The self-consistent calculation of Si δ-doped GaAs structures

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Applied Physics A

Materials

Science & Processing

The self-consistent calculation of Si δ-doped GaAs structures

E. Ozturk^1,^∗, Y. Ergun¹, H. Sari¹, I. Sokmen²

1Cumhuriyet University, Department of Physics, 58140 Sivas, Turkey

2Dokuzeylul University, Department of Physics, 35182 Izmir, Turkey

Received: 4 December 2000/Accepted: 31 January 2001/Published online: 26 April 2001 –  Springer-Verlag 2001

Abstract. In this study, we report results of a self-consistent calculation obtained for the sub-band structure of Si δ- doped GaAs material by using a new alternative method.

We will discuss the influence of the δ-doping concentration and the δ-layer thickness on the sub-band structure for a non-uniform distribution, which is taken as different from the known Gaussian distribution. The confining potential, the sub-band energies, the sub-band occupations, and the Fermi energy have been calculated by solving the Schrödinger and Poisson equations by using the Airy functions self-consistently.

PACS: 73.20.Dx; 73.20.At

Recently, rapid advanced material-growth technology such as the molecular beam epitaxy (MBE) technique enables the growth of abrupt high-concentration doping profiles. The confinement of the donors has been shown to be very narrow and the concept of planar or δ-doping has been introduced.

In a semiconductor a one-dimensional doping profile can be considered to be δ-function-like, if the thickness of the doped layer is smaller than other relevant length scales. Such narrow doping profiles can be mathematically described by Dirac’s delta function.

When the Si donors are located in an atomic mono- layer of GaAs host material, the ionized donors provide a continuous positive sheet of charge. Due to electrostatic attraction the electrons remain close to their parent ionized donors and form a quasi two-dimensional electron gas (2DEG) in the V-shaped potential well produced by the continuous positive sheet of charge. In this narrow potential well the electron energies for motion perpendicular to the growth surface are quantized into two-dimensional (2D) sub-bands. This sub-band structure has been calculated self-consistently by solving the Schrödinger and Poisson equations.

Employment of the δ-doped method results in a signifi- cant improvement of a number of semiconductor structures

∗Corresponding author. (E-mail: [email protected])

and devices. δ-doping is of great importance for device ap- plications, especially for field-effect-transistor structures [1], which are increasingly used for fast logic. This doping technique enables the ultimate control of a dopant profile and will play an important role in future photonic and quantum electronic device research. Profiles with full widths at half maximum (FWHMs) of 30 and 20 Å determined by a capacitance–voltage measurement technique have been reported for Si- and Be-doped structures, respectively [2, 3]. Due to this property of localizing impurities in space, δ-doping is used in devices to give rise to quantum confinement of carriers [1, 4]. Single and multiple δ-doping have been studied theoretically by several authors using different approximations [5–10].

In the present paper, we investigated theoretically the sub- band structure of n-type Si δ-doped GaAs by using a new alternative method, at low temperature. We have studied the influence of the δ-doping concentration and the δ-layer thickness on the electronic structure for a non-uniform distribution. The effect of diffusion of donor impurities along the growth direction has been compared in [9] for both the uniform distribution and the non-uniform distribution. In this study, we have seen that a system with donor impurities dis- tributed uniformly in a sheet with a thickness equal to 50 Å has approximately the same behavior as a system with a non- uniform distribution with the FWHM equal to 30 Å (as different from [10]). In addition, the results obtained by the present method will be compared with model calculations based on the wave function of the infinite quantum well [9]

(herein called the infinite-well model (IWM)). In the IWM,

the δ-doped layer is assumed to be in an infinite quantum

well whose thickness is L

0

. In the IWM, L

0

is a variational

parameter that the Poisson and Schrödinger equations con-

verge to fixed potential profile and fixed sub-band energies for

a fixed doping concentration and thickness that describe the

sub-band populations. But where, an unphysical behavior has

been appeared; the last filled state seats over the δ-potential

height [8, 9], that is, nearest filled confined state to the Fermi

level must be lower than the δ-potential height in the energy

scale. Although this state must be a confined state in the well,

in the IWM it is over the potential height of the δ-potential.

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This is a contradiction and thus an alternative method for the self solution of the Poisson and Schrödinger equations is needed to avoid the contradiction. Thus the last filled state (near to the Fermi level) is confined in the well, as expected.

Since the last filled state first responds to the external fields (electrical and magnetic fields), the correct finding of its energy level and population is very important.

