Superlattices and Microstructures, Vol. 4, No. 415, 1988 623

Hole Subbands in One-Dimensional Quantum Well Wires

Mark Sweeny and Jingming Xu Department of Electrical Engineering, University of Toronto, Toronto, Canada M5S lA4

Michael Shur Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455

(Received 18 August, 1987)

A new method for calculating the effects of size quantization upon the valence subbands in the zincblende semiconductors has been developed. A brief description of the method is presented, together with the calculated subband structure of a one dimensional quantum well wire. The subband structure is more complex than the case of the conduction band, possessing large nonparabolicities and effective masses different from the bulk. Our results show that the subbands can have a lower effective mass than the heavy hole mass. This may allow for the implementation of faster P-type devices and complementary circuits. Better computational methods for the subband structure of the valence subbands are useful for the design and modeling of lasers, light detectors, bipolar transistors, and any other devices which utilize holes in quantum wells.

The valence bands, or holes, of the zincblende semiconductors are less studied than the conduction band. This is due to the fact that they have lower mobilities in the bulk, and so are often thought to be unimportant to technology. The success of silicon CMOS technology demonstrates the utility of complementary devices, where both p and n channel devices are used in a circuit. A great deal of effort is now being devoted to the development of GaAs comulementarv devices. One of the maior problems involved is’ that while ultra high sped& n- channel devices such as Modulation Dooed Field-Effect Transistors have heen developed, the p-khannel devices to date are too slow because of the heavy hole effective mass and consequent low mobility. Furthermore, electrooptic devices and bipolar transistors require both types of carriers for their operation. Interest in the physics of holes is expected to grow, both because of the increasing realization that it is important for microelectronics, and because the physics is rich with phenomena of scientific importance.

The theory of the valence bands is based upon the Luttinger-Kahn Hamihonian [Il. Unlike the conduction band, which involves the scalar, or one component Schrodinger equation, the Luttinger-Kohn Hamiltonian tightly co\plesfour field components, which transform under rotations as the 4 dimensional irreducible (spin 3/2) representation of SU(2). The increased mathematical complexity has slowed down progress in this research area. Tavger, in a very early paper [21, predicted that size quantization would break the degeneracy of the valence bands at the Gamma point. Tavger’s result, which was arrived at by use of latter, Nedorezov [

oup theory, was qualitative. Much 3f formulated the equations for the

uroblem of size ouantization in a two dimensional sheet -and obtained preiiminary numerical results. The many papers on the subject published after Nedorezov’s have all used similar methods.

We have applied coordinate free methods and notation to the study of the valence subbands. This reduces the algebraic complexity and makes it easier to use other coordinate systems. Band structure equations similar to those of Nedorezov, wilJ be given for the case of confinement in a cylindrical quantum well wire. To our knowledge, this is the first time that size quantization of the valence bands has been studied in a geometry other than a two dimensional sheet. Typical numerical results for the GaAs one dimensional wire will also be given. These calculations show that quantum confinement can result in effective masses much less than the bulk heavy hole mass. Considering that the impurity scattering in one dimensional structures is greatly suppressed due to the momentum conservation L41, we would expect that devices of one-dimensional p-type wires could be substantiahy faster than existent p-channel devices. One dimensional structures offer other advantages for use in electron and electrooptic devices [5$61.

One dimensional confinement can be attained by means of electrical fields, doping and/or heterostructures as proposed in [4~71. With the steady improvement of epitaxial growth techniques and lithography, the realization of one dimensional devices should soon be within the reach of many research and development laboratories.

Following Luttinger 181, the isotropic Hamiltonian can be written as:

** H = --- (yt+ -;- y)(-V*) +2y(JV)* ) (1)

where Jx, J, and Jz are the angular momentum operators of spin 3/2 . If an explicit representation for J is used, based upon the eigenvalues of Jz, then this becomes a

0749-6036/88/040623+04$02.00/0 0 1988 Academic Press Limited

624 SuperlattIces and Microstructures, Vol 4, No. 415, 1988

coupled set of e uations. such equations[l 91

Luttinger and Kohn present in a rectilinear coordinate system

aligned along the crystal axies. Nedorezov solved these equations with the boundary condition, that I@ at z=fd/2 where d is the film thickness. However, it is difficult to generalize the calculation to other coordinate systems.

