NASA/TMm2003-211983

Basic Equations for the Modeling of Gallium

Nitride (GaN) High Electron Mobility

Transistors (HEMTs)

Jon C. Freeman

Glenn Research Center, Cleveland, Ohio

February 2003

https://ntrs.nasa.gov/search.jsp?R=20030014725 2018-05-15T15:37:23+00:00Z
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NASA/TM--2003-211983

Basic Equations for the Modeling of Gallium

Nitride (GaN) High Electron Mobility

Transistors (HEMTs)

Jon C. Freeman

Glenn Research Center, Cleveland, Ohio

National Aeronautics and

Space Administration

Glenn Research Center

February 2003
NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov
Basic Equations for the Modeling of Gallium Nitride (GaN)

High Electron Mobility Transistors (HEMTs)

Jon C. Freeman

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Summary

Gallium nilride (GaN) is a most promising wide band-gap semiconductor for use in high-power

microwave devices. It has functioned at 320 °C, and higher values are well within theoretical limits. By

combining four devices, 20 W has been developed at X-band. GaN High Electron Mobility Transistors

(HEMTs) are unique in that the two-dimensional electron gas (2DEG) is supported not by intentional

doping, but instead by polarization charge developed at the interface between the bulk GaN region

and the A1GaN epitaxial layer. The polarization charge is composed of two parts: spontaneous and

piezoelectric. This behavior is unlike other semiconductors, and for that reason, no commercially

available modeling software exists. The theme of this document is to develop a self-consistent approach

to developing the pertinent equations to be solved. A Space Act Agreement, "Effects in A1GaN/GaN

HEMT Semiconductors" with Silvaco Data Systems to implement this approach into their existing

software for III-V semiconductors, is in place (summer of 2002).

Introduction

This report describes a method to model gallium nitride (GaN) with the Silvaco product ATLAS

and its subroutine BLAZE. Unlike other semiconductor systems, GaN requires the addition of strains

at all parts of the device to include the effects of piezoelectricity. One must also include a spontaneous

polarization at certain interfaces. For these reasons this report consists of 11 sections. Section 1 introduces

notations and definitions, in the area of linear elasticity. Section 2 discusses piezoelectricity and gives the

complete set of basic equations to be solved in a general GaN device. Section 3 introduces spontaneous

polarization. Section 4 indicates the additions needed in the Poisson-Schrodinger solver for GaN analysis.

Section 5 gives some critical and basic details on the layer structure ofAIGaN/GaN HEMTs. Section 6

covers band-gap variations, while the remaining sections treat necessary issues to complete the analysis.

Discussion

Our goal is to model A1GaN/GaN HEMTs using Silvaco's existing software platform. For the

A1GaN/GaN system and other III-V nitrides, the effects of both spontaneous and piezoelectric

polarization fields must be included in an analysis. These fields dominate the properties of the 2DEG, and

this report serves as a starting point to incorporate them. The polarizations enter the problem via the

constitutive equation relating D, E, and P :

D=_o_rE +P

NASA/TM--2003-211983 1
WhereD is the electric flux density, E the electric field, and P the polarization of the material.

This then influences Poisson's equation as follows. Taking the divergence of the previous equation we

have

or

Then Poisson's equation is

V' +V'F

V.D=p

=0

-eoe_ V2_ =p-V.P

Thus the form using the electrostatic potential has an "extra" term due to polarizations not included

in the relative dielectric constant e r. The added polarization P is composed of both the spontaneous and

piezoelectric parts. The spontaneous part is modeled with an ab initio result of a theoretical calculation,

while the piezoelectric portion must be determined from the stress/strain fields of the device. The

constitutive equations for the stress/strain and D, E, and P are"

"_=dr .-E +S E :_

--6=, s.2 +a = 7=a :f

The quantities _, S, S E, d, _ s, are the stress, strain, compliance, piezoelectric, and permittivity

tensors, respectively.

The fact that P modifies Poisson's equation means that it also modifies the energy levels for the

2DEG as obtained from the Schrodinger equation. The strain also modifies the band gaps, via the

deformation potential terms.

This report consists of eleven sections. Section 1 introduces notations in the area of linear elasticity.

Section 2 discusses piezoelectricity. Section 3 treats the concepts and properties of spontaneous

polarization, as this is a dominant term, yet it is not well understood. Section 4 develops the forms for

Poisson's and Schrodinger's equations. Here, the inclusion of dipoles and dangling bonds at the interface

is addressed. Section 5 gives an account of the issues concerning the stresses and strains in AIGaN/GaN

systems. Section 6 discusses the very important band-gap issues. Section 7 is just one page covering the

variation in thermal conductivity with temperature. Section 8 gives numerical values for many of the

terms discussed in other sections. It shows that the polarization terms are indeed dominant, as shown in

published works. The effects of residual strains, strains due to interface mismatch, and oxide and metal

pads are covered. Section 9 briefly visits some of the issues concerning heterojunctions. Section 10 gives

data on many of the known traps discussed in the open literature. Finally, section ! 1 is a discussion of

amplifiers and the best power levels reported to date. The appendix gives the closed-form expression for

the field due to the spontaneous polarization.

NASAfI'M--2003-211983 2
1. Elasticity

This section serves to introduce the needed definitions in elasticity theory that will be utilized later.

Let u = mechanical displacement, then the strain is defined as (ref. 1)

_o. (r,t) = I(OU t O//j]

-2_-_ra+--_r, ) i,j=x,y,z

(1)

where Sv means i = direction of displacement (or force)

j = normal to surface (face)

Strain is commonly given the symbols e=, U_x,...

Now

S= Sxy Syy sxj Sy z A iisSl 6 5 2 ss1 -}s4I

s= s,__s= L_s_{s4 S 3 J

(2)

Following Auld (ref. 1), Nye (ref. 2), and many others, we use the convention that compressive strain

is negative, while tensile is positive. Note that the factors of 1⁄2 and sign convention vary in the literature.

Write

S1

$2

_= $3

$4

$5

.$6

(3)

The stress is given as

= = T6

rx: ry: r.. r5 r4 r3

(4)

Alternate symbols for Tj are cry,, rw....

NASA/TM--2003-211983 3
Write

rl

r2

 =r3

7-4

r5

T6_

Tj < 0 for compression

Ti > 0 for tension

Assume linear theory is applicable, then Hooke's law is valid, so

(5)

TO.= co.laSki (6)

or

S_jkl = So._ Tkl (7)

where Cvklare the stiffness constants (N/m 2) and svkl are the compliance constants (m2/N).

The compact notation is

T = c : S (6a)

S = s : T (7a)

where : stands for the tensor dyadic product. GaN and A1N form in either the Wurtzite (Wz)(hexagonal)

or Zincblende (cubic) crystal classes. For Wz with symmetry notations 6 mm, ca ....

cO =

-Cll C12 C13 0 0 0

C12 ell ¢13 0 0 0

¢13 ¢13 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 ¢44 0

0 0 0 0 0 Cli --C12

(8)

To summarize

rl

r3

r5

.T6.

¢11

C12

c13

= 0

0

0

¢12 ¢13 0 0 0

ell ¢13 0 0 0

¢13 ¢33 0 0 0

0 0 C44 0 0

0 0 0 ¢44 0

0 0 0 0 Cll -- C12

2

!S1

_$2

$3

$4

$5

_s6J

(9)

NASA/TM--2003-211983 4
$1

S 2

$3

S 4

$5

_$6

SI1 S12 S13 0 0 0

S12 Sll SI3 0 0 0

=/S13 S13 $33 0 0 0

L! 0 0 $44 0 0

0 0 0 $44 0

0 0 0 0 2(s11 - s12

I

r3

r4

rs

_r6J

(10)

The determination of the five c,a values is rather difficult. Presently GaN must be grown on a substrate

(Sapphire, SiC, ZnO, Si, etc.). The epitaxial growth may be MOCVD, MBE, HVPE, among others; and

the growth rates, gas pressures, and temperatures vary considerably. Under the present situation the films

vary greatly. They differ in stoichiometry, defects, crystal faults, surface morphology, and film thickness

(nm to lam). For these reasons, among others, one may only expect ranges of values for the fundamental

constants for GaN or A1N. Table I gives two sets of values for the c,a.The first column was derived from

acoustic velocity measurements (ref. 3) on plasma-induced MBE films on Sapphire without a nucleation

layer, with thickness between 800 and 1300 nm. The second column is from best guesses after averaging

entries from the literature (refs. 4 to 8).

GaN

Cll

C12

c13

c3_

C44

C66

TABLE I.--STIFFNESS CONSTANTS FOR GaN

c v in GPa

370

145

110

390

9O

112.5

cy (average.

from literature)

382

145

106

389

103

123

IDifference

(percent)[

3.2

3.6

0.26

14

AIN

Cll

C12

C13

¢33

TABLE II.--STIFFNESS CONSTANTS FOR A1N

c,j in GPa

410

140

100

c,/ (average.

from literature)

390

125

120

[ Difference

(percent) I

4.9

10.7

2O

390 395 1.28

C_ 120 118 1.7

Given the cv the s,j are (ref. 1)

let

CA = C33 (Cll + C12 ) -- 2 c23 (11)

then

,Fc33 1 I

(12)

NASA/TM--2003-211983 5
s12 - (13)

C33

------Sll

CA

C13

S13 -

C A

(14)

Cll +c12

$33 -

c A

1

$44 =--

C44

Table III gives the s v using best guess values for the c v.

(15)

(16)

TABLE m.--COMPLIANCE COEFFICIENTS FOR GaN AND A1N.

GaN A1N

2 _ 3.175 3.5319

Sll m x10-12

N

s12 -1.044 -1.0136

sl3 -0.58072 -0.76506

$33 2.887

$44 9.7087

2.9965(2.8 _omexperiment)

8.474

While GaN, AIN, and Alx Gal_x N are anisotropic, many papers treat them as isotropic for simplicity. For

an isotropic material, we have

c_ =

r3,+2g 3. 3. 0 0 0-

_. _,+2g _, 0 0 0

_. 2_ ;_+2g 0 0 0

0 0 0 I.t 0 0

0 0 0 0 g 0

0 0 0 0 0

(17)

Where 3. and p are the Lam6 constants. Two other constants are normally used instead of 3. and/.t;

namely E and v, Young's modulus and Poisson's ratio, respectively. They are given by

(18)

(19)

NASA/TM--2003-211983 6
Thebulkmodulusis defined as

3_, + 2_t B3

The T/S relationships in terms of E and v are (ref. 9)

E

r==0+,,)0- 20[0-,,)Sx_+_(s.¢+s::)]

E [0-,,)S_y+,,(s::+s_)]

Tyy = (1 + vX1 - 2v)

E

T:: - (1 + vX1 - 2v)[(1 - v)s= + v(S_ + Syy )]

E

Ty: = T. v =-_ Sy:.