1 Theory

We use the electronic structure on the basis of the effective- mass approximation. In order to obtain the confining potential, the density profile, the sub-band energies, the sub- band occupations, and the Fermi energy we have calculated self-consistently the Schrödinger and Poisson equations. The Schrödinger equation for the system is given by

− h

²

2m

^∗

d

²

dz

²

+ V(z) + V

xc

(z)

Ψ

i

(z) = E

i

Ψ

i

(z) (1) where m

^∗

is the effective mass, z is the direction perpendicular to the δ-doped layer, V(z) is the guessed V-shaped potential profile with different slopes (Fig. 1), and V

xc

(z) is the exchange-correlation potential. The exchange-correlation interaction on the electronic confinement and the sub-band structure is very weak in δ-doped structures with high doping concentrations [7, 11]. Hence we can confidently neglect this effect. The wave functions in the different regions are ex- pressed as

φ

i

=

 

 

 

 



Aexp [Kz] I. region

A

₁

Ai [÷

1

(z)] + B

1

Bi [÷

1

(z)] II. region A

2

Ai [÷

²

(z)] + B

²

Bi [÷

²

(z)] III. region A

3

Ai [÷

3

(z)] + B

3

Bi [÷

3

(z)] IV. region A

4

Ai [÷

4

(z)] + B

4

Bi [÷

4

(z)] V. region A

5

Ai [÷

5

(z)] + B

5

Bi [÷

5

(z)] VI. region A

6

Ai [÷

6

(z)] + B

6

Bi [÷

6

(z)] VII. region

Bexp [−Kz] VIII. region

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Fig. 1. Schematic illustration of the guessed V-shaped potential profile with different slope at each region. The inset represents the interface for two different potential regions

where Ai [χ(z)] and Bi[χ(z)] are Airy and Bairy functions, respectively, and

K =

2m

^∗

(V

0

− E

i

) h

²

χ

1

(z) =

m

^∗

(L

0

− L

1

)

²

2h

²

V

₁²

1/3

×

V

0

− 2V

1

L

0

− L

1

z + L

0

2 − E

i

χ

2

(z) =

m

^∗

(L

1

− L

2

)

²

2h

²

V

₂²

1/3

×

V

0

− V

1

− 2V

2

L

1

− L

2

z + L

1

2 − E

i

χ

3

(z) =

m

^∗

(L

2

)

²

2h

²

V

₃²

1/3

×

V

0

− V

1

− V

2

− 2V

₃

L

2

z + L

₂

2 − E

i

χ

4

(z) =

m

^∗

(L

²

)

²

2h

²

V

₃²

1/3

×

V

0

− V

1

− V

2

+ 2V

3

L

2

z − L

2

2 − E

i

χ

5

(z) =

m

^∗

(L

1

− L

2

)

²

2h

²

V

₂²

1/3

×

V

₀

− V

1

+ 2V

2

L

1

− L

2

z − L

1

2 − E

i

χ

6

(z) =

m

^∗

(L

0

− L

1

)

²

2h

²

V

₁²

1/3

×

V

0

+ 2V

1

L

0

− L

1

z − L

0

2 − E

i

,

By using the boundary conditions and the normalization condition for the wave functions, the energy levels and the wave functions can be obtained. Thus, we can calculate the Fermi level E

F

, from the condition that the total number of electrons must equal the total number of donors. The temperature- dependent number of electrons per unit area in the ith sub- band is given by

n

i

= m

^∗

k

B

T

πh

²

ln {1 + exp[(E

F

− E

i

)/k

B

T ]} , (3) where k

B

is the Boltzmann constant, E

F

represents the Fermi energy, and i is the sub-band index. The Poisson equation for the confining potential V (z) in the δ-doped semiconductor is written as

d

²

V (z)

dz

²

= − 4 πe

²

ε [N(z) − N

d

(z)] , (4)

with N (z) =

nd

i=1

n

i

|φ

i

(z)|

²

, (5)

where ε is the GaAs dielectric constant, n

d

is the number of

filled states, and N

d

(z) is the total density of ionized dopants.

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The Dirac delta function is used for the description of the charge distribution, i.e., N

d

(z) = N

d^2D

ä (z), where N

d^2D

is the 2D donor concentration. For a non-uniform distribution we used

N

d

(z) = N

_d^2D

∆z

1 − e

⁻^L0^∆z

e

⁻^∆z^2z

(6)

where L

0

is the top width of the delta potential, which is approximately equal to the diffusion length of the donor atoms ( ≈ 200 Å) in GaAs and ∆z is the doping width of the Si atoms. We do not expect the distribution of the donor atoms to coincide with the Gaussian distribution, because they are very heavy when compared with both the electron and hole masses. Thus, we are the first to introduce (6) to identify the segregation of the donor atoms.

A flow chart of the self-consistent calculation is shown in Fig. 2. At each region the calculation of the Poisson equa-

Fig. 2. Flow chart of the self-consistent calculation

tion is given in the appendix. The self-consistent solution of (1)–(6) gives the confining potential profile, the density profile, the sub-band energies, the sub-band occupations, and the Fermi level.

