit your manuscripts athttpwwwhindawicom

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CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


AlGaN GaN

R

O 120588Lw

Lb

z

Figure 1Thediagramof a cylindrical wurtziteGaNAlxGa1minusxNQD

(radius and height) of QD and piezoelectric polarizations areconsidered as a function of hydrostatic pressureThe paper isorganized as follows In Section 2 we describe the theoreticalframework Then the pressure and the strain coefficients ofGaN and AlxGa1minus119909N are discussed in Section 3 In Section 4the numerical results are discussed Finally conclusions aredrawn from the present study in Section 5

2 Theoretical Framework

In Figure 1 the schematic view of a cylindrical WZGaNAlxGa1minus119909N QD is depicted with a detailed descrip-tion of the different dimensions of the QD (dot radius 119877height 119871

119908 and barrier thickness 119871

119887) Additionally the GaN

quantum dot is embedded in an AlxGa1minus119909N host matrixmaterial and the 119911-axis is defined to be the growth directionWithin the frame of the effective mass approximation theHamiltonian for a hydrogenic donor impurity in cylindricalWZ GaNAlxGa1minus119909N QD under the influence of hydrostaticpressure can be written [4] as

119867 =

119867

0minus

119890

2

4120587120576

0120576 (119901)

1003816

1003816

1003816

1003816

1003816

997888

119903 minus

997888

119903

0

1003816

1003816

1003816

1003816

1003816

(1)

where 997888119903 (9978881199030) denotes the position vector of the electron

(impurity ion) 119890 is the absolute value of the electron charge120576

0is the permittivity of free space and 120576(119901) is the pressure-

dependent effective mean relative dielectric constant of GaNandAlxGa1minus119909NmaterialsTheHamiltonian 119867

0is given by [11]

119867

0= minus

ℏ

2

2119898

perp(119901)

(

1

120588

120597

120597120588

(120588

120597

120597120588

) +

1

120588

2

120597

2

120597120593

2)

minus

ℏ

2

2119898

(119901)

120597

2

120597119911

2+ 119881 (120588 119911 119901) minus 119890119865

119908119887(119901) 119911

(2)

where 119898

and 119898perpare the pressure- and strain-dependent

effective masses of electron along and perpendicular to the

[0001]-direction And 119881(120588 119911 119901) is the electron confinementpotential due to the band offset (119876

119895= 0765) and is given by

[13]

119881 (119911 119901)

=

119876

119895[119864

119892Al119909Ga1minus119909

N minus 119864119892GaN] 119871

119908

2

lt 119911 lt

119871

119908

2

+

119871

119887

2

minus

119871

119908

2

minus

119871

119887

2

lt 119911 lt minus

119871

119908

2

0 |119911| le

119871

119908

2

(3)

119881 (120588 119901) =

0 120588 le 119877

119876

119895[119864

119892Al119909Ga1minus119909

N minus 119864119892GaN] others(4)

In (2) F(p) is the pressure-dependent built-in electricfield (BEF) in the finitely thick barrier layer for WZGaNAlxGa1minus119909NQDThe values of the BEF along the growthdirection in the well (119865GaN) and barrier (119865Al

119909Ga1minus119909

N) that resultfrom the difference in the total electric polarizations in eachregion are given by simple formulas [14]

119865GaN (119901) =

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119871

119887(119875

AlGaNsp + 119875

AlGaNpe minus 119875

GaNsp minus 119875

GaNpe )

120576

0(119871

119887120576

GaN119890+ 119871

119908120576

AlGaN119890

)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119865AlGaN (119901) = minus119871

119908

119871

119887

119865GaN (119901)

(5)

where 119871119908 119871119887and 120576GaN119890

120576AlGaN119890

are the thicknesses and elec-tronic dielectric constants for GaN and AlxGa1minus119909Nmaterials119875

GaNsp 119875AlGaNsp and 119875GaNpe 119875AlGaNpe are the spontaneous and

piezoelectric polarizations forGaN andAlxGa1minus119909Nmaterialsand the bowing parameter 120578 is chosen as minus0019 Cm2 [14]

119875

AlGaNsp = 119909119875

AlNsp + (1 minus 119909) 119875

GaNsp minus 120578119909 (1 minus 119909) (6)