E

T:_ =Tx: =-o-_S:x

E

ray = Tyx =-_--_ Sxy

(20)

(21)

(22)

(23)

(24)

(25)

(26)

or

1 T ._

1

_ =_ [,-_y-,#= +_:)]

(27)

(28)

1

s,_- (l_)r_.

s_ -_7;_(1+ v)_

(29)

(30)

(31)

(32)

NASA/TM--2003-211983 7
There are several special cases:

Plane strain (6'.-:= 0)

or

E

r=:,_0+vXl-2v) ,. [(,__ +%]

_ g

T - Ev

E

(33)

(34)

(35)

(36)

(37)

Plane stress (T= = 0)

S..=O

Sy: =S_: =0

08)

(39)

(40)

(41)

(42)

NASAFFM_2003.211983

E

E

T:_--0

E

8

(43)

(44)

(45)

(46)
Ty. = Tx_ = 0 (47)

or

1 T sxx=7(xx-v ,)

1 T VTxx)

S,y =_-( _,-

= +T._

(48)

(49)

(5O)

sx,,-(1Ev)Y y

Sy: = Sx: = 0 (52)

We may approximate the constants _, _t, v, E, B, and M (the biaxial modulus) as follows. From

equations (9) and (17) and table I, we see

Cll = _.+ 2p = 370 = 382 (53)

c12 = _. = 145 = 145 (54)

c13 = _.= 110 = 106 (55)

¢33 = _"+ 2_t = 390 = 389 (56)

C44 = }.1,= 90 = 103 (57)

c66 = }J. = 112.5 = 123 (58)

From visual inspection it appears that a good choice is

_.+ 2_ - 383 (59)

and

_.- 126.5 (60)

la-128 (61)

NASA/TM--2003-211983 9
Withthischoice,andusingequations(18)to (20),wefind(averagevaluesdenotedwithacarat(^))

(62)

(63)

/_=212 (64)

- -- = 427 (63a)

In the literature I have found the following formulas

B= (¢11 +c12)c33 - 2c23 _ E

Cll + c12 + 2c33 - 4c13 3(1 - 2v)

(65)

E 2c23

M- -ell +c12 ---

1 - v c33

(66)

Note, equations (65) and (66) permit the determination of E and v, as the cu are known, that is, two

equations in the unknowns E and v. From the values in table I, we find

Columnl J_=357 _=0.21 /_=206.6 )V/=453 (67)

Column2 E=377 (,=0.20 /_=207.2 /Q=469 (68)

From the literature I have found the following reported values.

B: 117, 160, 180, 188, 200, 204, 206, 210, 220, 268

These yield and average/1 = 195 GPa.

v:0.23, 0.25, 0.38, b= 0.29

E:196, 290 E=243

M:413, 450, 479, /_/= 447

(69)

(70)

(71)

(72)

NASA/TM--2003-211983 10
FromtableI ofreference 10 we fred

CI1 = 363

C12 =128

C13 = 99

C33 =- 3 85

C44 = 85

/_=195

/_=273

= 0.27

and the experimental values given (ref. 10) were

B = 200 + 20

E = 290

v = 0.23 + 0.06

(73)

(74)

C_l

c12

£'13

£'33

c44

TABLE IV.--ELASTIC CONSTANTS FOR AIN FROM REF. 10

From ref. 3

410

Best guess

390

[ Difference

(percent)[

4.9

140 125 10.7

100 120 20

390 395 1.3

120 118 1.7

/_ = 201

(, = 0.21

A_/= 442

The dispersion in the basic parameters reflects the wide variation in the films due to a variety of

factors, as well as attempting to forge an approximate isotropic model.

In most cases the films are assumed to be in "biaxial stress" which means

r= =% ,and%:=o

then equation (27) gives

S=E= - VTyr =Y=- vr=

E

T_- S_ 1-v

(75)

(76)

NASA/TM--2003-211983 11
which is apparently the reason for the deffmition of M in equation (66). The literature often uses the

notation

Oxx = M0001 exx (77)

where the subscript 0001 denotes the z-direction (parallel with the c-axis of the film). The crystal is then

defined as having a Ga face. From equation (9)

T 3 = ¢13S1 + c13S 2 + c33S 3

If T3 = 0, a "free surface," then if $I = $2 (biaxial strain), then

0 = 2c13S 1 + c33S 3

2c13

u= - Uxx (78)

C33

This result will be used often later. It can be shown that

Tx_ = 2c23 - c33(Cll + c2) S.. (79)

2c13 --

where it is assumed that T_ = Tin, and T.-: = 0. This corresponds to equation (50)

E

Tx_ =---S..

2v ""

Using equations (60) and (61), _. = 126.5, _t = 128, then -E/2v =-639.2. Using the values in column 2 of

table I, equation (79) becomes

T_ = -861 s_... [percent difference [ = 26 percent (80)

This demonstrates the uncertainty in the magnitudes of the quantities in this section.

Estimation of Stress and Strain

We assume the stress in the layers may be conceptually described by the superposition of three

components:

_TOTAL = OEXTERNAL + _THERMAL + GINTRINSIC

The thermal part is the most straightforward, it is

OTHERMA L = OTH = ActAT--

E

1-v

(81)

(82)

NASA/TM--2003-211983 12
Where Aot is the difference in thermal expansion coefficients (TEC), between the layers. Also AT is

the difference between the growth temperature and the sample temperature (generally room temperature

(RT)). The above assumes the substrate and epilayer are relaxed at the growth temperature (not true in

general), and that no relaxation during cooling has taken place.

The external portion (aEXV) is due to loading by adjacent layers; lattice mismatch, metal pads, and

any recognizable force. The last term is referred to as intrinsic or residual stress; it acts as a catch-all for

stresses that do not fit into the two previous categories. The residual stresses are difficult to handle

because they depend on growth parameters, which differ greatly for the basic growth methods. The basic

methods are MBE, MOCVD, HVPE, and variations of them. Some of the parameters are growth

temperature, pressure, gas flow rates, substrate crystal orientation, surface conditions, and preparation

procedure. The GaN is often grown on a nucleation layer of either GaN or AIN. The properties of the

nucleation layer include thickness, growth temperature, and doping. Finally, defects produced by

vacancies, cracks, misfit dislocations, threading dislocations, etc.

After the stress has been quantified the strain will still not be known exactly, as the material responds

to stress in at least two ways. One is to strain via Hooke's law, the other is to generate defects and

propagate them throughout the material (relaxation). Finally, cracks do occur in GaN and sapphire, and

many in the GaN are not visible by observation of the surface.

External Stress/Strain

Most often the strain due to lattice mismatch is assumed to be the dominant factor. Def'me the misfit

parameter

fm- a - a o (83)

a0

where a0 is the relaxed lattice parameter and a is the strained value. The strain is then

_xx=+ fm (84)

where the appropriate sign is chosen. Maeda (ref. 11) introduced a "lattice relaxation ratio, R" by

0 R) (85) E X,lt. _ -- __

a0

This is very useful, as it permits some relaxation at the interfaces. The corresponding stresses are

E Ac E

a m .... _.. (86)

2v co 2v "

E Aa

c_ - = M exx (87)

1-v a 0

where Ac/co is the strain along the c-axis, and Aa/ao is that in the basal plane. The composite stress in a

given film, or layers of film may be calculated at several levels of approximation. We start with Etzkorn

(ref. 12).

NASA/TM--2003-211983 13
R

/

Figure 1.--Epitaxial film of thickness hF on a substrate of thickness hs. Coordinate

axes in the layers are z and t, respectively. Bending is due to differing thermal

expansion coefficients.

The figure is for the case when the net stress is compressive. The subscripts F an S represent film and

substrate, respectively. The equations are as follows.

Def'me the parameter

r= [Es/O_Vs)]hs- Mshs (88)

Assume the thermal strain dominates

ActAT

(_F = -ME -- (89)

l+r

ActAT

o s = M s -- r (90)

l+r

e_v= (ct F -ct s )aT (91)

l+r

eRSr- (a F -a s )AT r (92)

l+r

t = hs - hFr

2(1 +r) (93)

NASA/TM--2003 -211983 14
AaAT 6(h s - 2t) _ 6(h F + 2t) + (hs + hF (94)

z+t

_EF (Z) =_RT nt (95)

R

Notice equation (95) gives the strain as a linear function of distance z from the interface. Equation

(94) gives the radius of curvature. Observe that this formulation only requires material parameters and the

thickness of the substrate and film.

Next, the equation of Skromme (ref. 13):

Msh +MFh - 3MFh (hs+

CSURFACE = 6(hs + h F )hF R

(96)

where R is the radius of curvature which must be measured. Kozawa (ref. 14) has a slightly simpler form

(assumes v the same in both layers)

Esh3s + EFh3F

c Y = 6(1 -- vXh s + h y )h y R

(97)

TABLE V.--DATA FROM REF. 14 WITH OTHER

ENTRIES IN PARENTHESES

E (GPa)

_c Oo-_)

v

Sapphire

300

425

GaN

2.5 to 48.8

196(295)

7.5 5.45

0.3 (0.25) 0.3 (0.25)

From Hearne (ref. 15) we have Stoney's formula:

Msh2s

(YF--

6hER

where for MOCVD at 1050 °C; sapphire: M = 450 or 479, depending on wafer; 6H-SiC: M = 586 GPa.

Then

I_F -= axx = MF _xx

The composite stress was measured to be tensile with range

0.14 < 6_ < 0.29 GPa

(98)

(99)

NASA/TM--2003-211983 15
Thethermalcomponent(bymeasurements) was-0.66+ 1 GPa, and the theoretical thermal

component is -1.4 + 0.7 GPa. Thus, the residual and perhaps some lattice portion dominates the stress

in the given films. The radius of curvature R is determined by surface profilometry or laser reflection.

Generally

2m<R<20m

and in most cases R < 6 m. However, R = 0.58 m for HVPE films.

Some experimental data (though incomplete) is helpful. From reference 16 a sample consisting of

a 6H-SiC substrate with 3p of porous GaN, then 3_t of HVPE GaN on top. The surface was in tension

(wafer like a cup that holds water). The radius of curvature R was -6.5 m, the in-plane stress was

0.64 GPa, and the strain along the c-axis was -1.12 x 10 -3. The substrate thickness was not given.

A common range of parameters for GaN on sapphire is: 250 btm of sapphire with 500 A of AIN

nucleation layer, and 1 to 5 lam of GaN. The residual strain in the GaN is of the order of 1.0 × 10-4.

General Case of Several Layers

For the general case of several layers, one may use the results of Olsen and Ettenberg (ref. 17). The

stress in any layer may be estimated from (see fig. 2)

t2

tl

1

R

Figure 2.---General layer structure with thickness tj for

each layer. Measured radiusof curvature is R.

+m

2R

(100)

NASA/TM--2003-211983 16
Hereit isassumedthattheelasticmoduli(Young'smoduli)ofall layersarethesameandequalto E. In

equation (100)y is measured from the bottom ofthej thlayer. The terms in equation (100) are

N

t=_t i

t=l

(lOl)

( /-I

N N

R=t 3 6_ _'tit J _g

k i=1 j>l

(]02)

_:, =1 i<j

_ci =0 i=j

_:i =-1 i>j

(103)

In equation (102) the strain at the interface of the ithandj thlayer _v is related to the measured radius

of curvature. However, it is assumed the strains e 0 are known, along with the thickness t, of each layer.

Ifw-L, then one may assume spherical bending, then a two-dimensional (2D) solution may be obtained

from the given one-dimensional (1D) by

/ \ _j (Y)ID

°i lY)2D - (104)

l-v

Reasonable choices for the interface strains are the larger of thermal, lattice mismatch, or residual

strains (ref. 18). The reviews in references 19 to 22 should be consulted for more information.