2 Results and discussion

Using the Airy functions we have studied the sub-band structure of Si δ-doped GaAs. In Fig. 3 we show the confining -potential and the sub-band energies with their squared envelope wave functions obtained for a single δ-doped GaAs layer with L

₀

= 500 Å, N

d^2D

= 5 × 10

¹²

cm

⁻²

, and ∆z = 20 Å for a non-uniform distribution, at low temperature. The solid curve represents the obtained confining potential profile and the dashed curve represent the guessed V-shaped potential profile.

In (Fig. 4a and b) the confining potential and the electronic density profile are shown for different doping concentrations for a non-uniform distribution, respectively. In Fig. 4a, the confining potential and the sub-band energies are referred to the Fermi-energy level and not to the top of the potential. At high doping concentrations, an increasing charge density in the layer leads to more band bending and gives rise to the formation of a deeper quantum well. This feature could be of use in controlling the confinement of carriers in devices using this type of doping. With increase of the δ-doping concentration both the depth of the confining potentials and the density profiles are importantly changed. For a non-uniform distribution we show the confining potential and the density profile for different doping thicknesses in Fig. 5a and b, respectively. As seen in these figures, the confining potential and the density profile are quite sensitive to changes in the thickness of the donor distribution.

Fig. 3. The confiningδ-potential, the sub-band energies with their squared envelope wave functions, and the Fermi-energy level obtained for a sin- gleδ-doped GaAs layer with L0= 500 Å, Nd^2D= 5× 10¹²cm⁻², and∆z = 20 Å, at low temperature. The dashed curve shows the guessed V-shaped potential profile and the solid curve represents the obtained confined poten- tial profile

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Fig. 4. a The confining potentials and b the electronic density profiles for three different doping concentrations with L0= 500 Å and ∆z = 20 Å. The donor concentration is equal to a from top to bottom and b from bottom to top 1, 3, 5 × 10¹²cm⁻². All potential profiles are relative to the Fermi- energy level

For different doping concentration and different doping thickness, change of the sub-band energies and the sub-band occupations are summarized in Table 1. As expected, the number of filled states increases with donor concentration.

As shown in this table, changes of sub-band energies and sub-band occupations are very sensitive to the δ-doping concentration and the δ-layer thickness. It is clear that the distribution of the electrons over the levels is dependent on the donor concentration and the thickness of the donor distribution. The thickness of the donor distribution can thus be used as a fitting parameter in the calculations to obtain the same sub-band population in this system.

Using a complete-set method, instead of Airy functions, one is led to little-different results as illustrated in Table 1. As shown in Fig. 6, while for the complete-set method (IWM)

Fig. 5. a The confining potentials and b the electronic density pro- files for three different doping thicknesses with L0= 500 Å and Nd^2D= 5× 10¹²cm⁻². The thickness of the donor distribution is equal to a from bottom to top and b from top to bottom 10, 50, 100 Å. All potential profiles are relative to the Fermi energy

only the lowest three sub-bands are confined in the confining potential, for the Airy-function method all sub-bands are in the confining potential. On the other hand, this method has an advantage of convergence to true values of sub-band populations and energies; after 3–5 loops the calculation is completed (convergence achieved), which is a tremendous improvement when compared with a fully numerical method which requires a few hundred tries.

3 Conclusions

We have investigated the changes of the sub-band struc-

ture in a Si δ-doped layer in GaAs as a function of the

δ-doping concentration and the δ-layer thickness for a non-

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Table 1. The obtained values of the sub-band energies and the sub-band occupations as dependent on different doping concentration and different donor thickness are compared for two different methods

N_d^2D ∆z V0 EF− Ei (meV) ni(10¹²cm⁻²)

Method

(10¹²cm⁻²) (Å) (meV) i= 1 i= 2 i= 3 i= 4 i= 1 i= 2 i= 3 i= 4

Complete set 33.96 27.41 8.22 – – 0.768 0.231 – –

1 20

Airy function 39.7 26.92 8.81 – – 0.752 0.247 – –

Complete set 88.6 69.41 26.71 11.05 – 1.942 0.745 0.309 –

3 20

Airy function 94 68.81 27.86 12.44 – 1.907 0.761 0.330 –

Complete set 136.48 106.8 45.2 21.09 5.54 2.985 1.26 0.590 0.154

5 20

Airy function 138 102.6 45.21 20.22 10.53 2.845 1.265 0.565 0.322

Complete set 148 109.5 43.68 20.58 4.87 3.063 1.222 0.575 0.135

5 10

Airy function 148.2 106.2 44.66 18.29 9.5 2.972 1.249 0.511 0.265

Complete set 107.8 96.9 48.89 24.3 8.3 2.71 1.369 0.683 0.233

5 50

Airy function 113 92.07 48.41 24.54 13.39 2.578 1.356 0.688 0.376

Complete set 72 83.22 50.13 30.47 14.83 2.33 1.4 0.851 0.414

5 100

Airy function 78 77.23 49.37 31.15 20.88 2.161 1.381 0.872 0.584

Fig. 6. Calculation of the confining potential and the sub-band energies by using two different methods. The solid lines represent the Airy- function method and the dashed lines indicate the infinite-well method;

