ECE236A Semiconductor Heterostructure Materials Group III Nitride Semiconductors Lecture 17, Nov. 30, 2017

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1 ECE236A Semiconductor Heterostructure Materials Group III Nitride Semiconductors Lecture 17, Nov. 30, 2017 Spontaneous and Piezoelectric Polarization Effects on 2DEG in HFETs Effects of Polarization on InGaN quantum wells Impact of non-polar planes (m-plane, and a-plane) Ref.: Nitride Semiconductor Devices: Fundamentals and Applications, 1 st Ed., Hadis Morkoc, Wiley-VCH Verlag, Beware typos 1

2 High-degree of ionicity à wurtzite crystal structure: Wurtzite vs. Cubic Zinc Blende Structures - Non-zero ontaneous polarization - Large piezoelectric coefficients - Large bandgap for many (but not all) nitride and oxide semiconductors. Cubic Zinc blende: Lattice basis vectors (fcc) a = a 2 x + y ( ) b = a 2 x + z ( ) c Primitive Unit Cell: - Atom A at Origin - Atom B at a 4 x + y + z ( ) = a 2 y + z ( ) Wurtzite: Lattice parameters: z c - a in basal plane y - c in out of plane x - internal parameter u (bond length/c) Lattice basis vectors a = ax b = a 2 x ay c = cz Primitive Unit Cell: - Atom A at O 1 3 a b c 1 3 a b! + 1 " 2 + u $ - Atom B at uc c % In Cartesian coordinates: 1 2 a b c 1 2 a b! + u + 1 $ c " 2 % B z A A b B a y x B A a c b b 60 a a a b 1/3 1/3 A B c = 2 u = 3 8 a c/2 b=u.c Ideal structure: 2 3 a 2 a

3 Polarization Fields and Charge Densities The different symmetry of the wurtzite crystal structure leads to the presence of non-zero ontaneous polarization, i.e. a polarization field that exists in the absence of any external influence such as strain or an external electric fields. The piezoelectric coefficients for wurtzite semiconductors are typically very large compared to those of cubic zinc blende semiconductors. Polarization fields can be understood as arising from dipole moments that exist at the unit cell level in a crystal. Integration of these dipole moments yields the overall polarization field: 1 P = V ρ ( r ) r d 3 r Recall that polarization fields that arise upon application of an external field are reonsible for a material s dielectric reonse. E ext D = ε 0 E + P = ε 0 E +ε 0 χ e E = ε 0 ( 1+ χ e )E = εe P E ext = 0, P = 0 E ext 0, P 0 Polarization fields can be thought of as arising from electrostatic sheet charge densities within the crystal lattice or at the crystal surface. More generally, the electrostatic charge density associated with a polarization field can be computed form the relation: P = ρ pol dielectric susceptibility E ext dielectric constant +σ pol P σ pol P = σ pol 3

4 Spontaneous Polarization Effects Convention: Ga terminated (Ga-polarity), with single bonds formed from cation (Ga) to anion (N) à c-axis [0001] orientation. Polarization is in the direction from cation (Ga) to anion (N) at the tetrahedron bond (3 effective positive dipoles opposing 1 dipole in the single bonded Ga to N) à polarization is opposite to [0001] axis. Polarization exists at surfaces and interfaces (no long range polarization field). Polarization field in the AlN is larger than in the GaN. At the interface, there is a net polarization vector, ΔP 0 pointing in the [0001] for the single Al-N bond, and in the [000-1] axis 4 for the single Ga-N bond.

5 P = σ pol, Spontaneous Polarization Effects σ pol, = 0.034C / m 2 σ pol, = cm 2 The equivalent electric field is:. E = σ ε =10ε 0 for GaN ε 0.034C / m 2 E = F / m E = V / m Thus, ontaneous polarization charges that are unscreened by mobile charge in semiconductors can lead to very large electrostatic charge densities and electric fields. Using Vegard s linear interpolation law (not quite, see right panel): P Alx Ga 1 x N P Inx Ga 1 x N P Alx Ga 1 x N ( ) ( ) ( ) = 0.09x x = 0.042x x = 0.09x x Material P (C/m 2 ) GaN AlN InN ZnO BaTiO 0.26 LiNbO