2. Piezoelectricity

This section details equations we will use later. It is important to note that this discussion treats

dielectrics, not semiconductors, that is, effects of cartier screening are ignored. We start with the defining

equation for D, E, and P.

Here

D=e 0 E+P (105)

B

P = PELECTRONIC + PIONIC + POTHER (106)

In most cases only the first two terms in equation (106) are assumed to exist, and POTHER is assumed

equal to zero. This case then has the constitutive assumption that the polarization is linearly related to the

E field,

PELECTRONIC + PIONIC_ C 0 _e _ (107)

NASA/TM--2003-211983 17
where)_eis the polarizability. Then

D=E 0 E+E 0 Ze _

=EOE r g (P-oTHER ____O)

which is the normal case. For anisotropic material we write

1 D = Kyx K_ Ky:

K zx Kzy K zz

We use K since c is used so often for strain. Thus

1

--D=_c.E

CO

(108)

(109)

(11o)

or

1

C o

-- Oj = KijE j i,j = x,y,z

for GaN and A1N, we have (ref. 20)

KI1 0 0 1

0 r:11 0 0 K33

(111)

for GaN, KI_ = 9.5 and _¢33= 10.4

for AIN, Kll ---- 9.0 and K33 ---- 10.7

Now when the dielectric material is also piezoelectric (following Auld, (ref. 1))

(112)

which are the piezoelectric strain equations. Another form is

T=-e:E+c E :S

D=c o Ks .E+e:S

(113)

which are the corresponding stress equations. The superscripts mean constant electric field E, constant

stress T, or constant strain S. By inspection we observe (now ffOTI-mR = ffPlZZOELECTmC)

P=d:7 _ (114)

NASA/TM--2003-211983 18
where d,j are the piezoelectric strain constants. We also have

P=e;S

where e v are the stress constants. For our Wz crystals we have (refs. 1 and 23)

Py = 0 0 d15 0

P.. d31 d31 d33 0 0

r4

r5

56J

(115)

(116)

with the associations T1 = 01 = on, Tz = 02 = Oyy, T3 = 03 = 0=, T4 = o4 = 0., T5 = o5 = ox.-, T6 = (_6 = (_xy

As in section 1, the range of the coefficients is large.

The following strain constants are in units of 10-l° cm/V (see refs. 22 to 26):

d3_

d_

dis

TABLE VI.--PIEZOELECTRIC STRAIN CONSTANTS OF GaN

Measured

-0.9

Calculated

-1.16

Estimated

-1.7

Bulk

-1.9

Clamped

-1.4

Polycrystal

-1.0

2.39 3.7 2.8 2.0

-3.1 -2.75

The following strain constants are in units of 10 -1° cm/V:

d31

d33

dis

TABLE VII.--PIEZOELECTRIC STRAIN CONSTANTS OF A1N

Measured

-2.0

5.0

Calculated

-2.65

5.53

-3.4

Estimated

-2.71

6.72

Bulk

-2.8

5.6

Clamped

-2.0

4.0 (3.2 film)

Polycrystal

-2.71

-3.6

Theory says d31 = -(1/2) d33. The alternate representation is

P=e:S

or

Py = 00 00 el50e15!] 0

P: e31 e31 e33 0 0

f.u¢

tyy

ty=

tzx

_xy.

(117)

NASA/TM--2003-211983 19
Thefollowingstressconstantsareinunitsof C/m2:

TABLEVIII.--PIEZOELECTRIC STRESSCONSTANTS OFGaN

el5

e31

e33

Literature

-0.21

-0.3

-0.33

-0.22

-0.33

-O.36

-0.49

0.44

0.65

0.73

1.0

Bulk

-0.55

1.12

Crystal

-0.41

Clamped

0.85

Polycrystal

-0.3

0.6

The following stress constants are in units of C/m2:

TABLE IX.--PIEZOELECTRIC STRESS CONSTANTS OF A1N

e31

e33

Literature Bulk Crystal

-0.6

1.46 1.5 1.06

0.85

Clamped

-0.63

Polycrystal

-0.5

Also

It can be shown that

d r •E =

0 0 d31

0 0 d31

0 0 d33

0 d15 0

d15 0 0

0 0 0

eaj = Cald o

Ey

E:

(118)

(119)

NASA/TM---2003-211983 20
Using this relationship we may determine the d,I

1,0 ,_0 1,0

d31 =e31Sll +e32s21 +e33s31 +e34 $41 +e35 $51 +e36 $61

= e31 (Sll +s21)+e33s31 = (--0.35)(2.13 1 x 10 -12 )+(0.705)(- 5.8072 x 10 -12 )

= -1.16x 10 -1° cm/V = -1.16x10 -12 C/N

1" 1, o 1, o

d33 = e31s13 + e32s23 + e33s33 + ej4°$43 + e35 $53 + e36 $63

= 2e31s13 + e33s33 = 2.39 x 10 -12 C/N

to 1, o _o ?o 1,o to

d15 =ell s15 +el2 $25 +el3 $35 +el4 s45 +el5 s55 +el6 $65

= e15s44 = -2.75 × 10-12 C/N

In our application, one often assumes the GaN to be completely relaxed, and the AIGaN to be in

tensile biaxial strain. The resulting polarization field at the AIGaN/GaN interface is assumed to be as

shown below. Start with

P. = e31Uxx + e31Uyy + e33uz: (120)

Next assume a free surface (this implies the top of the A1GaN layer is free). This is a questionable

assumption as the gate metallization loads the layer. Next assume biaxial strain, Ux_= Uyy

.'. u:: =-2 c13 Uxx (121)

C33

which is equation (78) of section 1. Then

I cJ-L] (122) P- = 2e31Uxx -2 c13 Uxxe33=2 e31 -e33 Uxx

c31 c33

The term in brackets has been measured to be -0.3 C/m 2. An alternate representation starts with

P, = d31Gxx + d31(yyy + d33_. .. (123)

but

(Yxx = C11//xx + C12Uyy + Cl3Uz:

and assume biaxial stress conditions hold O_x= oyy. Then

P. = 2d31 (Cll + c12 )Uxx + d31Cl3U = + d33t_:-

Next assume a free surface, so _z = 0,

(124)

(125)

u.. = -2 c13 Ux.x (126)

¢33

NASA/TM--2003-211983 21
then

P= = 2d31(Cll + Cl2)Uxx + Cl3I- 2 Cl31uxxd31c33 fl

(127)

fmally

E c2]

_ l__L3Uxx

P. = 2d31 Cll + C12 C33 _J

(128)

The corresponding electric field is (assuming D = 0, which means no charge is present)

E. P" 2d31 I c2 ]

-- -- ell + C12 ---- 13 Uxx (129)

E0E r E0E r C33 .J

This field is about 6x106 V/cm. equations 128 and 129 are found in the literature, (refs. 5, 27, and 28),

and many have a factor of two before the c_3/c33 term. I interpret them as errors by attempting to add

u_ and u_ together as the "net strain" in the basal plane. For our case ofAlx Gal_xN, some authors

multiply the P= in equation (128) by the mole fraction x to obtain the effective value.

With the preceding concepts in place, we may introduce the entire set of equations to be solved in the

GaN/A1GaN system. The first group consists of the mechanical equations of equilibrium (ref. 29)

OTyx OT._

_x + Oy + Oz -qn-_x=

0

OTxy + _t_yy + t_]"=y _qn_d_ =O

Ox Oy Oz

OTx=OT,,x

Ox + --_-- + --0_-z"- q" -0--zz=0

(130)

The last term in each equation is the "body force," which we assume is the charge density times the

electric field. Next, the electromechanical constitutive equations are

(Yxx

(Yyy

r.x

Cll

C12

C13

= 0

0

0

C12 C13 0 0 0

ell C13 0 0 0

C13 C33 0 0 0

0 0 C44 0 0

0 0 0 C44 0

0 0 0 0 C11 - C12

2

Exx

Eyy

Yy=

_Y_yJ

0 0 e31 -

0 0 e31

0 0 e33

0 e15 0

e15 0 0

0 0 0

I ]

Ey

E:

(131)

NASA/TM--2003-211983 22
[ox] Dy

D:

=E 0 [Ka0 0

0

Kll

0 K33

0 E

E_

v +

e31 e3100e3300 el500

1

Exx

0 0

Ezz

o _'y:J

0

_Yxy

Ou 8v 8w

=_-xx, =_-y),, E

8x 8z ""

8v 8w 8u Ow 8u o%,

-- + --Oy = Y"v:" --Oz + --Ox = Y:X"Sz-_ --Ox+= YxY

which couple the stress, strain, electric field, and electric displacement. The latter expressions are the

compatibility expressions (ref. 29) relating strains to the material displacements. Notice the notation for

mechanical displacements uses u, v and w; rather than just u with a subscript. The last expressions to be

used are the continuity, Gauss, and current equations.

a, aj_+aJ_ aJ. +---= = 0 EX _--m

G - a, J_=_,x.(_)Ex+D. a.(___))

oy ax

On(_ )

E. =_L* Jy = g:(_)ey + D.

" az ay

J. =....(_)E:+z_.o.(_)

" 8z

(132)

(133)

(134)

019x + ODy 019. + -----_"= qn(_) (135)

Ox oy az

_ en = n(_)= n(x, y, z) _mm3 (136)

where we have assumed just electrons are present. Similar terms for holes may be added for the most

general case. Notice, we have 20 equations in 20 unknowns. This is an extension of the general elasticity

problem to include a piezoelectric semiconductor. In elasticity theory, one has 15 equations in

15 unknowns.

The total set of equations has been provided for the sake of completeness. In principle they would

be solved along with Poisson's and Schrodinger's equations in the Poisson-Schrodinger solver which

presently exists in the software package.

NASA/TM--2003-211983 23
3. Spontaneous Polarization

There is some uncertainty about spontaneous polarization, PSPON, (see refs. 30 to 41). Bernardini

et al., have performed ab initio calculations for spontaneous polarization in nitrides (refs. 30 to 34).

Reference 34 discusses some of the controversial issues on the subject. On page 912 of (ref. 37)we quote

"The latter distortion--henceforth called internal strain--determines a preferred polarity of the tetragonal

axis and is responsible for the occurrence of spontaneous polarization." I interpret the discussion to mean

that there is an alteration of the position of atoms in a cell, which is deemed an internal strain (not

apparent macroscopically), which causes the spontaneous polarization. From reference 38, "The so-called

pyroelectric fields then arise from the temperature dependence of the spontaneous polarization when the

temperature is changed." Figure 1 in reference 39 shows both a macroscopic strain _ and an internal

strain u. Apparently u may exist when _ = 0, so a polarization may exist due solely to u; which would be

a (macroscopically) strain-free polarization (spontaneous). A procedure to measure PSPON is given in

reference 41.

PsPoN is screened by charges adsorbed onto the surfaces of the material, as well as being screened

by free charge in the bulk. It is assumed (or defined) to be either uniform or zero in the bulk. One paper,

not referenced here, stated it was zero in the bulk. Its presence is known only at interfaces with other

materials (air being one of them). Pyroelectricity is the production of an electric field between two

surfaces of the material held at different temperatures.