The wave function of the electron confined in the WZGaNAlxGa1minus119909N QD can be written as

120595 (120588 120593 119911) = 119891 (120588) ℎ (119911) 119890

119894119898120593 119898 = 0 plusmn1 plusmn2

119891 (120588) =

119869

0(120573120588) 120588 le 119877

119869

0(120573119877)

119870

0(120573119877)

119870

0(120572120588) 120588 gt 119877

(7)

where the constants (120572 and 120573) are determined by the con-tinuity of the derivative of the radial wave function at theQD boundary and 119898 is the electron 119911-component angularmomentum quantum numberThe radial wave function119891(120588)of the electron can be obtained using the Bessel function 119869

119898




ℎ (119911) =

119862

1Ai (1205761) + 119863

1Bi (1205761) minus

119871

119908

2

minus 119871

119887lt 119911 lt minus

119871

119908

2

119862

2Ai (1205762) + 119863

2Bi (1205762) |119911| le

119871

119908

2

119862

3Ai (1205763) + 119863

3Bi (1205763)

119871

119908

2

lt 119911 lt

119871

119908

2

+ 119871

119887

(8)

Here 120576119895= ((2119898

lowast

119895ℏ

2)119890119865)

13(119911 minus 119864

119911minus 119881(119911 119901)119890119865) The coef-

ficients 119862119895and 119863




120601 (120588 120593 119911) = 119873120595 (120588 120593 119911) 119890

minus120572(120588minus1205880)2

minus120573(119911minus1199110)2

(9)




119864

119887= 119864

0120588+ 119864

0119911minusmin120572120573

⟨120595

1003816

1003816

1003816

1003816

119867

1003816

1003816

1003816

1003816

120595⟩

⟨120595

1003816

1003816

1003816

1003816

120595⟩

(10)



120576


119886GaN (119875)

120576


119886AlGaN (119875)

(11)


119886eq (119901) =119886GaN119871119908 + 119886GaAlN119871119887

119871

119908+ 119871

119887

(12)


119886] (119875) = 1198860] (1 minus119901

3119861

0]) ] = GaNAlN (13)



120576

119911119911GaN = 119877119867

GaN120576119909119909GaN

120576

119911119911AlGaN = 119877119867

AlGaN120576119909119909AlGaN(14)


119877

119867

] =119862

11] (119875) + 11986212] (119875) minus 211986213] (119875)

119862

33] (119875) minus 11986213] (119875) ] = GaNAlN

(15)



120590




119864


+ (119886

2GaN + 1198872GaN) 120576119911119911GaN

119864


(16)




119864

119892] (119901) = 119864119892] (0) + 120594]119901 (17)


119864



119898

0

119898

perp

] (119901)

= 1 +

119862

119895]

119864

119892] (119901) (19)







120576

]120585(119901) = 120576

]infin120585(119901)(

120596

]LO120585 (119875)

120596

]TO120585 (119875)

)

2

(20)



120596

]119895120585= 120596

]119895120585(119901) + 2119870

]119895120585119909119909

120576

119909119909] (119901) + 119870]119895120585119911119911

120576

119911119911] (119901) (21)


canbe given by [25]

120574

]119895120585= 119861

0]1

120596

]119895120585(0)

(

120597120596

]119895120585(119901)

120597119901

) (22)


119895120585(0) is the zone-




119895120585119909119909and 119870]



120597120576

]infin120585(119901)

120597119901

= minus

5 (120576


3119861

0](09 minus 119891

]ion) (23)



119861

0] =(119862

11] (119875) + 11986212] (119875)) 11986233] (119875) minus 21198622

13] (119875)

119862

11] (119875) + 11986212] (119875) + 211986233] (119875) minus 411986213] (119875)

(24)



120576] (119901) =2

3

120576

]120585perp(119901) +

1

3

120576

]120585119911(119901)

(25)


119875

]pe = 119890

]33(119901) 120576

]119911119911+ 2119890

]31(119901) 120576

]119909119909 (26)

where 119890]31

and 119890]33




3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

03

06

09

12

15

18

F (m

Vc

m)

FGaN

Lb (nm)

FAlGaN


Lw = 3nmx = 03





119887in


119887






119887


119887increases


119887 The reason




31and 119890

33(in



119886 119898

perp119898

119890

31119890

33119886

1119887

1119886

2119887

2




11 11986212 11986213 and 119862

33(in units of GPa)