6 Piezoelectric Polarization Piezoelectricity refers to the generation of a polarization field in reonse to the application of stress in a crystal. The relationship between the piezoelectric polarization vector, P pz,i and the stress tensor σ jk is given by a third rank tensor d ijk : P i pz = d ijk σ jk piezoelectric moduli In matrix notation, this becomes: P i pz = d ij σ j with i=1,2,3 and j=1,2,.., 6 For wurtzite structures, this becomes: d ij ( WZ) =! " d d d 31 d 31 d We can also formulate the piezoelectric effect in terms of the relationship: P i pz = d ij σ j = d ij c jk ε k e ik ε k $ % e ik is the piezoelectric tensor 6

7 Piezoelectric Polarization P pz,i e ik ε k! " P x P y P z $! = % " The out of plane strain : e e e 31 e 31 e ε zz = ε 3 = c c 0 c 0 P pz i = e ij ε j with i=1,2,3 and j=1,2,.., 6 j P z = e 31 ε xx + e 31 ε xx + e 33 ε zz = 2e 31 ε xx + e 33 ε zz! $ % " ε xx ε yy ε zz ε yx ε zx ε xz $ % Hexagonal symmetry: e 24 = e 15 Without shear: ε yx = ε zx = ε xz For biaxial strain only: ε xx = ε 1 = ε yy = ε 2 = a a 0 a 0 a for buffer layer or substrate a 0 for epi layer P z = 2e 31 ε // + e 33 ε zz " ε zz = 2 c % " 13 $ 'ε xx = 2 c % 13 $ 'ε // c 33 c 33 pz c P z = % e 31 2e 13 P Alx Ga 1 x N 33 (ε // $ ' c 33 = " xe AlN + ( 1 x)e GaN $ % ε ( x) 7

8 Effect of Spontaneous Polarization in GaN/AlGaN 2DEG In a GaN/AlGaN 2DEG structure, the presence of polarization induced sheet charges at the GaN/AlGaN interface has a large effect on the 2DEG density: [ 0001] d P AlGaN P GaN P pz AlGaN z= qφ B σ pol > 0 0 d E c E F σ pol qn d qn s d 2 φ dz = ρ = qn d 2 dφ dz = qn d z + C 1 ξ GaN z=d = qn s ε GaN ξ AlGaN z=d +ε GaN ξ GaN z=d = σ pol ξ AlGaN z=d = qn s σ pol ξ AlGaN z=d = qn s σ pol dφ dz = qn d ( z d) + σ qn pol s 8

9 [ 0001] d Effect of Spontaneous Polarization in GaN/AlGaN 2DEG P AlGaN P GaN P pz AlGaN qφ B σ pol > 0 qn d σ pol E c E F dφ dz = qn d ( z d) + σ qn pol s φ = qn d 2 z=0 z=d qn s ( z d) 2 + σ pol qn s % $ (z + C 2 ' φ ( z = 0) = φ b C 2 = qn d d 2 φ b 2 φ ( z) = qn d 2 ( z d) 2 + σ qn pol s % (z + qn dd 2 φ b $ ' 2 9

10 Effect of Spontaneous Polarization in GaN/AlGaN 2DEG At the GaN/AlGaN interface, we can equate the energies with reect to the E F reference as follows: E c E F qφ ( z = d) = ΔE c + E F qφ ( z = d) = ΔE c + E 1 + n s g 2D ( ) ΔE c = E 1 + n s qφ z = d ΔE g c 2D E 1 $ ΔE c = % " $ π 2 m GaN * + q2 d 2 * 2m GaN ' ) ( 1/3 % " 'n s $ " E 1 = $ $ q 2 n ' s ) % ( ε GaN 2 * 2m GaN 2/3 2 * 2m GaN % ' % ' 1/3 " q 2 n % $ s ' ε GaN 2/3 a 1 ( ) g 2D = m * GaN π 2 a 1 + π 2 n * s q σ pol qn s d + qn dd 2 -, φ * b / m GaN +, 2./ 1/3 q 2 2/3 " n % $ s ' a 1 = ΔE c qφ b + q2 N d d 2 ε GaN 2 + qdσ pol For σ pol =10 13 q/cm 2, d=20nm, =10ε 0, N d =10 18 cm ev 3.6 ev Thus n s is dominated by the term undoped material qdσ pol and can be very large (~10 13 cm -2 ) even in 10

