Observation of the Random-to-Correlated Transition of the Ionized Impurity Distribution in Compensated Semiconductors

Observation of

the Random-to-Correlated Transition of the

Ionized Impurity Distribution in Compensated Semiconductors

Jiro Kato ^a , Kohei M. Itoh ^a , and Eugene E. Haller ^b

a Dept. Applied Physics and Physico-Informatics, Keio University Yokohama, 223-8522 Japan

b UC Berkeley and Lawrence Berkeley National Labs, Berkeley, CA 94720 USA

Compensated Semiconductor

Ge:(As;Ga)

major impurity is As minor impurity is Ga

Conduction band

Valence band 　　　　 Ionized impurity concentration: N

= 2[Ga]

(our samples are K=0.6) Neutral impurity concentration: N

= [As] – [Ga]

Compensation ratio: K = [Ga]/[As]

1. Introduction

Distributions of Ionized Impurities

Temperature (Thermal Energy)

Random distribution Correlated distribution

Random Theory :

D. M. Larsen, Phys. Rev B 13, 1681 (1976)

Correlated Theory :

S. M. Kogan and N. Van Lien, Sov.

Phys. Semicond. 15, 26 (1981) n-type

Ge:As,Ga

Ionized Impurity Concentration (Coulombic Energy)

Can we change distribution of ionized impurities

by changing T and N

?

Coulomb gap width vs. k _B T and N _I

6 1 2 3 4

2 



 



 ⋅

⋅ K

N

e π _I

κ

= k

T N

=7.5x10

cm

at T=4 K Random

Distribution

Partially

Random Completely Random

k

T

Lager Δ for lager N

Conduction Band

Correlated Distribution

Electrons Density of States

Energy

Donor Level

Δ

K=0 K=0.1 K=0.5

Width of the

Coulomb gap Δ= ^N

Objectives

Experimental observation of

the correlated to random ionized impurity distribution as ionized impurity concentration

and functions of the temperature

Temperature

Ionized Impurity Concentration Correlated Distribution Random Distribution

Experiment 1 Experiment 2

Determination of the ionized impurity distribution

2: Experiment

As : 1s →2p

Absorption

Correlated

Correlated distribution Random distribution

Large Electric Field Small Electrical Field

Random

1s 1s

2p 2p

Position

0 1 C om pe ns ati on R ati o

Compensation Ratio

K=N

/N

As Concentration

Ga Concentration

Im pu ri ty C on ce nt ra ti on

Samples

Position 0

1 Ge Ingot CZ-grown Ge

Samples were cut from several points

Ge with As and Ga Fig.1 Impurity concentration

vs. position of the ingot

Theoretically expected linewidth for the two distributions at fixed T

Far infrared absorption of As:1s

2p

line

Hall effect 300K～10K

BOMEM DA-8 Si:B Bolometer

0.0 5.0x10¹³ 1.0x10¹⁴ 1.5x10¹⁴ 2.0x10¹⁴ 2.5x10¹⁴ 3.0x10¹⁴ 0.0

0.2 0.4 0.6 0.8 1.0 1.2

FW HM [cm

]

Ionized impurity concentration [cm

]

Fig.2 Theoretical predictions

Random distribution

Correlated

distribution

Determination of

the horizontal axis of Fig.2



 



 Ν

+ −







 

 



 + Ν

+



 



 + Ν

= −

T k

E N N T

k E N T

k E N

N n N

B D C

A D B

D C

A B

D C

A

A D

)exp (

exp 4 1

exp 1

) ( 2

2 β

β β

N : Carrier concentration N_D: Donor concentration N_A: Acceptor concentration E_D: Ionize energy of donors k_B: Boltzmann constant T : Temperature β=2

Find N _D , N _A , and K for each sample (Compensation ratio : K= N

/N

)

N

(Ionized Impurity Concentration)

N

=2 × N

5 6 7 8 9 10 11 12 13 14 15

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Free Carrier Concentration [cm-3 ]

1/Temperature [K^-1] 10

10 10 10 10 10 10 10 10 10 10

Determination of the vertical axis of Fig.2

Temp : 4K As : 1s→2p±

Absorption Coefficient [cm-1]

Wavenumber [cm

]

Lorenztian Fitting

Full Width at Half Maximum of As:1s → 2p

line

( ^{k k}

) ( ^FWHM

^{FWHM / 2 / 2} )

^.

A − +

Experimental Results

0.0 5.0x10¹³ 1.0x10¹⁴ 1.5x10¹⁴ 2.0x10¹⁴ 2.5x10¹⁴ 3.0x10¹⁴ 0.0

0.2 0.4 0.6 0.8 1.0 1.2

FW HM [cm

]

Ionized Impurity Concentration [cm

] T = 4K

0.0 5.0x10¹³ 1.0x10¹⁴ 1.5x10¹⁴ 2.0x10¹⁴ 2.5x10¹⁴ 3.0x10¹⁴ 0.0

0.2 0.4 0.6 0.8 1.0 1.2

FWHM [cm-1 ]

Ionized impurity concentration [cm^-3]

3. Monte Carlo simulation of the linewidth for correlated and random distributions

100 100.2 100.4 100.6 100.8 101

Counts of Data

Wavenumber [cm^-1]

･ Correlated

･ Random

N

= 3.50x10

cm

K = 0.6

Number of Donors = 200 Number of Acceptors = 120 N

= 3.50x10

cm

K = 0.6

Number of Donors = 200 Number of Acceptors = 120

T = 0 K Random

distribution

Correlated

distribution

0 1x10

2x10

3x10

0.0

0.2 0.4 0.6 0.8 1.0 1.2

Correlated Distribution Random Distribution

FWHM [c m

]