119864

119892119901

sp119862

11119862

12119862

13119862

33119891ion 120576

infinperp120576

infin119911



3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

03

06

09

12

15

F (m

Vc

m)

FGaN

Lb (nm)

FAlGaN

x = 03

x = 02

x = 01

P = 0GPaLw = 3nm





119909119909119887and 120576




119908= 2 nm 119871

119887= 5 nm and





119871



0= 0 and 119911

0= 0 As expected










and 119870]119895120585119911119911






119895120585


GaN 39a 757b 748b 568b 540b 091b 082b 118b 102b


0 2 4 6 8 10 1220

40

60

80

100

120

140

R (nm)

Eb

(meV

)




119908= 2 nm 119871


pressure 119875




1 2 3 4 5 6 7 830

35

40

45

50

55

60

65

Eb

(meV

)


Lw (nm)

R = 10nmLb = 20nmx = 03


119908) for


pressure 119875




010 015 020 025 030 035 04035

40

45

50

55

60

x

Eb

(meV

)

Lw = 2nmLb = 5nm

6nmR =



119908= 2 nm 119871



0 2 4P (GPa)

6 8 1035

40

45

50

55

60

65

70

75

Eb

(meV

)




0= 0 nm and 119911



4 6 8 10 12 14 16 18 2030

35

40

45

50

55

Eb

(meV

)

Lb (nm)

R = 10nmLw = 3nm

x = 02




119908= 3 nm 119909 =






120588

0= 0 nm 119909 = 02 and 119911









425

450

475

500

525

550

z0 (nm)

Eb

(meV

)

R = 6nm

Lw = 3nmLb = 5nm

x = 03



0for119877 = 6 nm 119871

119908= 3 nm 119871

119887= 5 nm 119909 = 03 and several





0along


119908= 3 nm 119909 = 03 and 120588






119908in the dot




119908 119877




5 Conclusions




Acknowledgments


References






119909Ga1minus119909





119909Ga1minus119909





119909Ga1minus119909






119909Ga1minus119909














119909Ga1minus119909














CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






ℎ (119911) =

119862

1Ai (1205761) + 119863

1Bi (1205761) minus

119871

119908

2

minus 119871

119887lt 119911 lt minus

119871

119908

2

119862

2Ai (1205762) + 119863

2Bi (1205762) |119911| le

119871

119908

2

119862

3Ai (1205763) + 119863

3Bi (1205763)

119871

119908

2

lt 119911 lt

119871

119908

2

+ 119871

119887

(8)

Here 120576119895= ((2119898

lowast

119895ℏ

2)119890119865)

13(119911 minus 119864

119911minus 119881(119911 119901)119890119865) The coef-

ficients 119862119895and 119863




120601 (120588 120593 119911) = 119873120595 (120588 120593 119911) 119890

minus120572(120588minus1205880)2

minus120573(119911minus1199110)2

(9)




119864

119887= 119864

0120588+ 119864

0119911minusmin120572120573

⟨120595

1003816

1003816

1003816

1003816

119867

1003816

1003816

1003816

1003816

120595⟩

⟨120595

1003816

1003816

1003816

1003816

120595⟩

(10)



120576


119886GaN (119875)

120576


119886AlGaN (119875)

(11)


119886eq (119901) =119886GaN119871119908 + 119886GaAlN119871119887

119871

119908+ 119871

119887

(12)


119886] (119875) = 1198860] (1 minus119901

3119861

0]) ] = GaNAlN (13)



120576

119911119911GaN = 119877119867

GaN120576119909119909GaN

120576

119911119911AlGaN = 119877119867

AlGaN120576119909119909AlGaN(14)


119877

119867

] =119862

11] (119875) + 11986212] (119875) minus 211986213] (119875)

119862

33] (119875) minus 11986213] (119875) ] = GaNAlN

(15)



120590




119864


+ (119886

2GaN + 1198872GaN) 120576119911119911GaN

119864


(16)




119864

119892] (119901) = 119864119892] (0) + 120594]119901 (17)


119864



119898

0

119898

perp

] (119901)

= 1 +

119862

119895]

119864

119892] (119901) (19)