11 Sheet Carrier Concentration with Al Molar Fraction 11

12 Lattice constant and Bandgap for AlInGaN 12

13 Lattice Constants and Band-edge Energies for Nitrides g E AlN = 6eV b = 2eV 13

14 Polarization and Internal Electric Field in In y Ga 1-y N/GaN Quantum Wells In an In y Ga 1-y N quantum well with GaN barriers coherently strained to GaN (0001), the piezoelectric polarization in the InGaN layer can produce a substantial internal electric field within the quantum well. Polarization fields neglected: [0001] E c Polarization fields included: P P P P pz E c E v GaN InGaN GaN GaN InGaN GaN The internal electric field in the quantum well leads to: - Decrease in energy of luminescence peak - Increase in radiative recombination lifetime (decrease in radiative recombination rate. We can calculate the internal polarization field in the InGaN quantum well and also estimate the resulting potential drop across the quantum well: In the following, we will consider an InGaN/GaN [0001] quantum well of thickness d. E v 14

15 Polarization and Internal Electric Field in In y Ga 1-y N/GaN Quantum Wells P Iny Ga 1 y N P GaN = ( 0.034C / m 2 )( 1 y) ( 0.042C / m 2 ) = 0.034C / m 2 ( 0.008C / m 2 ) y = 0.034C / m 2 $ ΔP = % ' $ σ = % ' ( 0.008C / m 2 ) y ( ) y 0.008C / m 2 ( q / cm 2 ) y ( q / cm 2 ) y P pz at GaN on In y Ga 1-y N interface at In y Ga 1-y N on GaN interface at GaN on In y Ga 1-y N interface at In y Ga 1-y N on GaN interface To obtain in the In y Ga 1-y N layer, we first compute the strain and stress tensor components: ε 1 = ε 2 = a a GaN InGaN = ε // ε 3 = 2 c " 13 ε // ε σ 3 = 0 σ 1 = σ 2 = c 11 + c 12 2 c 2 % 13 $ a InGaN c 33 c 33 * The piezoelectric polarization in the In y Ga 1-y N layer is then: $ 2d 31 c 11 + c 12 2 c 2 ', 13 )ε " P pz 3 = d 31 ( σ 1 +σ 2 ) = 2d 31 c 11 + c 12 2 c, // 2 % % c 13 ΔP $ 'ε pz 33 ( = + // $ c 33 2d 31 c 11 + c 12 2 c 2, ' 13 )ε σ, // pz = ΔP pz 15 - % c 33 ( 'ε // GaN on In y Ga 1-y N In y Ga 1-y N on GaN

16 Polarization and Internal Electric Field in In y Ga 1-y N/GaN Quantum Wells If we assume an In y Ga 1-y N quantum well with composition y=0.25, we obtain: % σ pz = $ % q / cm q / cm 2 ( ) InGaN on GaN ξ InGaN = σ +σ pz ε InGaN GaN on In y Ga 1-y N In y Ga 1-y N on GaN In comparison, for the same quantum well, we have: % σ pz = $ % q / cm q / cm 2 GaN on In y Ga 1-y N In y Ga 1-y N on GaN V / cm Piezoelectric polarization effects are much larger than those of ontaneous polarization. The internal electric field in the quantum well is then given, in the absence of screening of mobile carriers, by: for In 0.25 Ga 0.75 N/GaN quantum well For typical quantum well thickness, d=3nm, the potential drop across the quantum well, ΔV InGaN is then: ΔV InGaN = ξ InGaN d 1.2V However in a real device structure, this will be partially screened by free carriers and ionized dopants. 16