Ionized Impurity Concentration [cm

]

4: Comparison:

Experiment and Theory (1)

T=4K

0 1x10

2x10

3x10

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Correlated Distribution Random Distribution

FWHM [cm

]

Comparison:

Experiment and Theory (2)

Completely

Random Partially Correlated

Transition N

from Random to Correlated

Partial Randamization due to the thermal energy

by T = 4 K

Effect of raising Temperature

Further thermal randamization

0 1x10

2x10

3x10

0.0 0.2 0.4 0.6 0.8 1.0 1.2

FW HM [c m

]

Ionizd Impurity Concentration [cm

] 7.5x10

cm

10K 4K

( )

Relation between critical T and N _I

κ

= 16.1 K = 0.66

(T

K)

Thermal Energy

Thermal Energy > Coulomb Gap : complete random distribution Thermal Energy < Coulomb Gap : correlated distribution

6 1 2 4

3

2  

 



 ⋅

⋅

=

∆ K

N e π

κ

For N _I =7.5x10 ¹³ cm ^-3

Coulomb Gap linewidth

＝ 0.33 meV N

k

T= 0.34 meV

Excellent agreement between Δ and k _B T

0 1x10¹⁴ 2x10¹⁴ 3x10¹⁴

0.0 0.2 0.4 0.6 0.8 1.0 1.2

FWHM [cm-1]

Ionizd Impurity Concentration [cm^-3]

7x10

cm

10K

4K

Temperature - Induced Transition

[ ]

( )

[ ] [ ]

( ) N ( Ry kT)

n Ga As

Ga n n

B

Cexp *

1 −

− =

− +

β

( )

I

n T N

N = + 2

Thermal Ionization of donors

Completely Random Correlated to Random Transition

Partially Correlated Temperature

T

≅5K in agreement

with the prediction T

=4.1K

4 6 8 10 12 14 16 18

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

N_I=2.26×10¹⁴[cm^-3]

NI= 4 . 3 2×1 01 3[ c m- 3]

Ionized Im pu rity Concentration [c m

]

10.5x10¹³ 10.0x10¹³ 9.5x10¹³ 9.0x10¹³ -

-

-

- 8.5x10¹³ -

4 6 8 10 12 14 16 18

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

FWHM [cm-1]

Temperature [K]

N_I=7.80×10¹³[cm^-3]

FWH M [ cm

]

Temperature [K]

N_I=2.26×10¹⁴[cm^-3]

N_I=4.32×10¹³[cm^-3] N_I=7.80×10¹³[cm^-3]

Conclusion

・Experimentally observed Random-Correlated transition for the first time

・ The transition occurs around N

= 7.5x10

cm

at 4K in excellent agreement with the theoretical prediction

・ thermal energy > coulomb gap : completely random distribution

thermal energy < coulomb gap : correlated distribution

Create

correlated distribution

Appendix 1 :

Monte Carlo calculation of the linewidth for the correlated and random distributions

Repeat for 8000 central donors to calculate the inhomogeneous broadening

Monte Carlo Method

Random distribution

Correlated distribution

Distribute more than 100 donors and acceptors in the cell randomly

Create distribution of

neutral and ionized impurities randomly

Calculate electric field

for the neutral donor at the center of the cell

Creation of the Correlated Distribution (1)

Change the position of 1 electron

Compare the total coulombic energy

Accept the new distribution

Reduced Increased

Reject the

new distribution

Repeat for more than 20,000 times

Creation of the Correlated Distribution (2)

Pseudoground States

-1.4 10⁷ -1.2 10⁷ -1 10⁷ -8 10⁶ -6 10⁶ -4 10⁶ -2 10⁶ 0 2 10⁶

0 5000 10000 15000 20000

N_I=10¹⁴[cm^-3] 800 Donors 400 Acceptors

Number of Trial Random

distribution

Correlated distribution

Distribution in a unit cell

N_d=800,N_a=400 Ionized donor

Neutral donor Acceptor

Y Axis

Z Axis

N_d=800,N_a=400 Ionized donor

Neutral donor Acceptor

Y Axis

Z Axis

Random Distribution Correlated Distribution

Calculational Results

0 200 400 600 800

-1000 -500 0 500 1000

Counts of Data

Coulombic Energy at Neutral Impurity center [Na*3Ry*]

Distribution of the Stark Energies in case of the Random distribution

0.0 5.0x10¹³ 1.0x10¹⁴ 1.5x10¹⁴ 2.0x10¹⁴ 2.5x10¹⁴ 3.0x10¹⁴ 0.0

0.2 0.4 0.6 0.8 1.0 1.2

FWHM [cm-1 ]

Ionized impurity concentration [cm

]

Random Distribution

Correlated Distribution FWHM

Appendix 2 :

Other broadening mechanism

1: Phonon life broadening ・・・ 0.066 cm ^-1 2: Concentration broadening ・・・ 0.04 cm ^-1 3: Dislocation broadening ・・・ 0 cm ^-1 4: Electric field broadening

Broadening of electron states

Electric field broadening

イオン化不純物濃度

線形スタルク効果 二次のスタルク効果

四重極相互作用

Full width at half maximum of 1s→2p absorption peaks [cm-1]

Ionized impurity concentration

Linear Stark effect Quadratic Stark effect

Quadrupole interaction

(1) Linear Stark effect (

E) Δε∝ N

(2) Quadratic Stark effect (

E

) Δε∝ N

(3) Quadrupole interaction (

dE/dz) Δε∝ N

E : electric field strength Δε: The increase of linewidth

N

: Ionized impurity concentration

1s→2p_±: Linear Stark effect does not occur

Important for low N