120576

]120585(119901) = 120576

]infin120585(119901)(

120596

]LO120585 (119875)

120596

]TO120585 (119875)

)

2

(20)



120596

]119895120585= 120596

]119895120585(119901) + 2119870

]119895120585119909119909

120576

119909119909] (119901) + 119870]119895120585119911119911

120576

119911119911] (119901) (21)


canbe given by [25]

120574

]119895120585= 119861

0]1

120596

]119895120585(0)

(

120597120596

]119895120585(119901)

120597119901

) (22)


119895120585(0) is the zone-




119895120585119909119909and 119870]



120597120576

]infin120585(119901)

120597119901

= minus

5 (120576


3119861

0](09 minus 119891

]ion) (23)



119861

0] =(119862

11] (119875) + 11986212] (119875)) 11986233] (119875) minus 21198622

13] (119875)

119862

11] (119875) + 11986212] (119875) + 211986233] (119875) minus 411986213] (119875)

(24)



120576] (119901) =2

3

120576

]120585perp(119901) +

1

3

120576

]120585119911(119901)

(25)


119875

]pe = 119890

]33(119901) 120576

]119911119911+ 2119890

]31(119901) 120576

]119909119909 (26)

where 119890]31

and 119890]33




3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

03

06

09

12

15

18

F (m

Vc

m)

FGaN

Lb (nm)

FAlGaN


Lw = 3nmx = 03





119887in


119887






119887


119887increases


119887 The reason




31and 119890

33(in



119886 119898

perp119898

119890

31119890

33119886

1119887

1119886

2119887

2




11 11986212 11986213 and 119862

33(in units of GPa)


119864

119892119901

sp119862

11119862

12119862

13119862

33119891ion 120576

infinperp120576

infin119911



3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

03

06

09

12

15

F (m

Vc

m)

FGaN

Lb (nm)

FAlGaN

x = 03

x = 02

x = 01

P = 0GPaLw = 3nm





119909119909119887and 120576




119908= 2 nm 119871

119887= 5 nm and





119871



0= 0 and 119911

0= 0 As expected










and 119870]119895120585119911119911






119895120585


GaN 39a 757b 748b 568b 540b 091b 082b 118b 102b


0 2 4 6 8 10 1220

40

60

80

100

120

140

R (nm)

Eb

(meV

)




119908= 2 nm 119871


pressure 119875




1 2 3 4 5 6 7 830

35

40

45

50

55

60

65

Eb

(meV

)


Lw (nm)

R = 10nmLb = 20nmx = 03


119908) for


pressure 119875




010 015 020 025 030 035 04035

40

45

50

55

60

x

Eb

(meV

)

Lw = 2nmLb = 5nm

6nmR =



119908= 2 nm 119871



0 2 4P (GPa)

6 8 1035

40

45

50

55

60

65

70

75

Eb

(meV

)




0= 0 nm and 119911



4 6 8 10 12 14 16 18 2030

35

40

45

50

55

Eb

(meV

)

Lb (nm)

R = 10nmLw = 3nm

x = 02




119908= 3 nm 119909 =






120588

0= 0 nm 119909 = 02 and 119911









425

450

475

500

525

550

z0 (nm)

Eb

(meV

)

R = 6nm

Lw = 3nmLb = 5nm

x = 03



0for119877 = 6 nm 119871

119908= 3 nm 119871

119887= 5 nm 119909 = 03 and several





0along


119908= 3 nm 119909 = 03 and 120588






119908in the dot




119908 119877




5 Conclusions




Acknowledgments


References






119909Ga1minus119909





119909Ga1minus119909





119909Ga1minus119909






119909Ga1minus119909














119909Ga1minus119909














CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





120596

]119895120585= 120596

]119895120585(119901) + 2119870

]119895120585119909119909

120576

119909119909] (119901) + 119870]119895120585119911119911

120576

119911119911] (119901) (21)


canbe given by [25]

120574

]119895120585= 119861

0]1

120596

]119895120585(0)

(

120597120596

]119895120585(119901)

120597119901

) (22)


119895120585(0) is the zone-




119895120585119909119909and 119870]



120597120576

]infin120585(119901)

120597119901

= minus

5 (120576


3119861

0](09 minus 119891

]ion) (23)



119861

0] =(119862

11] (119875) + 11986212] (119875)) 11986233] (119875) minus 21198622

13] (119875)