PI'VRO = P A Temperature (137)

=(Px_x +pyay + (138)

For GaN

Px = Py = 104 V basal plane

m.K

p: = 7 x 105 V along the c-axis

m.K

F

Both Wz AlxGal_xN and GaN possess PsPoN, and at their interface a positive bound surface charge exists

(cubic GaN is piezoelectric but does not possess spontaneous polarization). This assumes a Ga face (only

case we consider). This immobile charge is assumed (calculated) to reside in a layer 3 to 4 ,_ thick, (I will

draw it in the AIxGal__N region for clarity). Figure 3 depicts two ideal samples with their bound

polarization charges (no screening).

NASA/TM--2003-211983 24
I PSPON AIN

i (0001)

C-axis

+ + + + + + + +

PSPON(AIN)= -8)<10 .2 C/m2az I PSPON

GaN

÷ ÷ ÷

PSPON(GaN)= _2.9×10-2 C/m2 az

Figure 3•mldeal samples of AIN and GaN illustratingspontaneouspolarizations.

Note the z-axis and the c-axis are collinear. This relationship was assumed by Bernardini et al., and is

important with respect to algebraic signs. Recall that the polarization vector P is defined to originate on

negative bound charge and terminate on positive bound charge. The materials are assumed to grow from

the bottom and increase in thickness as one moves up the c-axis (0001). From a simple charge superposition argument, a net positive bound charge density a(SURF) exists at the surface• Analytically, we

have

t_SLrRF = (8- 2•9)x 10-2 C/m 2 = 5•1 x 10-2 C/m 2

= 3.19 x 1013 e/cm 2

(139)

Recall that at a boundary (see fig. 4)

Pn2 - Pnl =-o( su#ac_ •Bound @

Figure4.--Definition for bound surfacecharge at

interface between two polarized dielectrics.

Here P2 "-_= Pn2....

For our case

Pn2 = _ •n=(-8xl0-2a:)•a- - =-8x10-2 C/m2

Pr_ = fill "n=(- 2•9× 10-2a:)•a= =-2.9x 10-2 C/m2

• Pn2 -Pr_ = -8x10-2 + 2-9x10-2 = -5"1×10-2 C/m2

• -5.1 × 10-2 C/m2 _,_(SUR_ACE)

= _"BOUND

• 9_o_CE)_c(SURF)= 5.1× 10 -2 C/m 2

(140)

NASA/TM--2003-211983 25
Figure 5 summarizes the situation.

I

I

, AIN

--[ ...... [-- GaN

I

" I

8

_ I#SPONIXIO-2

AIN or

AIxGal_x N

GaN

PSPON

I f

\

\ // t

!

/

/t

i I

I

/

i/

//

/

i

Detail

PSPON

0001

T

3to4A

.... r Ga-Face

l GaN

Figure 5.--Ideal situation at AIN-GaN interface. Positive surface of AIN is reduced by negative charge

on GaN surface, Resultis a net positivecharge at interface,

Notice that the bound surface charge density is of the order of magnitude of the surface density of

electrons in HEMT structures.

Several papers, and in particular reference 40, have stated that the ab initio values should be reduced

by at least a factor of 2. One reason put forth is that there may exist inversion domains at the interface,

as well as islands of zincblende (cubic), observed experimentally, dispersed throughout the Wz phases.

Other causes can of course be put forth; impurities and surface defects. It is also assumed (stated in one

paper) that

V "PsPoN = 0 (141)

as no bound volume charge density exists. This assumption seems reasonable if the internal strain is

uniform and hence has no divergence. However, note that this assumption may be subject to future

criticism.

4. Poisson-Sehrodinger Solver

Figure 6 gives the coarse, initial band diagram in the region near the heterojunction. A key parameter

is the step height AEc at the interface. Anderson's model is presently used in some sottware, which gives

the step the value

NASA/TM--2003-211983 26
AIxGal_x N GaN

Z

E C

Figure 6.mldeal triangular well for confinement of 2DEG at interface

of AIxGa l_x N and GaN layers. Conduction band edges are denoted

as E c.

AEc = _ AlxGa__xN --_,GaN (142)

where ;( is the electron aff'mity. We will discuss alternative ways to estimate AEc in later sections. The

triangular well is composed of the step and the sloping side. The linear variation means a constant electric

field is present. Apparently the establishment of the field comes about as follows. At the heterojunction,

the step in Ec along with the position of the Fermi level causes electrons to accumulate in the notch. The

region into the bulk on the left is therefore depleted, and the existing donors establish the field. The field

is modeled by the standard depletion approximation

E(o):q [ND :'qNs ] (143)

where

No = depletion region sheet density

Ns = free carrier sheet density

11 = a weighting factor (-0.5 to 0.8)

GaAs value at the heterojunction is

E(0)= 2.39 x 106 V/m (144)

For our case of AlxGal_xN/GaN, the situation is apparently quite different. From Sacconi (ref. 42), if

we initially neglect all polarization, then the step height and Fermi levels (for typical doping) are not

sufficient for a notch to form. Then electrons do not accumulate and create the 2DEG. After adding the

spontaneous and piezoelectric factors, the notch forms, and the 2DEG is present. Apparently a depletion

region does not form, as the polarization charge is sufficient to establish the field. If one does form,

perhaps it is swamped by the polarization charge. One may look at the situation as follows: the 2DEG

NASAfI'M---2003-211983 27
forms to screen the bound spontaneous and piezoelectric charge residing at the interface. This introduces

the question, "where do the electrons come from that establish the 2DEG?" There are valid reasons that

state they cannot come from the bulk GuN, so apparently they come from donors somewhere in the

AlxGaI__,N region. This is not a trivial issue, as charge neutrality must hold, but the bookkeeping is not

generally agreed upon in the literature. Instead of using the field expression developed from the depletion

approximation, authors assume it should be represented by equation (129) in section 2,

[ 2] E. 2d31 c13

-- Cll +C12 ---- U_r

E 0 E r C33

(145)

If the strain in the AlxGal__N is tensile Uxx> 0, then the field is oriented toward the bulk just as in the

case of the depletion approximation. Figure 7 summarizes the discussion.

AIGaAs

Donors --_

r -X .....

I I

I i

I---I _-- 2DEG

(a)

Depletion

model

AIxGal_x N

Polarization r_

charge _ _ J+l II

(b)

Figure 7.m2DEG formation. (a) By intentional doping. (b) By polarization charge.

Poissou's Equation

Without spontaneous or piezoelectric polarization, Poisson's equation is

vErb= P- q[N/_-N]+N +-N t-+p-n]

E E

(146)

where No,+ N -A , N _+, Nt-- are ionized donor/acceptor densities, and ionized donor/acceptor trap densities,

respectively. Also

E=EOE r

(147)

q=lel= +l.6×lO-19C

The relative dielectric constant _ r considers the standard electronic plus ionic polarizabilities. When

both spontaneous and piezoelectric polarizations are included, the form is as follows.

Start with

V-D =p

v.( o

NASA/TM--2003-211983 28
NeglectV e for simplicity

then

v (e0_,E+p_._,o+_o_)=

V "[E (-- V(_)]+ V" (PPIEZO + PSPON )= [3

V _--_ 0

v_=--_+l(v._zo+v.Go_)

E E

= __P_ ! (Pb (PIEZO) + Pb (SPON))

E

q + +p-n] V2dp=--_[ND-N] + Nt-N t

1 [Pb(I'IEZO) + Pb(SPON)]

E

We see that the potential now also depends on the divergence of the polarizations. Since

-P=d:T =d:6

(148)

(149)

(150)

(151a)

or

ff=e:S =e:ff (151b)

and the components of P are linearly related to the components of the stress or strain tensors; an accurate

analytical model for the strain variation with position is mandatory.

Sehrodinger Equation

Now we consider the effect of the polarization on the energy levels for the 2DEG. Assume the wave

functions have the form

V j(x,Y,Z)=¢i(g) ejOzejk'x+jk2y

then write Schrodinger's equation as

h22 Oza( lm, O_')Oz+h2(k}+k2)2m* _,+V*(z)_ i=E_,

(152)

(153)

NASA/TM--2003-211983 29
Weusean asterisk on the potential energy (PE) term V*(z), to denote that it represents the total potential

energy field that establishes the well. The potential energy near the well is given in figure 8. The figure

needs to be discussed in some detail. First, recall the relationships between the electric field E(z), the

electrostatic potential dp(z) and the electron potential energy band diagram (ref. 43)

Ec(AIxGal-xN) __

Ec(GaN)

I

I

Figure 8.--First-order approximationof potential energy well at heterojunction.

I dE, (154)

E(z)=-_dp(z)= q dz

Here E(z) is the magnitude of the component of electric field of interest, and E, the energy band edge.

Here E, is a convenient reference energy level, and q is positive as given in equation (147). If only

electrostatic forces are considered (the normal case in semiconductors), then the energy band edges take

the shape of d?(z), with the negative sign treated appropriately. For heterojunctions, the band edges and

d?(z) do not track one another. This is due to the differences in electron affmities on either side, and how

they change with position as one moves away from the interface. An electric field is generated with a

magnitude of dz(z)/dz. The variation of Z(z) is difficult to determine thus some standard approximations

have been used (ref. 44). Another force field is due to the spatial variation of the density of states (see

refs. 45 to 47 for more information). The portion of the well near the notch is assumed to model the entire

(or total) potential energy, and as one moves away, the field decreases and reaches fiat-band deep into the

GaN region. Most authors express V*(z) as

V*(z)=-ed_(z)+ AEcU(z ) (155)

where U(z) is the unit step function. These two terms are sufficient to defme the well. The first should

include all charges in the vicinity, while the second is of a quantum nature. Another quantum term is the

exchange-correlation potential Vxc, but it is generally small, and may be neglected. Not all of the charge

has been included in equations 146 or 150; namely, that which forms dangling bonds and interface

dipoles. Some authors have also included the classic image-force potential. The exchange-correlation is

(ref. 48)

Vxc =- l+0.7734xln l+x)l[,nCXrs) y

(156)

NASA/TM--2003-211983 30
where

C/

4x c o rO_2

m*q 2

(157)

_:-er (GaN)= er (AlxGa,-x N) (158)

r s =rs(Z)=I-_-(G*_FI(Z)1-1/3

(159)

x(z)= z)

21

(]60)

(161)

(162)

The classic image potential is

VIM --

K c _ Kb )q2

16x_0 _Cc(_Cc+ Kt,)l_l

(163)

where z is in the channel (GaN). The subscripts c and b stand for channel and barrier (A1GaN) regions.

VIM=

K c _ Kb )q2

16x_ 0 Kc(_ c +Kb)(-z)

(164)

where z is in the barrier (z > 0 ). In practice V_has been scaled by factors of 0.43, 0.53, and 0.75 for

better fits to measured results. The dangling bonds have been represented by the ad hoc term (ref. 49)

q2

Vat, - (z < 0, channel) (165)

4n c o_c dB

where ec = _:c. The value for ds is as.vr3/4 where as is the lattice constant in the barrier. In the barrier

q2,

Vat, - (z > 0, barrier) (166)

4x _0_b d B

NASAfI'M--2003-211983 31
whered c = ac _ / 4 where ac is the lattice constant in the channel (GaN region, z < 0). Apparently these

are special cases of a more general form which varies with position. Assume

Vab (Z): n a q2

4n_O_r (Zo +lzl) (167)

where na is the number of dangling bonds at the interface.