L0= 500 Å, Nd^2D= 5× 10¹²cm⁻² and∆z = 20 Å. The potential profiles and the sub-band energies are relative to the Fermi energy

uniform distribution by using the Airy functions. In this study, the non-uniform distribution is different from the Gaussian distribution used by other authors. The electronic structure has been calculated by solving the Schrödinger and Poisson equations self-consistently. It is shown that the sub-band structure depends strongly on doping concentration and donor thickness. With increase of the δ- doping concentration, the electronic structure is dependent

on N

_d^2D

; hence N

_d^2D

can be used as a tunable parameter for these systems. The distribution of the electrons over the levels is quite sensitive to the thickness of the donor distribution. Thus, the thickness of the donor distribution can be used as a fitting parameter in the calculations to obtain the same sub-band population in this system.

The δ-doping technique enables the ultimate limit scaling of doping profiles. Such scaling of the dimensions of a semiconductor structure is quite important for semiconductor devices. Future research and development of semiconductor structures will make use of the advances and the knowledge gained in the course of research in the field of δ-doping.

This method used for the calculation of Poisson equations is also extended to obtain the unknown part of the potential.

The procedure is applied only for the known carrier distri- butions as explained in the appendix. Also, our method will be extended to structures under an applied electric field on the electronic sub-bands in a V-shaped potential, in future studies.

Appendix:

To avoid the difficulty of the double integration of the Airy functions to solve the Poisson equation we reduced this double integration to the single integration as explained below. In Fig. 1 the inset shows the interface (say the z

j

point) for two different potential regions. At the region ( j + 1) the Poisson equation can be written as (A1)

d

²

V

j+1

(z)

dz

²

= −4ð

j+1

(z) . (A.1)

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At the point z = z

j

, by using the continuity of potentials, the first derivative of the unknown potential V

j+1

(z) and the unknown potential V

_j+1

(z) can be written as, respectively

dV

_j+1

(z) dz = −4ð

z

zj

dz

j+1

(z

) + dV

j

(z) dz

z=zj

, (A.2)

and

V

_j+1

(z) = − 4ð

z

z_j

dz

z

z_j

dz

j+1

(z

) + dV

j

(z) dz

z=zj

× (z − z

j

) + V

j

(z)|

z=zj

. (A.3) The double integral on the right-hand side of (A.3) can be reduced to the single integral as follows

V

j+1

(z) = − 4ð

z

zj

dz

(z − z

)

j+1

(z

) + dV

j

(z) dz

_z=z

j

× (z − z

j

) + V

j

(z)|

z=zj

. (A.4)

Thus, the unknown potential can be obtained from the known one. Under the same conditions, by using the continuity of potentials at each region, the unknown potentials can be found.

References

1. E.F. Schubert, A. Ficher, K. Ploog: IEEE Trans. Electron Dev. 33, 625 (1986)

2. E.F. Schubert, K. Ploog: Jpn. J. Appl. Phys. 24, L608 (1985) 3. E.F. Schubert, J.M. Kuo, R.F. Kopf, H.S. Luftman, L.C. Hopkins,

N.J. Sauer: J. Appl. Phys. 67, 1969 (1990)

4. S.M. Shibli, L.M. Scolfaro, J.R. Leite, C.A.C. Mendonça, F. Plentz, E.A. Meneses: Appl. Phys. Lett. 60, 2895 (1992)

5. A. Zrenner, F. Koch, K. Ploog: Surf. Sci. 196, 671 (1988) 6. L. Ioriatti: Phys. Rev. B 41, 8340 (1990)

7. M.H. Degani: Phys. Rev. B 44, 5580 (1991)

8. A. Ben Jazia, H. Mejri, H. Maaref, K. Souissi: Semicond. Sci. Technol.

12, 1388 (1997)

9. E. Ozturk, Y. Ergun, H. Sari, I. Sokmen: Superlattices Microstruct. 28, 35 (2000)

10. M.H. Degani: J. Appl. Phys. 70, 4362 (1991)

11. P.M. Koenraad, F.A.P. Blom, C.J.G.M. Langerak, M.R. Leys, J.A.A.J. Perenboom, J. Singleton, S.J.R.M. Spermon, W.C. Van der Vleuten, A.P.J. Voncken, J.H. Wolter: Semicond. Sci. Technol. 5, 861 (1990)