119862

11] (119875) + 11986212] (119875) + 211986233] (119875) minus 411986213] (119875)

(24)



120576] (119901) =2

3

120576

]120585perp(119901) +

1

3

120576

]120585119911(119901)

(25)


119875

]pe = 119890

]33(119901) 120576

]119911119911+ 2119890

]31(119901) 120576

]119909119909 (26)

where 119890]31

and 119890]33




3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

03

06

09

12

15

18

F (m

Vc

m)

FGaN

Lb (nm)

FAlGaN


Lw = 3nmx = 03





119887in


119887






119887


119887increases


119887 The reason




31and 119890

33(in



119886 119898

perp119898

119890

31119890

33119886

1119887

1119886

2119887

2




11 11986212 11986213 and 119862

33(in units of GPa)


119864

119892119901

sp119862

11119862

12119862

13119862

33119891ion 120576

infinperp120576

infin119911



3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

03

06

09

12

15

F (m

Vc

m)

FGaN

Lb (nm)

FAlGaN

x = 03

x = 02

x = 01

P = 0GPaLw = 3nm





119909119909119887and 120576




119908= 2 nm 119871

119887= 5 nm and





119871



0= 0 and 119911

0= 0 As expected










and 119870]119895120585119911119911






119895120585


GaN 39a 757b 748b 568b 540b 091b 082b 118b 102b


0 2 4 6 8 10 1220

40

60

80

100

120

140

R (nm)

Eb

(meV

)




119908= 2 nm 119871


pressure 119875




1 2 3 4 5 6 7 830

35

40

45

50

55

60

65

Eb

(meV

)


Lw (nm)

R = 10nmLb = 20nmx = 03


119908) for


pressure 119875




010 015 020 025 030 035 04035

40

45

50

55

60

x

Eb

(meV

)

Lw = 2nmLb = 5nm

6nmR =



119908= 2 nm 119871



0 2 4P (GPa)

6 8 1035

40

45

50

55

60

65

70

75

Eb

(meV

)




0= 0 nm and 119911



4 6 8 10 12 14 16 18 2030

35

40

45

50

55

Eb

(meV

)

Lb (nm)

R = 10nmLw = 3nm

x = 02




119908= 3 nm 119909 =






120588

0= 0 nm 119909 = 02 and 119911









425

450

475

500

525

550

z0 (nm)

Eb

(meV

)

R = 6nm

Lw = 3nmLb = 5nm

x = 03



0for119877 = 6 nm 119871

119908= 3 nm 119871

119887= 5 nm 119909 = 03 and several





0along


119908= 3 nm 119909 = 03 and 120588






119908in the dot




119908 119877




5 Conclusions




Acknowledgments


References






119909Ga1minus119909





119909Ga1minus119909





119909Ga1minus119909






119909Ga1minus119909














119909Ga1minus119909














CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






31and 119890

33(in



119886 119898

perp119898

119890

31119890

33119886

1119887

1119886

2119887

2




11 11986212 11986213 and 119862

33(in units of GPa)


119864

119892119901

sp119862

11119862

12119862

13119862

33119891ion 120576

infinperp120576

infin119911



3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

03

06

09

12

15

F (m

Vc

m)

FGaN

Lb (nm)

FAlGaN

x = 03

x = 02

x = 01

P = 0GPaLw = 3nm





119909119909119887and 120576




119908= 2 nm 119871

119887= 5 nm and





119871



0= 0 and 119911

0= 0 As expected










and 119870]119895120585119911119911






119895120585


GaN 39a 757b 748b 568b 540b 091b 082b 118b 102b


0 2 4 6 8 10 1220

40

60

80

100

120

140

R (nm)

Eb

(meV

)




119908= 2 nm 119871


pressure 119875




1 2 3 4 5 6 7 830

35

40

45

50

55

60

65

Eb

(meV

)


Lw (nm)

R = 10nmLb = 20nmx = 03


119908) for


pressure 119875




010 015 020 025 030 035 04035

40

45

50

55

60

x

Eb

(meV

)

Lw = 2nmLb = 5nm

6nmR =



119908= 2 nm 119871



0 2 4P (GPa)

6 8 1035

40

45

50

55

60

65

70

75

Eb

(meV

)