Since the junction is strained, the band gaps on either side will be perturbed via the deformation

potential. Write the perturbation as (ref. 50)

%,(z)+ u==(z)] (168)

where the negative sign reflects the experimental fact that hydrostatic compression (u. < O) causes the

gap to widen. The deformation potential is ac. Therefore, from the existing literature, the most general

expression for V* (z) is

V* (z) = Es(z)+ AEg(z)+ AEcu(z)-er_(z)+ Vxc(Z)+ VIM(Z)+ Vas(z ) (169)

where Es(z) is the conduction band edge (ref. 50). This term is assumed piecewise constant on either side

of the junction, and differs by the step height AEc. In effect, it reflects the difference in electron aff'mities

between the two sides. The choice of assuming it is constant with position neglects the field due to dx/dz

(ref. 44). Only one paper included Vdband many neglect Vx_and V_ as they are usually small with respect

to AEc and _b(z). Reference 50 is the only one to my knowledge to include Es(z) and AEg in V*(z).

The previous paragraphs assumed no interface charge was present, and that _b(z)is continuous across

the interface. However, dipoles may exist at the junction (refs. 46, 51 to 58), and depending on the

horizontal mesh size in a simulation, their effects should be included. Consider a 2D dipole formed by

nearly adjacent layers of positive and negative charge as shown in figure 9.

(_(z)

+ --

+ --

+ --

+ --

Figure 9.mDiscontinuity in potential

due to dipole (double layer).

NASA/TM--2003-211983 32
The step change in dp(z)is (ref. 59)

od

A_b= -- (170)

E o

where o = sheet charge density and d = separation between the sheets.

If the horizontal step size in a simulation is greater than at, then the dipole charge would be

overlooked. Equation (170) allows its inclusion without an exceedingly small step (d is on the order of

5 A). Some heterojunctions have negligible interface dipoles (ref. 58), but this may not be the case for

the A1GaN/GaN system. Recall that a dipole (or double layer) is an assembly of charge as indicated

above and d is less than other dimensions in the problem. Notice dipoles can add or subtract from AEc

depending on their polarity. This may be used to adjust AEc for a given simulation. From experiments,

AEc appears to vary from about 0.6 to 2.4 eV. Some of this variation may be due to variations in )_(z), or

doping, but a portion may indeed be due to dipoles.

At the beginning of this section we showed the change in Poisson's equation when polarization was

included; now we show how it modifies V*(z). Rewrite equation (169):

V*(z)= Es(z)+ AEg(z)+ AEcU(z)-ed_n(z)+ Vxc(z)+ VIM(z)+ Vdb(z)+ VDIPOLE(Z)--eUp(Z ) (171)

where we have added two more terms; VomoLe and -eUp(Z), the dipole and polarization potentials

respectively. It is very important to notice that the -_b,(z) now given the subscript n, in equation 171 is

that given by equation 146, and not that given in equation 150. We do this to conceptually separate the

effects of the normal charges included in Poisson's right hand side, and the image, dangling bond, dipole,

and polarization terms.

The potential energy for a dipole is

VDIPOLE (Z) = --e_DIPOL E (172)

where (ref. 60)

_)DIPOLE -- _/f2 (173)

47_ E0E r

and _ = OTOT/d,_

(3rTOT = charge on the positive side of the dipole

de = separation between sheets

f2 = the solid angle subtended by the patch at the observation point (see fig. 10)

Before we introduce polarization effects, recall the definition of the electrostatic potential

(174)

NASA/TM--2003-211983 33
O,oo,e,oato,,

Figure lO._Geometry for determining potential of finite patch of double

layer(dipole).

Observation point

When a polarization field exists, its potential is (ref. 61)

Up(_)= 1 P(U). (_- U)dv'

4xe0 ]_' 7UI _

(175)

which may be written as

v'. Pdv'

up(F)=4_% 4_'___., I 4rc-_0

4x c 0 IF - F'I _-4x _-_---0-IF- F'I

(176)

where

Psb = P" h = Pn (178)

9 vb = -V. P (179)

which are the bound surface and volume charge densities, respectively. The surface integrals are taken

over surfaces bounding the polarization distribution.

Now we attempt to ascertain the effects Up(F ) may have on the final energy levels for the 2DEG via

V*(z). Apparently, the spontaneous polarization has the distribution of surface bound charge density as

shown in figure 11.

The values of the charges are (from left to right)

C ,5x10_ 2 C , C

-8×10-2 m--T _i- and 2"9 ×10-2 m-T

These charges are Psb and would contribute to Up(F ) by the first term in equation 177. Assuming

V. PSPON = 0, we have no further contributions from PSPON• We neglect any pyroelectric terms, as no

temperature difference can exist across 4 A.

For the piezoelectric term, we have

Pemzo =d:6=e:S 080)

where the stress/strain may be decomposed into residual, thermal, and lattice mismatch components.

It can also include stresses due to metal bonding pads. These terms will be dealt with in a later section.

NASA/TM--2003-211983 34
Fornow,wecansaythatequation171givestheproperpotentialenergyfieldandonemustnext

evaluateup( F).

AIxGal_xN/GaN GaN Substrate

z .11 I

I

-I I

i-I

I I

i_1I - I

-- I-ql--

I -I

I I

i._ /

I+

I +

I+

I

i +

I+

I+

I

~2A

I

n.4_ -4 A

I+

I+

I

~2b,

Figure11.---Assumed bound charge densityat three mainsurfacesof AIxGal_xN/GaNsubstrate HEMT. Charge

is assumed to be developed by spontaneouspolarizationin AIxGal_xN and GaN layers.

5. Material Configuration

Figure 12 gives the general layer structure for our device. Our primary goal is to determine the strain

at all parts of the structure. Starting at the nucleation layer we treat GaN/A1N/AI203. The A1N adheres to

the sapphire and then allows the GaN to form on its surface. The A1N cracks and forms crystallites

(sometimes) during its deposition. The GaN forms threading defects as it nucleates, and these threads

spiral up through the material (never stopping) and continue to the A1GaN layer. It is unknown if new

ones are initiated and propagate through the A1GaN. The defects have a density between 108 and

1012 cm -2. The sheet value of 10]°cm -2 corresponds to 2x 1017cm-3 defects (assuming a sheet 5 A thick).

Figure 13 gives a top view of the 2DEG and the negatively charged threads. The field of a charged thread

varies from about 120 V/m to zero in a radius of 10 btrn. Perhaps they should be modeled as filled traps;

however, there is no obvious way to determine the equivalent trap density.

Eastman's research group at Cornell University state that a Ga face growth along the c-axis is

guaranteed if an A1N buffer (nucleation) is used. The GaN is usually in compression and the A1203 in

tension. Figure 14 shows the stress at the interface.

We neglect the fact that sapphire and SiC are piezoelectric, and only consider the strain in the GaN.

First of all, the bound positive charge that supports the spontaneous polarization should exist at this

interface. Most authors neglect this fact, and implicitly assume the charge is neutralized by immobile

negative charge. This may be partially true, but some have noted a conducting channel at this interface.

Eastman et al., observe "leakage" at this plane when using SiC and assume it starts to leak due to its

relatively small band gap. In our modeling we may assume complete or partial neutralization with fixed

charge with the remainder composed of mobile electrons.

Often the strain at the GaN/sapphire interface is assumed to be due to lattice mismatch and/or TEC

mismatch. Since the A1N deforms to relieve the -16 percent lattice mismatch between GaN and sapphire,

it is questionable if one can accurately calculate the stress or strain using the misfit parameter method.

NASA/TM--2003-211983 35
Instead, we will use measured in-plane stress and strain. For reference, the misfit parameter for A1N on

sapphire is (ref. 5)

Gate

.s

,.-'""" Si3N4 passivation

AIxGal_x N

Threading (," _ ( GaN

defects--_"_"\I" -" - "- "" "_

0.05 I_m _ AIN nucleation layer

&

T

C-axis

(0001)

z

Substrate: Sapphire (AI203)

SiC -250 I_m

Figure 12.--HEMT structure showing nearly all relevant portions.

x

2DEG

t" -, ...... -z-- Depleted region

/ _....... / in 2DEG due to

I _ /

! | /

/ thread charge

\ \

Figure 13.reView looking toward GaN region from

AIxGal_xN layer. 2DEG is assumed to be uniform

with depleted regions around negatively charged

threading defects.

Compression ¢ _ Tension

Stress

neutral

plane -_

\

/

GaN

I

I

Figure 14.mApproximate stress profile near

GaN/sapphire interface.

AI203

NASA/TM--2003-211983 36
a 0 A1N = 3.112/_ fro= xf_(3"112)- 4"7589

3.112 (181)

a 0 A1203 = 4.7589 A = 20.3 percent

The 43 factor comes about because sapphire is rhombohedral, and the hexagons for the basal plane

are rotated 30 °. For GaN on AIN we find

3.1892-3.112

fm= = 2.4 percent

3.1892 (182)

E_ = -fro = -2.4 percent

and finally for GaN directly on sapphire we have (c-plane)

f,, _ x/-3(3.1892)- 4.7589 = 24 percent

3.1892

or (183)

fm = ,f3(3.1892)- 4.7589 = 16 percent

4.7589

This latter number is the one most ot_en quoted, however, it is not really appropriate (ref. 5). These are

the results from measurements:

For GaN, the in-plane stress (refs. 62 to 65) Oxx = Cyy = Oll

0.2 GPa < loll] _< 0.91GPa

From reference 15, ,16111,= 0.66 + 0.1 GPa. The lowest and highest values found in literature were

[O'II[LoW = 0.18 GPa, [oI1[_GH =0.91 GPa

The in-plane strain is u_ = Uyy = Ull ------Ell

lO-4-<lu=l-<5×10-3

The ranges found, including highest and lowest, are

0.058percent-<lu==l_<0.125percent, lU=lLOW= 0.016percent,

The calculated thermal stresses are GaN/sapphire Act = 1.71 x 10"5/1(

300

OTH = ActAT - 0.7 GPa 1-0.25

U.._ [HIGH = 0.57 percent

AT-1030 K

(184)

NASA/TM---2003-211983 37
Recallequation50of section 1, and repeated after equation 79 reads

E

Txx =---S.. 2v --

(185)

Using the measured (_11 and letting E = 300 GPa, we fmd

e..: = 0.33 percent

which is within the experimental limits.

For AlxGal_xN, the measured stress and strains are

0.8 GPa _<Crxx< 1.4 GPa

0.016GPa _<Uxx <0.026GPa

0.0012 GPa _<u.. < 0.0057 GPa

Moving up through the structure, we may state that the GaN is seldom intentionally doped, however,

the minimum n-type carrier density is in the low 1017cm -3 range. We have discussed the AlxGal_xN/GaN

interface in section 4, and in general, the AI_Ga1__N is in tensile strain. Most often, authors have assumed

the GaN to be fully relaxed, which we will show is not true. The interface between the AlxGal_xN and the

Si3N4 passivation layer has not been reported on extensively, however, it is assumed to play an integral

role in the problems presently plaguing the HEMT. It is assumed that the negative charge that supports

the spontaneous polarization is neutralized by adsorbed ions. It is also assumed that donors may be

present that actually deliver the charge that forms the 2DEG. There is disagreement as to the source of the

2DEG charge. This interface is assumed by some to contain traps that interact with the gate, which causes

the unexplained current collapse when radio frequency is applied. Some attribute the collapse phenomena

to dielectric loss and bulk traps in the Si3N4.

