NSM Archive - Gallium Indium Arsenide GaInAs)

Band structure and carrier concentration

Basic Parameters
Band structure
Intrinsic carrier concentration
Lasing wavelength
Effective Density of States in the Conduction and Valence Band
Temperature Dependences
Dependences on Hydrostatic Pressure
Band Discontinuities at Heterointerfaces
Energy gap narrowing at high doping levels
Effective Masses and Density of States
Donors and Acceptors

Basic Parameters for Ga_xIn_1-xAs_yP_1-y

Zinc Blende crystal structure

	Ga_0.47In_0.53As	Ga_xIn_1-xAs	Remarks	Referens
Energy gaps, E_g	0.74 eV	(0.36+0.63x+0.43x²) eV	300 K	Goetz et al.(1983)
Energy gaps, E_g		(0.4105+0.6337x+0.475x²) eV	2 K	Goetz et al.(1983)
Electron affinity	4.5 eV	(4.9-0.83x) eV	300 K
*Conduction band*
Energy separation between X valley and top of the valence band E_X	1.33 eV	(1.37-0.63x+1.16x²) eV	300 K	Goetz et al.(1983)
Energy separation between L valley and top of the valence band E_L	1.2 eV	(1.08-0.02x+0.65x²) eV	300 K	Goetz et al.(1983)
Effective conduction band density of states	2.1·10^{17 cm^-3}	see Temerature dependences
*Valence band*
Energy separation of spin-orbital splitting E_so	***	***
Effective valence band density of states	7.7·10^{18 cm^-3}	see Temerature dependences
Intrinsic carrier concentration	6.3·10^{11 cm^-3}	see Temerature dependences

Band structure for Ga_xIn_1-xAs

	Ga_xIn_1-xAs (zinc blende, cubic). Band structure Important minima of the conduction band and maxima of the valence band.. For details see Goldberg Yu.A. & N.M. Schmidt (1999) .
	Ga_xIn_1-xAs. Energy gap E_g Energy separations between Γ- ,X-, and L -conduction band minima and top of the valence band vs. composition parameter x. Porod and Ferry (1983)

Interfacial elastic strain induced by lattice parameter mismatch between epilayer and substrate results in significant band-gap shifts:

Ga_xIn_1-xAs. Energy band gap Eg of unstrained (solid line) and strained (dashed line and experimental points) vs. composition parameter x.
Solid line is calculated according to Eg= (0.4105+0.6337x+0.475x²) eV.
Experimental points are obtained at 4K.
Kuo et al.(1985)

Brillouin zone of the face centered cubic lattice, the Bravais lattice of the diamond and zincblende structures.

Brillouin zone of the hexagonal lattice.

Temperature Dependences


E_g (x,T)= 0.42 + 0.625x -[5.8/(T+300)-4.19 /(T+271)]·10^-4T²x- - 4.19·10^-4T²/(T+271) +0.475x² (eV)	(eV)	Ga_xIn_1-xAs	Paul et al.(1991)
E_g (x,T)= E_g (0) + (6x²- 8.6x +5.2)·10^-4T^2/(337x²- 455x +196)		Ga_xIn_1-xAs on Ga As	Karachevtseva et al.(1994)
E_g (x,T)= 0.42 + 0.625x -[5.8/(T+300)-4.19 /(T+271)]·10^-4T²x- - 4.19·10^-4T²/(T+271) +0.475x² (eV)	(eV)	Ga_xIn_1-xAs
where T is temperature in degrees K

	Ga_0.47In0.53As. Energy gap E_g of vs. temperature Points are experimental data. Solid line is theoretical calculation. E_g(0)=821.5 ± 0.2 meV. Zielinski et al.(1986)
	Ga_0.87In0.13As. Energy gap E_g of vs. temperature Points are experimental data. Solid line -- 1.321 - 4.1·10^-4 T²/(T+139) Karachevtseva et al.(1994)

Lasing wavelength λ₀

Intrinsic carrier concentration:

n_i = (N_c·N_v)^1/2exp(-E_g/(2k_BT)) ~= 4.82 x 10¹⁵ · [(0.41-0.09x)^3/2+(0.027+0.047x)^3/2]1^/2 x
x(0.025+0.043x)^3/4[T^3/2 exp(-ν/2)(1+3.75/ν +3.28/ν² -2.46/ν³)¹^/2 ] (cm^-3) ,
where ν=E(x,T)/2kT
Paul et al.(1991).

Ga_xIn_1-xAs. Intrinsic carrier concentration vs. temperature for Ga_xIn_1-xAs.
T = 100K; 200K; 300K; 400K; 500K;
Paul et al.(1991)

n_i = 6.3x1011 cm-3 for Ga_0.47In_0.53As at 300K

Effective density of states in the conduction band: N_c

N_c ~= 4.82 x 10¹⁵ · (m_Γ/m₀)^3/2T^3/2 (cm^-3) ~= 4.82 x 10¹⁵ · (0.023+0.037x+0.003x²)^3/2T^3/2 (cm^-3) :

Effective density of states in the valence band: N_v

N_v ~= 4.82 x 10¹⁵ · (m_h/m₀)^3/2T^3/2 (cm^-3)= 4.82 x 10¹⁵ · (0.41-0.1x)^3/2T^3/2 (cm^-3) :

Dependence on Hydrostatic Pressure


E_g (0.47,P)~= (0.796+10.9x 10^-3 ·P -30x10^-6 ·P²) eV	80K, Ga_0.47In_0.53As	x=0.47	Lambkin and Dunstan (1988)
E_g (0.47,P)~= (0.733+11.0x 10^-3 ·P -27x10^-6 ·P²) eV	300K, Ga_0.47In_0.53As	x=0.47
E_g (0.0, P)~= (E_g (0)+4.8x 10^-3 ·P) eV	300K, InAs	x=0.
E_g (1.0, P)~= (E_g (0)+12.6x 10^-3 ·P -37.7x10^-6 ·P²) eV	300K, GaAs	x=1.

where P is pressure in kbar.