0= 0 nm and 119911



4 6 8 10 12 14 16 18 2030

35

40

45

50

55

Eb

(meV

)

Lb (nm)

R = 10nmLw = 3nm

x = 02




119908= 3 nm 119909 =






120588

0= 0 nm 119909 = 02 and 119911









425

450

475

500

525

550

z0 (nm)

Eb

(meV

)

R = 6nm

Lw = 3nmLb = 5nm

x = 03



0for119877 = 6 nm 119871

119908= 3 nm 119871

119887= 5 nm 119909 = 03 and several





0along


119908= 3 nm 119909 = 03 and 120588






119908in the dot




119908 119877




5 Conclusions




Acknowledgments


References






119909Ga1minus119909





119909Ga1minus119909





119909Ga1minus119909






119909Ga1minus119909














119909Ga1minus119909














CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





and 119870]119895120585119911119911






119895120585


GaN 39a 757b 748b 568b 540b 091b 082b 118b 102b


0 2 4 6 8 10 1220

40

60

80

100

120

140

R (nm)

Eb

(meV

)




119908= 2 nm 119871


pressure 119875




1 2 3 4 5 6 7 830

35

40

45

50

55

60

65

Eb

(meV

)


Lw (nm)

R = 10nmLb = 20nmx = 03


119908) for


pressure 119875




010 015 020 025 030 035 04035

40

45

50

55

60

x

Eb

(meV

)

Lw = 2nmLb = 5nm

6nmR =



119908= 2 nm 119871



0 2 4P (GPa)

6 8 1035

40

45

50

55

60

65

70

75

Eb

(meV

)




0= 0 nm and 119911



4 6 8 10 12 14 16 18 2030

35

40

45

50

55

Eb

(meV

)

Lb (nm)

R = 10nmLw = 3nm

x = 02




119908= 3 nm 119909 =






120588

0= 0 nm 119909 = 02 and 119911









425

450

475

500

525

550

z0 (nm)

Eb

(meV

)

R = 6nm

Lw = 3nmLb = 5nm

x = 03



0for119877 = 6 nm 119871

119908= 3 nm 119871

119887= 5 nm 119909 = 03 and several





0along


119908= 3 nm 119909 = 03 and 120588






119908in the dot




119908 119877




5 Conclusions




Acknowledgments


References






119909Ga1minus119909





119909Ga1minus119909





119909Ga1minus119909






119909Ga1minus119909














119909Ga1minus119909














CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




010 015 020 025 030 035 04035

40

45

50

55

60

x

Eb

(meV

)

Lw = 2nmLb = 5nm

6nmR =



119908= 2 nm 119871



0 2 4P (GPa)

6 8 1035

40

45

50

55

60

65

70

75

Eb

(meV

)




0= 0 nm and 119911



4 6 8 10 12 14 16 18 2030

35

40

45

50

55

Eb

(meV

)

Lb (nm)

R = 10nmLw = 3nm

x = 02




119908= 3 nm 119909 =






120588

0= 0 nm 119909 = 02 and 119911









425

450

475

500

525

550

z0 (nm)

Eb

(meV

)

R = 6nm

Lw = 3nmLb = 5nm

x = 03



0for119877 = 6 nm 119871

119908= 3 nm 119871

119887= 5 nm 119909 = 03 and several





0along


119908= 3 nm 119909 = 03 and 120588






119908in the dot




119908 119877




5 Conclusions




Acknowledgments


References






119909Ga1minus119909





119909Ga1minus119909





119909Ga1minus119909






119909Ga1minus119909














119909Ga1minus119909














CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





425

450

475

500

525

550

z0 (nm)

Eb

(meV

)

R = 6nm

Lw = 3nmLb = 5nm

x = 03



0for119877 = 6 nm 119871

119908= 3 nm 119871

119887= 5 nm 119909 = 03 and several





0along


119908= 3 nm 119909 = 03 and 120588






119908in the dot




119908 119877




5 Conclusions




Acknowledgments


References






119909Ga1minus119909





119909Ga1minus119909





119909Ga1minus119909






119909Ga1minus119909














119909Ga1minus119909














CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





119909Ga1minus119909





119909Ga1minus119909





119909Ga1minus119909






119909Ga1minus119909














119909Ga1minus119909














CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



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