17 Polarization and Internal Electric Field in In y Ga 1-y N/GaN Quantum Wells Fluctuations in growth rates or in-plane during growth (interface is not perfectly flat) can lead to variation of quantum well thicknesses. With 1 monolayer variation (c/2=0.28nm), the incremental potential drop across the quantum-well layer is: ξ InGaN d 0.11V for In 0.25 Ga 0.75 N This can result in a local potential well for electrons or holes of depth ~ 0.1eV sufficient to localize carriers and suppress diffusion for non-radiative recombination centers associated with defects such as dislocationsà high radiative recombination efficiency even in defective nitride semiconductor materials (in addition to/instead of clustering). In a 2DEG structure, e.g. for a field effect transistor, the polarization induced charges are generally desirable since high 2DEG electron density leads to high current-carrying capacity, high power, etc. In an InGaN/GaN quantum well structure, typically used in a visible light emitter, increased radiative recombination lifetime (decreased rate) arising from polarization fields is often undesirable, particularly if high brightness in LED is desired. Is there a way to eliminate the undesirable effects of polarization fields in nitride semiconductor quantum well structures? Yes, the quantum well interface can be oriented in a direction for which is no polarization-induced charge or internal electric field. P = 0 so there 17

18 Commonly Used Plane Nomenclature A. Konar et al. Semicond. Sci. Tech , le/91897-scaling-semi-polar-substrates.html 18

19 Growth of InGaN/GaN Quantum Well on m-plane/[1-100] orientation GaN InGaN GaN! " 1100 $! " 1120 $ y x [ 0001] z! P z = " $ P P InGaN P GaN = P x % P z ' x + P y y + P z z = 0 Thus ontaneous polarization does not yield polarization-induced charge or internal electric fields in the direction of quantum confinement. Piezoelectric Polarization: For structures grown on a GaN substrate or thick GaN layer, we can assume is nonzero only in the InGaN layer. σ 1 = 0 ε 2 = a GaN a InGaN ( ) / a InGaN ε 3 = ( c GaN c InGaN ) / c InGaN ε 1 = ( c 12 ε 2 + c 13 ε 3 ) / c 11 σ 2 = c 21 ε 1 + c 22 ε 2 + c 23 ε 3 = c 12 ε 1 + c 11 ε 2 + c 13 ε 3 = c c 12 σ 3 = c 31 ε 1 + c 32 ε 2 + c 33 ε 3 = c 13 ε 1 + c 13 ε 2 + c 33 ε 3 = c 11 c 12 P pz ( ) / c 11 ε 2 + ( c 11 c 12 )c 13 / c 11 ε 3 2 ( )c 13 / c 11 ε 2 + ( c 11 c 33 c 13 ) / c 11 ε 3 P pz z = d 31 σ 1 + d 31 σ 2 + d 33 σ 3 = d 31 σ 2 + d 33 σ 3 bu P pz = P pz x t x + P pz y y + P pz z = 0 à no polarization induced charges and no z polarization induced internal electric field. 19

20 Growth of InGaN/GaN Quantum Well on a-plane/[11-20] orientation (continuation for m-plane) For large area planar devices, the piezoelectric polarization field would ideally be screened by surface charges at the edge of the device. However, near the edges or in the vicinity of inhomogeneities in the material, variation in P pz in the z-direction can occur and can lead to localized polarization-induced charges and electric fields. GaN$ InGaN$ GaN$! 1120 y " $! 1100 " $ 0001 [ ] x z ( ) / a InGaN ( ) / c InGaN ε 1 = a GaN a InGaN ε 3 = c GaN c InGaN ε 2 from σ 2 = 0 P sz = P pz = 0 à no polarization induced charges and no polarization induced internal electric field in the direction of crystal growth. For m-plane and a-plane growth of InGaN/GaN quantum well structures, the detrimental effects of polarization field for light emitters, primarily physical separation of electrons and holes in the quantum well, can be largely avoided. 20

21 Obtaining semi-polar substrates Figure 2. The conventional fabrication process for semi-polar GaN substrates: A GaN layer more than a few millimetres thick is grown, by HVPE, on c-plane GaN template and sliced in intended semi-polar directions Figure 3. Growth process of a {2021} GaN layer on patterned sapphire. (a) Formation of c-plane sapphire sidewall by dry etching, which is 74.6Â inclined from the {2243} plane of the sapphire. (b) Nucleation of GaN stripes from the c-plane sapphire side-walls by MOVPE. (c) Formation of the {2021} GaN film by the coalescence of neighbouring GaN stripes. (d) Crosssectional scanning electron microscopy of a {2021} GaN layer. 21