Energy gap narrowing at high doping levels

Ga_0.47In0.53As. Energy gap narrowing E_g vs. donor (solid line) and acceptor (dashed line) doping density
solid line -- donor doping density;
dashed line -- acceptor doping density
Jain et al. (1990)


ΔE_g ~= (A ·N^1/310^-9 +B ·N^1/410^-7 +C ·N^1/210^-12) meV	300K, Ga_0.47In_0.53As	x=0.47
where
n : A=15.5; B=1.95; C=159	300K, Ga_0.47In_0.53As	x=0.47
p : A= 9.2; B=3.57; C=3.65	300K, Ga_0.47In_0.53As	x=0.47
N -- carrier concentration in cm^-3

Band Discontinuities at Heterointerfaces

Band discontinuities at Ga_xIn_1-xAs/Al_yGa_1-yAs heterointerface Shur (1990).

			Referens
Conduction band discontinuity	ΔE_v =(ΔE_g -ΔE_v) eV		Shur (1990)
Valence band discontinuity	ΔE_c = (0.44 ΔE_gg) eV		Shur (1990)
where ΔE_gg (eV) = [1.247y + 1.5(1-x) - 0.4(1-x)²] (eV) is the difference between Γ-valleys in Ga_xIn_1-xAs and Al_yGA1-yAs .

Energy gap E_g discontinuity :	ΔE_g = ΔE_gg	for y<0.45
Energy gap E_g discontinuity :	ΔE_g = 0.476 +0.125y + 0.143y² +1.5(1-x) - 0.4(1-x)²	for y>0.45

Band discontinuities	ΔE_v ~= 0.38 eV ΔE_c ~= 0.22 eV	at Ga_0.47In_0.53As/InP heterointerface	Adachi (1992); Hybertsen (1991)
Band discontinuities	ΔE_v ~= 0.2 eV ΔE_c ~= 0.52 eV	at Ga_0.47In_0.53As/Al_0.48In_0.52As heterointerface	Adachi (1992); Hybertsen (1991)

	ΔE_c /ΔE_g = [0.653 + 0.1(1-x)] eV	at Ga_xIn_1-xAs/Al_xIn_1-xAs heterointerface	Wolak et al.(1991)

Effective Masses and Density of States:

Electrons

For wurtzite crystal structure the surfaces of equal energy in Γ valley should be ellipsoids, but effective masses in z direction and perpendicular directions are estimated to be approximately the same:

Effective Electron Masses		Remarks	Referens
Effective electron mass m_e= m_Γ	0.023 -0.037x +0.003x² m_o	Ga_xIn_1-xAs; 300K; for Γ - valley	Goldberg Yu.A. & N.M. Schmidt (1999)
Effective electron mass m_e	m_Γ= 0.041 m_o at n= 2x·10^{17 cm^-3} m_Γ= 0.074 m_o at n= 6x·10^{18 cm^-3}	Ga_0.47In_0.53As; x=0.47	Pearsall (1982)
	m_L = 0.29 m_o ; ( L - valley ) m_X = 0.68 m_o ; ( X - valley )	Ga_0.47In_0.53As; x=0.47	Pearsall (1982)

Ga_xIn_1-xAs. Electron effective mass vs. concentration x for Ga_xIn_1-xAs;
300K
Adachi (1992)

Holes

Effective Masses for Zinc Blende GaN		Remarks	Referens

Effective hole masses (heavy) m_h	m_h ~= (0.41 -0.1x) m_o	Ga_xIn_1-xAs; 300K;	Goldberg Yu.A. & N.M. Schmidt (1999)
Effective hole masses (light) m_lp	m_lp ~= (0.026 -0.056x ) m_o	Ga_xIn_1-xAs; 300K;
Effective hole masses (split-off band) m_s	m_so ~= 0.15 m_o	Ga_xIn_1-xAs; 300K;

Donors and Acceptors

Ionization energies of Shallow Donors		Remarks
Sn, Ge, Si, C	~ 5 meV	Ga_0.47In_0.53As; x=0.47	Goldberg Yu.A. & N.M. Schmidt (1999)
Sn, Ge, Si, S, Se, Te	> 1 meV	InAs; x=0
Sn, Ge, Si, S, Se, Te	~ 6 meV	GaAs; x=1
Ionization energies of Shallow Acceptor
Mg	~ 25 meV	Ga_0.47In_0.53As; x=0.47	Goldberg Yu.A. & N.M. Schmidt (1999)
Zn	~ 20 meV	Ga_0.47In_0.53As; x=0.47
Cd	~ 30 meV	Ga_0.47In_0.53As; x=0.47
Mn	~ 50 meV	Ga_0.47In_0.53As; x=0.47
Fe	~ 150 meV	Ga_0.47In_0.53As; x=0.47
(above valence band), 280, 370, and 440 below conduction band
Mg	~ 25 meV	Ga_xIn_1-xAs; 0<x<1
Be	~ 25 meV	Ga_xIn_1-xAs; 0<x<1
Cd	~ 8-20 meV	Ga_xIn_1-xAs; 0<x<1

(above valence band), 280, 370, and 440 below conduction band
Sn-10; Ge-14; Si-20; Cd-15; Zn-10 meV		InAs; x=0
C - 20, Si - three acceptor levels ~ 30, 100, and 220, Ge - 30, Zn - 25, Sn - 20.		GaAs; x=1