Shockley[o] pointed out the analogy between holes in a semiconductor, and sound in an elastic medium. Both are described by a vector field, and exhibit longittdinal waves (light holes) which travel faster than transverse waves (heavy holes).

The Hamiltonian can be written as

?i* HA=-- V*A - h*cl V(V.A)

2mH (2)

As stated, A is a vector wavefunction. The heavy holes obey

V-A = 0

while the light holes can be written as

(3a)

A=W (3b)

for a scalar field 0.

Even without electron spin and spin orbit coupling, the subband structure in a quantum well is complex, as the boundary conditions mix transverse and longitudinal waves.

Real holes can be regarded as a tensor product between a vector A and a spin l/2 field 6, but the tensor product includes both a 4 dimensional spin 3/2 iomponent and a 2 dimensional spin 1B component. Physically, the spin 3/2 components are the heavy and light holes, while the spin l/2 components are the split off band. In &As, the separation is - 300meV. The Luttinger-Kohn Hamiltonian, eq.(l), takes into account the suin orbit snlittina bv assuming that the spin l/2 split off b&rd is far iway,-& writing C& equation‘ for the _ remaining 4 degrees of freedom. The approach we used begins with a Schrodinger equation like eq.(2) for X, a tensor product of spin 1 and spin l/2,

X = (xx. xy, xz) (4)

where Xx, Xy, and Xz are two component spin l/2 wave functions. The condition that X be pure spin 3/2 is equivalent to the constraint

cr.X=O (5)

where (T = ( ox, (TV, C&Z) are the Pauli matrices. Because some operators ineluding the Hamiltonian, eq.(2), will mix in spin l/2 components, one also needs the projection operator

which projects only the spin 3/2 component, x indicates vector cross product. Pure heavy hole solutions obey the further constraint

v.x HH=() (6a)

while pure light hole solutions can be written as

where ‘PLH is a spin l/2 field.

XHH and Y’LH each obeys a simple Schrodinger equation

fi*

- -V*~HH~LH)=E~HH~LH) (7) 2m

with m=heavy hole mass mB for XHH and light hole mass mL for YLB. Equations 6a, 6b and 7 are very similar to the equations for sound in an elastic medium discussed by Shockley.

Defining Fn(r,z) to be the eigenfunctions of V* in cylindrical coordinates:

F,,(r,z) = exp(ikzz).exp(it@J,(k,r)

one can write the light hole solutions as

(8)

XLH = (42V - i(Vxo))F,(r,z)Yo

Yo=(?>orI.l,> aconstant

The heavy hole solutions can written as

(9)

XHH = ~M,z)xno (10)

where Xnu is independent of r, Cl and z, and obeys

cr.XnO=O (11)

The condition

v.xm = 0 (12)

imposes additional constraints and one finds the two solutions

XHH* = tlFn-tX3R + iM%Xt/:! + t3Fn+tX-t/2 (JW

XHHB = ttFn+2X-3/2 + it&+tX-tn + t3FnXtR (13b)

where X&, and X&r are eigenstates of Jz and

tt = -(k,* +4kz*)/d3

t2 = -2kzkr

tg = k* = k,* +k,*. P3/2XLH = (2+ia x)X/3

Superlattices and Microstructures, Vol. 4, No. 415, 7988

With the cylindrically symmebic solutions in hand, we can solve for the wave functions in a quantum well if we can make a linear combination for which all

four components @_3~, x-1~ etc) vanish at r=rg the wire radius. This means that the determinant vanishes:

where J&r) is the integer (n-th) order Bessel function, k~ and kH are defined by

E = %2(kL2 + kz2)/2mL and E = 6*(kH2 + kZ2)/2mH

The dispersion of the n-th set of hole subbands can be then obtained by solving equation (14) numerically for E, as a function of k,. The “O”th, “1”st and “2”nd sets of hole subbands of a GaAs wire of radius lOOA are plotted in Fig. 1, Fig.2 and Fig.3 respectively. The light and heavy hole effective masses have been set to equal to 0.08&x10 and 0.45n-113 respectively. These values are based upon GaAs averaged over all directions. The subband splittings scale inversely as the square of the radius, but the shape of the bands is independent of radius.