Stress/Strain With Position

This is a most important topic, but unfortunately, not discussed much in the literature. Only one

reference, Demarest (ref. 62), used the standard software package for stress/strain from ANSYS. The

paper shows the strain variation around metal pads on SiO2 layers. It should be mentioned that when GaN

is grown on sapphire and SiC with A1N nucleation layers, polymorphs of Wz and zincblende regions may

coexist. In some cases as much as 25 percent may be zincblende (on an Si wafer). During growth the GaN

is in tension, but during cooling it goes into compression in the basal plane. Misfits form during the entire

process, so the state of strain is highly variable from sample to sample due primarily to the different

temperatures for growth, cooldown rates, and other factors.

For reference, the basal plane strains at an interface are

e_=_vy = +fro = ± a° -astn3 (186)

aSUB

where astm is the lattice constant in the layer in which the strain is sought. The corresponding strain along

the c-axis is

NASA/TM--2003-211983 38
_--=- _. 2 --_xx cl3 (187)

C33

For GaN, c,l, and a'[', that is, as c decreases, a increases. The net strain is the combination of lattice

mismatch, thermal mismatch (TEC), and residual strain. The latter often dominates, so we should rely on

measured results as much as possible. For thin layers (up to about 2 _tm), we may assume the stress

is linear (ref. 63). Figure 15 depicts the measured stress for films of GaN on 6H-SiC (by MOCVD). In

the region near 2 _tm the material apparently relaxes and remains in horizontal tension for further growth.

In figure 16 from (ref. 64), which shows the biaxial stress o_ = o_v = o 11 to be negative

(compression).

0o6 --

IX

0,5 n

0.4--

0.3 --

0.2 --

/

/

0.1 -- /

/

/

/

0.0 _"

s

-0.1 :-_'_

--0.2

/ \

/

/

/ \

/

/ \

/

/

I I I I

0 1 2 3 4

t, _tm

Figure 15.--Measured stress in GaN epilayer grown on 6H-SiC by MOCVD. Note, stressfree region occurs near 0.5 I_m thickness (see ref. 66).

0.5

0.4

0 0.3

-_ 0.2

0.1

0.0 I I I

10 20 30

t, I_m

Figure 16.--Reduction in stress with thickness of GaN on sapphire.

Here (rxx = ¢ryy _--_rll which is negative (film in compression).

Absolute value is plotted.

This trend is common; GaN on sapphire is often in compression while on SiC it is in tension. This,

however, is not necessary, and the films may be in either state of strain on both substrates. It all depends

on the mode, and temperature of growth, the nucleation layer preparation, the thickness of both the

nucleation layer and the GaN layer, and the crystal orientation of the substrate and its surface condition.

For GaN on sapphire (A1203), it appears that the strain decreases with thickness to some power, or

almost linearly, or exponentially from the AIN layer (refs. 14, 65, 66). Table X gives the vertical strain

with thickness t in _tm (ref. 65).

NASA/TM--2003-211983 39
TABLE X.--STRAIN ALONG THE c-AXIS

VERSUS FILM THICKNESS

t, _tm Strain Fit equation result

1 12.3 12.3

5 7.3 7.3

20 4.6 4.66

60 1.5 3.26

The fit equation is

_: =[12.3×10-4]t -°.32416

Table XI gives the basal plane compressive stress versus thickness (ref. 14), and it looks almost

linear.

TABLE XI.--MEASURED STRESS VERSUS

FILM TH]CKNESS FROM REF. 14

_m

1

Compressive stress, GPA

-0.64

10 -0.59

20 -0.56

50 -0.47

(188)

The paper by Detchprohm (ref. 67) shows the vertical (c-axis) strain varies exponentially from the

nucleation surface. The strain does not relax until the layer is about 100 lam thick. This data was taken

with MOCVD and HVPE grown samples (see fig. 17).

ff

{5

5.191

5.185

5.179 I I I I I

0 50 100 150 200 250

t, _tm

Figure 17.--Relaxation of c-axis strain for GaN on sapphire.

NASA/TM--2003-211983 40
The c-axis lattice constant was fitted to an exponential

c (z) = 5.185 + (0.006)e -° 0692

then the strain with position is

+)- 5.185

"" 5.185

0.006 -0.0692

E..-- e

-" 5.185

c in A, z in B (189)

(190)

For Si3N4

Ac

c 2s13

Aa s11 +s12

a

- 1.2GPa < oll _<-0.02 GPa

(191)

6. Band-Gap Issues

The band diagram in figure 18 gives the ideal charge distributions at the Schottky barrier, the

AIGaN/GaN interface, and the nucleation/substrate plane.

We show positive charge on the Schottky barrier metallization that accumulates to neutralize the

negative charge of the spontaneous polarization. The positive polarization charge and the 2DEG are

shown at the heterojunction, and the spontaneous charge with some form of neutralization is shown at

the nucleation layer. This is the minimum charge distribution for the device. Figure 19 shows the charge

distributions for dipoles and dangling bonds that may exist at the AlxGa___N/GaN interface.

+ - - ÷ Substrate

(rPSPON + - (r2DEG

e#Pb + -

+

+

EF

_Ec E1

Figure 18.mMinimum charge distribution for an AIxGal_x N GaN HEMT. Charge at Schottky barrier,

heterojunction,and GaN/substrate interfaceshown left to right.

NASA/TM--2003-211983 41
I

Dipole that increases&Ec

Dipole that decreases &Ec

_ Dangling bonds

1 angling bonds

Figure 19.mDetail of possible charge

conditionsat heterojunction.

The charge state in the AlxGal_xN and the Si3N4 passivation layer are not considered here. The

AlxGal_xN barrier contains A1, but the total charge there is unknown. Often an Si donor layer is placed

a few hundred angstroms from the interface to provide the 2DEG carriers. This doping apparently has

a strong effect on the strain state of the barrier, which is a significant source of piezoelectric charge.

We can omit the step discontinuity in _ given in equation 170 of section 4, by combining it with the

step; that is, eA_b + AEc = AE'c . This keeps ¢ continuous, if that causes problems with the software

solvers. While this is convenient, it eliminates the force field of the dipole in the Schrodinger equation.

Recall that the step AEc is due to the absence of available states in the barrier region, and not some form

of repulsive field due to charge. The dangling bonds do not cause a jump in dpand pose no problem.

The range of AEc reported in the literature is about

0.17eV < AE c < 1.8eV (192)

Anderson's model assumes

AEc -- )_AlxGal_xN-- )_GaN (193)

However the uncertainties in Z are very large (refs. 68 and 69)

For GaN 2.1 < Z < 4.26 eV (194)

For AlxGal_x N Z(x) = 3- 4x for x < 0.75 (195)

or = 3.25- 2.95x x < 0.75

NASA/TM--2003-211983 42
Thestepsinthebandedgeshavebeenreportedtobeasfollows

AEv = 0.78x eV

= 0.81 (x = 1)and no strain

also 0.2 < AEv < 1.4 eV

For the nucleation sequence of GaN/A1N/A1203

AE v = 0.57 + 0.22 eV for 5 to 20 A layers

= 0.7 + 0.24 eV

= 0.8 + 0.3 eV

From these and knowing the band gaps, the step in Ec is known. For the following range

0.15 < AE c < 0.56 eV

the step is

(196)

(197)

AEc (x)= 0.75lEg (AlxGal_x N )- Eg (GaN)J- 1.96x (198)

For strain-free GaN, the following have been reported (see ref. 70)

AEv=1.36+0.07 AEc - 52 or AEc - 65 (199)

AEv 48 AE v 20

Thus, AEc is a fitting factor, and unfortunately it is one of the dominant terms in V*(z). Next we consider

the change in Eg(z) with strain. We write

Eg (STRAINED)= Eg (BULK)+ AEg (200)

where E s (bulk) means the unstrained/undoped material. The change may be expressed as

AEg = _" Do. e O. (201)

where D o is the deformation tensor and e¢ is the strain tensor. For GaN

D1 = -0.88, D2 = 4.52, D3 = 5.4, D4 = -2.7, D_ = -2.85, D6 = 3.82 eV (202)

A simpler form using in-plane stress a_ = Cyyis

AEg OEG (203)

= 0_1--_ al 1

where OEg/&_l_ = 21 to 27 meV/GPa

= 42 meV/GPa for hydrostatic stress (and Si doped)

NASA/TM--2003-211983 43
Sometimesone fmds (ref. 71)

dEe' - 42 meV/GPa

d Pressure

Other forms using both in-plane and c-axis strains are as follows:

Eg (STRAINED)

where

D3 (eV)

D4 (eV)

Another form from Yang (ref. 72) is

= Eg(RELAXED)+ C3Eg _..

E...z "

Ogg

- D 3 - D 4 c33

E:: C13

GaN A1N

5.8 8.84

-3.25 -3.92

AEg = AEHYDR O + z_kEsHEA R

AEHYDR0 =-a[e_ + _yy +2 ez: ]

AEsrma R =-b [Cx_ +_yy -2%:]

where a = -8.16 eV, and b = 3.71 eV. In some cases the hydrostatic and uniaxial strains are given by

1

- 11)

along [0001]

Another form is

AEg = C E ::

where C was determined by experiment to be 12 eV.

(204)

(205)

(206)

(207a)

(207b)

(208)

(209)

(210)

NASA/TM--2003-211983 44
Finally, one finds in terms of stress

in terms of strain

OEg = 10.1-15.4 eV

OEg _ -6.1 eV EII_ 0.0032

= -8.2 to - 13 eV (TENSILE)

= -16 eV el l > 0.0032

(211)

(212)

The expression for AEg in equation 168 in section 4 is that due to hydrostatic stress only.

In figure 20 (based on a figure in the literature); we show the energy gap of GaN with in-plane strain.

Some authors claim that the curve is symmetrical, that is, the gap increases for both compression and

tension.

e_

e-"

I,kl

4.0 --

3.8 --

3.6 --

3.4

0.00

f

/

f

J

/ I

t

I

I

/

,,o 1

/

I

.s

11

I

I

s 1

1

1

I

I I I I I

0.01 0.02 0.03 0.04 0.05

qEII

Figure 20.--Vadation of bandgap of GaN with basal plane strainell = exx = eyy

(see ref. 73).

Now we consider the change in band gaps with temperature (refs. 73 to 75).

For GaN

T 2

Eg = 3.503 + 5.08 x 10 -4 eV

T - 996 K

T 2

10 -4

or --/_g = 3.509 - 7.32 x

eV

T + 700 K

or Eg =Eg(T=O)-8.32xlO -4 T2

T+836K

eV

(213)

NASA/TM--2003-211983 45
ForAIN

Eg = 6.118-1.8x 10-3

T 2

T + 1462 K

eV

Eg(x,T)=(3.42x +0.068)-8.73×10 -4

T 2

T+830K

eV

(214)

The band gap of AlxGal_xN is assumed to vary with mole-fraction x as

Eg (x): x Eg (AIN) + (1 - x)Eg (GaN)-bx(1- x) (215)

where the bowing parameter "b" varies from 0.53 to 2.6 eV in the literature. Some have even stated it is 0,

+0.2, or +0.3 eV. The following forms are found.

Eg = 3.43 + 2x + 0.7x 2

(216)

= 3.43 + 2.56x

7. Thermal Conductivity

Let K(To) be the thermal conductivity at To (normally 20 °C). Then the variation with temperature is

K(T)= K(T0 cmwK

TABLE XII.--SUMMARIZES THE VALUES

FOR OUR MATERIALS

Material K r

Sapphire 0.25 to 0.50 0.544

SiC 3.3 SI, 0.5 0.524

GaN 1.3 to 1.7

Si3N4 0.37

A1N 2.0

Si 2.0

Some experimental results for AIN are below.

TABLE XIII.--THERMAL CONDUCTIVITY

VARIATION WITH TEMPERATURE FOR AIN

T Experimental value for K

300 2.85

400 1.80

600 0.96

1000 0.48

NASA/TM--2003-211983 46
8. Orders of Magnitude of Terms and Approximations

The exact form for the potential of a rectangle of charge is given in the appendix. It is tedious so we

will assume the potential of a disc of charge is a reasonable approximation. The potential of a disc of

radius "a" is

%(z)- °° I_a 2 z 2 ] (218) 2%% + -z2

where er0is the surface charge density. Assume the source-drain spacing is 2 _tm, then a = 1 _tm. For the

spontaneous polarization at the interface, the effective charge density is

o0 = 5.1x10_ 2 C

m2 (219)

Evaluate at 70 A from the surface

_b(70A) = 320V (220)

This would be the value if no screening existed due to mobile charge. We will assume the screening can

be modeled using the classic Debye-screening factor

fDS = e-= / eD (221)

where eD is the Debye length

_D = - f-_-_

"_29oq (222)

C

P0 =n0q =1.6×106 m3

The densities in the 2DEG are on the order of 1019 cm -3, which yields a length of about 8.25 A. The

evaluating point of 70 A is then about 8.5 Debye lengths, and the reduction factor becomes

fDS = 2 x 10-4 (223)

Thus, the potential is 64 mV, which is well within acceptable limits.

Now we estimate the terms in equation 171 in section 4, to ascertain their relative magnitudes. The

step in the conduction band is bounded as

0.17eV _<AE c < 1.8eV (224)

Therefore, all the other terms should be less than an eV or so. The change in the bandgap due to strain,

AE_, has a large dispersion. From section 6, we fmd

20 meV < AEg < 230 meV (225)

NASA/TM--2003-211983 47
Whilelarge,it isdefinitelywithinacceptableimits.Fortheelectrostaticpotential,westartwiththe

electricfieldfromequation129of section2.Itsvalueis4.8×10sV/m,whichisabout100timeslarger

thanthatforGaAs.It isverynearthebreakdownvalue(5to6xl 0sV/m).At 70A wehave

+(70A)= Es (0) (70A)= (4.8)< 108 V/m)(70 × 10 -1° m)

= 3.36 V .'. e+ = -3.36 eV

(226)

which is high. If we reduce the field by a factor of 10, which is reasonable, then the value is acceptable.

There are two justifications for the reduction by the factor of 10; one is the uncertainty in the stiffness

parameters while the other is the chosen order of magnitude of the strain. The exchange correlation is

-38.6 meV and the image force gives -0.146 meV. The dangling bond gives

Vab = 20.7n d meV (227)

where nd is the number of bonds. This would give a very large value for even one bond. Using the

screening factor reduces it by 10-4, which brings it into a reasonable range. The dipole contribution gives

-896 eV without screening (for just 100 electrons on a sheet). The screening reduces it to -89.6 meV,

which is excellent.

Now we estimate some of the strains that will contribute to Poisson's equation 150 in section 4.

We start with the strains due to metal pads and dielectric films on the AlxGal_xN top surface. Consider a

semi-inf'mite film of thickness h on a substrate. The stresses due to this "single edge" (refs. 76 and 77)

(see fig. 21).

-2 fE ( x31 (228)

,< t..-7-J

h=df

H

h<<H

fE = °'fdf

js

f

I"

/

/

f/

_ -y

z rl i

: rl"N_ilx I ,z)

Figure 21.--Geometry for calculationof stress in region z _>0. Single

stage at x = 0 developsstressesfor x >__0.

NASA/TM--2003-211983 48
-2vfE

T2 = t_ yy -

7t

-2f 

T 3 --t_.. -

T4 =¢_yz =0

T 5 =Gxz --

T6 =_xy =0

These are correct for x > 0. They are not correct for large negative x (which is out of our range).

When two edges are used we have the following situation (see fig. 22).

(229)

(230)

(231)

(232)

(233)

df '\ Idg ! of--,,

l Z _ I _ X

I

I

X1 _'_[

I

I

I

I

x2 _1

I

fE = °'NIf + _gdg

Figure 22.---Geometry for loading due to pad of width Lg.

Modeled as supe_osition of two single edges at x = 0

and x = Lg.

(x,z)

where of, Crg are stresses produced at z = 0.

T] =G_ --

fE = Old/+ Ggdg (234)

(235)

NASA/TM--2003-211983 49
T2 =_yy. - x2d

T3 =or.. - 2fE ( Xlz2 X222

-- _ _ra4 r4

T4 = _yx = 0 (238)

T 6 = CYxy = 0 (240)

This is known as the "effective edge" model. For AlxGal_xN or GaN we have

rl

i ][000.il P2 = 0 0 d]5 0 T3 (241)

P3 d31 d3! d33 0 0 T4

%

_T6.

P_-- dlsT5 (242)

P2 = 0 (243)

P3 = d31 (T1 + T2 )+ d31T3 (244)

then

Px =d15(@-)( x2--z xlzl (245) <4 4)

Py : 0 (246)

,.--_1(2J_ x_+_'_f2rJ__(Xlx_+d_[2--J_x,z x_z_

" k. 7_ .]< I"1 4 < n )_ r, 2 d n r; r_

Notice the coordinate system is not that used in other sections. To conform to that system observe the

following changes: z (here) goes to (-z) there, and y (here) goes to (-y) there. Thus, the only changes

required are the signs ofy and z. This has already been done in equations 245 and 247 above.

(236)

(237)

(247)

NASA/TM--2003-211983 50
A previousstudyusingtheaboveequationshasbeendoneintheAlxGal_xAs/GaAs heterojunction

system(ref.78).It wasobservedthatthestresschangedfromtensionto compressionatthemole-fraction

x = 0.03. The crystal type was zincblende (cubic) so

rz/11

--[!°o°°e'4o°0el4 °°1u44

U55

_U66 _

(248)

Note, only shearing stresses cause ff in this system. If grown in the [100] direction, then P = 0; if grown

in any other direction, then P ¢ 0. For GaAs

v = 0.23

d14 = 2.6× 10-l° cm

V

Px = dl4°'zx

(close to GaN) (249)

Py = dl4_ y.z. (250)

----kal,(_--O_) (251)

Figure 23 shows qualitatively the differences between the "edge" model and a more accurate

calculation using ANSYS (ref. 79). Figure 24 is from (ref. 80), which shows the piezoelectric charge near

a gate pad. Observe the charge density oscillates from positive to negative as one traverses the device on a

horizontal line. The densities are as large as 1017cm -s, which is the same order of magnitude as the

background density obtained in GaN.

NASA/TM--2003-211983 51
O3

E

O

t-

¢0

¢-

¢0

c"

1018

1017

1016

1015

1014

m

z

m

1013 I

o.o o.1

(.-)

_AAAA

A

A

q(N d - Na)

Ppz line load

A Ppz FEM

A

A

A

A A

(-)

AA _

A

A

I I I I

0.2 0.3 0.4 0.5

Depth, I_m

Figure23.--Doping profile and piezoelectric charge densityPpzcalculated by "edge" load

line method, and more accurate ANSYS finite element method (FEM) solution.

NASA/TM--2003-211983 52
16('-- --_

-'-. "0

-- [o111

(a)

i

,_... ,_...h_! (..J_ L7

15 0 15

,,_...15._.....

E

T

(b)

@7r@ __,

.---:_e.1_\...:,_ "7., (

I

[_oo]

=

=

15", I ,;15

IV--

II--

,._-.

®

*@ ' _:S"@ '

,s,47: '. . . ..j_!

.\ \.o./..I "-_ 0"- ii ;_ \

I o ,\ .I

-- [0il] [2111

(c) (d)

Figure 24.mCharacteristic "bifoleum loops" of piezoelectric charge generated by edges of pad on surface.

Numbers represent charge density; 17 means 1017cm-3. (a) [0i 1]-oriented field-effect transisters (FET) on

[100] plane. (b) [0i 1]-oriented FET on [011] plane. (c) [2i 1]-oriented FET on [i i i] Ga plane. (d) [041]-oriented

FET on [111] Ga plane.

9. Heterojunetions

In other sections various properties about heterojunctions were mentioned. This section seeks to

gather much of the information I have gathered that may prove useful. First of all, the assumption that the

interface strain is equal to that of the misfit parameter needs to be discussed. While it may serve as a

guide, its reliability is questionable. First of all, the intrinsic and thermal strain components must also be

calculated to estimate the strain at a junction. The most common interface is that of GaN on sapphire. The

misfit parameters in GaN and sapphire are

3.189qr3- 4.7589

For GaN f,, = 24 percent 3.189

3.189x/-3 - 4.7589

For sapphire fm= = 16 percent (252)

4.7589

NASA/TM--2003-211983 53
It wouldappearthattheGaNwouldbefialateral tension (and thus in vertical compression), and

visa-versa for the sapphire. Experimentally, while GaN is growing on sapphire, it is in tension. The

corresponding TEC are

For GaN ot = 5.59× 10-6/K

For sapphire tx = 7.3 × 10-6/K

This implies the GaN would be in compression. Theory would say the thermal stress is

I",l I-- A°tAT E(GaN)

l-v

320

1 - 0.25

= 0.74 GPa

(253)

Experimentally, the net stress is near 0.66 GPa, compression.

The next combination is A1N on sapphire (which is the standard nucleation layer). The misfit

parameters are

For A1N fm _ 13.189 -- 4.75891 _ 53 percent

3.112

For sapphire f,, _ ]3.112 -4.7589] _ 35 percent (254)

4.7589

The TEC for sapphire is a = 7.3 x 10-6/°C. It would appear that the A1N is in lateral tension latticewise and

compression from TECs.

Next we consider GaN/AIN

3.189-3.112 3.189-3.112

f,, = - 2.4 percent, fm= = 2.5 percent (255)

3.189 3.112

The GaN should be in lateral compression latticewise but in tension from the TECs. Whenjoined, the

measured lattice is 3.184 A. Experimentally, the GaN is in compression.

For GaN/6H-SiC

3.189 -3.08

fm- - 3.4 percent

3.189

ot(6H - SiC]

]

= 4.2 x 10-6 /°C

(256)

The GaN in compression latticewise and tension TEC-wise. In many experiments it is in tension, from

0.1 to 0.2 GPA (for 3-gm films).