The subband structure shows highly nonparabolic subbands which are anticrossing. Anticrossing means they repel when they approach each other just as the conduction and light hole bands repel due to K.P interaction in a narrow gap semiconductor [lo]. The effective masses at the Gamma point are different from the bulk hole masses, and from each other. They sometimes are electron like too. These qualitative features

0, I I 1

I I 1 I 0 5.0 10.0 12.5

K, (10*/m )

Fig.1 The “0”th (n=O) hole subbands of a GaAs wire of radius loOA.

I I

0 5.0 10.0 1:

KL (10*/m 1

Fig.2 The “1”st (n=l) hole subbands of a GaAs wire of radius lOOA.

;: -50

E

% L

-=

(Y = -100 W

-150 I I I I 0 5.0 10.0 12.5

K, (10*/m )

Fig.3 The “2”nd (n=2) hole subbands of a GaAs wire of radius lOOA.

appear already in calculations [I l-151 for two dimensional confinement, and in simple models, that ignore spin orbit coupling. An unusual feature of these subbands is the approximately linear slope of energy with k as k becomes large. Because of the severe nonparabolicity and the fact that each subband is different, it is difficult to make quantitative predictions regarding transport properties. The effective masses of the two top bands are less than the heavy hole mass, and if one averages over these bands for energies above the third band, then one can assign an average effective mass of one half the heavy hole mass. At such a filling, the first subband is filled to roughly kZ=l.5 (in 108/m) while the next subband is filled to only 0.5. Note that the next band to fill is the Jz=l band which has its maximum away

625

5

626 SuperlattIces and Mlcrostructures, Vol. 4. No. 415, 1988

from k,=O. For our 1OOA wire, this filling corresponds to a doping of 3x1017/cm3. The lower effective mass, together with the lower density of states available for scattering, suggests a higher mobility, but transport properties are an area where much work is needed before firm conclusions can be drawn.

Acknowledgement - This work was partially done while one of the authors (M. Sweeny) was with Unisys. We also thank the NSERC and 0.C.M.R for support.

References

[l] J.M. Luttinger and W. Kohn, Phys. Rev. 97, 869, (1955). [2] B. A. Tavger, Sov. Phys. -JETP 21, 125 (1965) [3] S.S. Nedorezov, Sov. Phys-Solid State 12, 1814, (1971).

141 H. Sakaki. Janan. Journal of Aoolied Phvsics. i9(12), (1986). *

& L ,

151 Y. Arakawa and A. Yariv. IEEE J. of Quantum Electronics, vol. QE-22, no.9, (1986). . [6] T. Mimura, S. Hiyamizu, T. Fujii and K. Nanku, Japan J. of Appl. Phys., 19, L225, (1980). 171 J. Xu, M. Sweeny and M. Shur, ‘Electronic and Optoelectronic devices utilizing light hole properties’, United States Patent Application. [8] J.M. Luttinger, Phys. Rev. 102, 1030, (1956). [9] W. Shockley, Phys. Rev. 78, 173 (1950) [lo] E. 0. Kane, J. Phys. Chem. Solids 1, 245 (1956) [ 1 l] M. Altarelli, Physics Review B 28, 842, (1983). 1121 A. Fasolino and M. Altarelli, in Two Dimensional Systems, Heterosbuctures and Superlattices, 176, eds., G. Bauer. F. Kuchar. and Heinrich. Sorintzer. Berlin.

.L “.

(1984). [13] Y.C. Chang, Appl. Phys. Letters, 46, 710, (1985). [14] J. Lee and M. 0. Vassell, Phys. Rev. B 34, no. 10, 7383 (1986) [15] D. A. Brodio and L J. Sham, Phys. Rev. B 31,888 (1985)