NASA/TM--2003-211983 54
ForGaN/4H-SiC

fr, _ 13.189 - 4.231 = 33 percent

3.189

cx(4H- SiC) = 4.2 x 10-6/°C

Both imply tension for GaN.

The case for AlxGan_xN/GaN, the lattice constant for AlxGal_xN is

A(x) = (-.077x + 3.189),&

Then its misfit parameter on GaN is

(257)

(258)

- 0.077x + 3.189 - 3.189

fm= (259)

- 0.077x + 3.189

The TEC for A1N is about 5.3, as compared to GaN's 5.59, so assume no added strains during

cooling. Experimentally, AI_Gal__N is in tension. The in-plane stress is between 0.82 and 1.33 GPa. The

in-plane strain _, is between 0.016 and 0.026. Table XIII gives the vertical strain with mole-fraction

TABLE XIH.--THE VERTICAL STRAIN WITH MOLE-FRACTION IN A1,Ga__xN

= (percent) ] 0.05 O. 12 0.24 0.1 0.15 0.35 ] 0.20 0.48 I 0.25 0.57 p

The c-axis lattice constant in GaN is

C(x)=(-0.2667x+5.17)A 0 <x <0.9

The total spontaneous polarization is

- C

PSI,ON(X)= --0.029(1-X)--0.081x m2

= (-- 0.029-- 0.052X) =:, 3.25X (1013) electrons/cm z

c_L

i(PIEZO) GaN = 0.0163x m2

if(total) GaN = PSPON(GaN) + Ppmzo (GaN)

C

=(-0.029 + 0.0163x) m2

(260)

(261 )

(262)

(263)

NASA/TM--2003-211983 55
alxoalxN)--426x( oc2

m

= (2.66 - 3XxX1013 ) electrons/cm 2

p (totalXbothmaterials)=(_8.4x_l.9x2)×lO_6 C2 m

C

- -0.096x --

m 2

(264)

(264a)

The stiffness coefficients are

C

2.6 x 10-2 < P(total) < 4 x 10-2

m 2

q3(x) = 5x + 103 - 1lx + 105 GPa

C33 (X) = -32x + 405 GPa

The stress coefficients are (refs. 5, 81, 82)

e31(x)= -0.1 lx -0.49 --

C

m 2

c

e33 (x) = -0.73x + 0.73 m2

The relative dielectric constant is

_(x)--0.sx+9.5

The Schottky barrier heights are (ref. 83)

ed_b = (1.3x + 0.84)eV

eO_b= (2.44x + 0.91)eV

The effective masses are

The electric field is

Ni

Ti/Al

me(X)= [O.3x + O.20- x)]m o

mhh(x)=[3.53x + 1.76(1- x)]m 0

(265)

(266)

(267)

(268)

(269)

(270)

(271)

(272)

(273)

(274)

NASA/TM--2003-211983 56
Finally,thesurfacesheetdensityis

e(x) = (7.9x 10-4 +l.15x 107 x)V/cm

ore(x)=(9.5x + 2.1x2 )x lO6 V/cm

or e(x)= 1.2x x 10 6 V/cm

(275a)

(275b)

(275c)

OTOT(X) = 0.068X _ (276)

m 2

ns(X)=6x×lO12 e (277a)

cm 2

Orns(X)_ 5.5xxlO13 e

cm 2

(277b)

10. Defects

For unintentionally doped AIGaN the cartier density is near 1.017cm 3. The carders are assumed to be

caused by N vacancies, but there is no general agreement. Listed are some speculations on defects.

1. Surface traps cause the radio frequency compression. These are emptied and filled by the gate

voltage. Some speculate that the gate voltage alters the strain in the barrier and passivation

layers and thus the amount of trapped charge.

2. Shallow traps cause I-V dc current loops.

3. Deep traps in the buffer are the cause for the I-V dc current collapse. Hot electrons are

introduced into the buffer for drain voltages above 20 V. Gate voltages have no effect on this

process.

A surface donor at 1.4 eV, 1.65 eV.

A deep trap at 1.91 eV.

A shallow trap at .95 to 1.0 eV with capture cross-section 4× 10-12cm 2.

For mole-fractions, the density of these traps are:

TABLE XIV.--APPROXIMATE TRAP

DENSITY IN A]xGal_xN

0.05

0.15

0.25

N, cn1-3

5x1017

2x10 TM

5×1018

For semi-insulating (SI) GaN a trap exists at 2.15 eV below Ec with density between 3 x 109 and

3 x 101° cm -2. For normal GaN, traps at 1.8 and 2.85 eV below Ec. A deep acceptor trap exists at 2.2 eV

below E¢ with density -1 x 101° cm -2.

NASA/TM--2003-211983 57
Thenitrogenvacancyhasastate40meVbelowEc, and the E2 trap exists 0.55 eV below Ec. The trap

that causes dc current collapse is 1.19 eV above Ev. The dominant Mg trap is 150 to 200 meV above Ev.

A trap exists at 0.95 eV above Ev and the characteristic yellow emission is due to a transition between

levels 1 eV below E¢ to one 0.2 eV above Ev. The following list gives the levels (I assume with respect

to Ev) of many of the standard impurities found in GaN (ref. 22).

Zn 0.34, 1.8-1.9, 2.2, 2.5-2.6, 2.8

Mg 0.25, 2.95, 3.2 (note, 0.15 to 0.2 in the paragraph above)

Cd 0.55, 2.7, 2.85

Be 0.7, 2.2

Hg 0.41, 2.9, 2.43

C 0.86, 2.15

Li 0.75, 2.23

P 2.85

As 2.58

Si 0.225

11. Amplifiers

The best ever performance: 9.1 W/mm at 10 GHz on 4H-SiC (refs. 84 to 86). The maximum available

is 12 W/ram, for then the channel goes above 300 °C and/or the SiC substrate starts to leak, and large

anomalous current starts.

Devices have been plagued by both dc and tuning-induced failures.

Eastman says degrade most theoreticalf by 30 percent. SiC has maximum thermal dissipation of

27.8 W/mm, but starts leaking at 10.9 W/mm. SiC leaks more than sapphire due to smaller bandgap.

Eastman 0.2 < x < 0.4

Channel temperature < 300 °C

P(max) at 10 GHz = 12.5 W/mm

The channel temperature is

rc. czL =r0(StmSrRArE)+ e,jDVz s

Rth = 24[l +(T- To)× lO-3] K'mm

W

(MAX)=500K

The devices are predominantly thermally limited, so thermal simulations would be important.

NASA/TM--2003-211983 58
Appendix

I i>_ _

"ZO - 1 1

11>-.. _ I

Figure 25.--Epitaxial AIxGa l_x N layer (z _>0) on GaN (z < 0) with spontaneous

polarization P and bound charge on plane z = 0 and z = z0.

The field due to the spontaneous polarization of the epitaxial layer of AlxGal_xN is developed.

Figure 25 gives the geometry. The spontaneous polarization is specified by

PSPON = -Poa: (A1)

Where P0 = 0.08 C/m 2 on the top surface and 0.051 C/m 2 on the bottom surface. This assumes no

neutralization of the negative charge on the top surface due to adsorbed charge. It also models the net

charge at the heterojunction. Starting with the In'st term of equation 176 of section 4

-fi. h ds'

l,ip (_) =

_r

Lr-"'l (A2)

On the upper surface h = -d:, so if- h = -P0. While on the lower surface we have h = --d:, P- h = P0.

Thus, the two surfaces at z = 0 and z = z0 are just uniform charge sheets.

On the lower surface we have

F' = x '-a x + y '-ay (A3)

F = x-dx + y-dy + z-d.. (A4)

= +O,-y')+ as':ax'ay' (A5)

NASA/TM--2003 -211983 59
Using elementary steps and reference 87-we f'md

u_(_):Io°Io° _ _o+'_'

4rt Co [(x-x')2 + (y + y')e +zZ l/2

From no. 200.01 (reference 87), we have

4_r c 0

4_r c o

where

Now for the first integral in equation (A7)

A 2 = (x- X') 2 + Z 2

Up(r-): -Po CnI(y__yo)+_/(Xo __X)2 +(y__yo)2 ..FZ2 1

4_ c 0

4_ c o

+ P0 IX/v 2 +bv_z 2 +bgn2_/v2 +bv_z 2 +2v+b-zsin -1 [vl_b2 bv-2z24+4z 2 }'_b 4_c° [ 2

v a

where b = 2(y o - y) and y < Yo always

Vb =(y-yo)+_(Xo-X) 2 +z 2 +(y-y0) 2

v a =(y-yo)+_/x 2 +z 2 +(y-yo) 2

where we have used 380.311,261.01,380.001, and 380.111 from reference 87. The second term of

equation (A7) is the negative of equation (A9) with Yo = 0.

On the upper surface we have

?-- ?--

r' = x a x + y ay + z Oa.

: x_+_ +z_:I_-_'1:[(x- x')_+_ - y')_+(_-zo)_l'_

ds'=dx'dy'

(A6)

(A7)

(A8)

(A9)

(AlO)

(All)

(A12)

NASA/TM--2003-211983 60
then

u_ 4_0_0_o _ __x/2___y,/2___._0/21 '2 (A13)

By observation of equation (A6), we see equation (A 13) only differs by a negative sign and the

replacement ofz 2 with (z-z0)2; thus only minor modifications are necessary on equation (A9), to evaluate

equation (A13). The superposition of the results of equation (A9) and (AI3) constitute the complete

potential of the two sheets of charge. The valid range is yo -<y, and all x and z.

NASA/TM--2003-211983 61
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NASA/TM--2003-211983 65
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

February 2003 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Basic Equations for the Modeling of Gallium Nitride (GaN) High Electron

Mobility Transistors (HEMTs)

6. AUTHOR(S)

Jon C. Freeman

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WBS-22-755-12-27

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REPORT NUMBER

E-13653

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM--2003-211983

11. SUPPLEMENTARY NOTES

Responsible person, Jon C. Freeman, organization code 5620, 216--433-3380.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Categories: 33, 76 and 31 Distribution: Nonstandard

Available eleclronically at http://gltrs.grc.nasa.gov

This publication is available from the NASA Center for AeroSpace Information, 301-621_)390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

Gallium nitride (GaN) is a most promising wide band-gap semiconductor for use in high-power microwave devices. It

has functioned at 320 °C, and higher values are well within theoretical limits. By combining four devices, 20 W has

been developed at X-band. GaN High Electron Mobility Transistors (HEMTs) are unique in that the two-dimensional

electron gas (2DEG) is supported not by intentional doping, but instead by polarization charge developed at the interface between the bulk GaN region and the A1GaN epitaxial layer. The polarization charge is composed of two parts:

spontaneous and piezoelectric. This behavior is unlike other semiconductors, and for that reason, no commercially

available modeling software exists. The theme of this document is to develop a self-consistent approach to developing

the pertinent equations to be solved. A Space Act Agreement, "Effects in A1GaN/GaN HEMT Semiconductors" with

Silvaco Data Systems to implement this approach into their existing software for III-V semiconductors, is in place

(summer of 2002).

14. SUBJECT TERMS

Semiconductor devices; FETs

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Unclassified

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