Many electron theory of 1/f Noise in doped semiconductors

Alexander Burin

Tulane University

Motivation (Fundamental)

Understanding the nature of anomalously strong 1/f noise in hopping conduction (e. g. McCammon, 2000-2006; Savchenko, 2000-2003 (Si-P-B); G. Deville, 2006, (Ga-As))

0.30.10.03 T (K) 1001010.1 (Hz)

(a.u.) 1)(

2

2

(a.u.) 1)(

62

2

T

10

100

1

0.110-2

100

102

104

106

Motivation (Practical)

1/f-noise affects a performance of semiconductor bolometers (McCammon, 2000-2006; Gershenson, 2000-2003 (Si-P))

Bolometers detect absorption of single X-ray or cosmic particle and can measure its energy by means of the change in temperature affecting the semiconductor conductivity

Universal low temperature conductivity in doped semiconductors (Shklovskii, Efros, 1978)

ln

T-1/2

7.2

,

,exp

2

0

2/1

00

Ca

eCT

T

T

Universal strong temperature dependence serves to define the small temperature variation induced by X-ray absorption

1/f noise in operation regime T~0.1K

6

0

2

2

50T(K)

T

law) s(Hooge' )()(

donorsN

T

Goal: Develop the general theory to account for the universal 1/f-noise

Previous work - 1

1. 1/f noise is caused by tunneling (McWorter, (1957))

f

df

k

dkdr

a

rk

2exp~

r

r/2

ka

rk

2/exp~'

Hopping through intermediate sites breaks down 1/f transition rate statistics

Previous work - 22. 1/f noise is caused by tunneling from traps (Shklovskii, (2003); Yu, (2003); Kozub (1996) occasional configurations with no intermediate sites)

0

320

ln2

34 , ,)(exp~)( ,2

exp

ar

π/vπvrrnvPa

r dd

r

E

Er

)/ln(

~)/exp(- ,)(~

,...2,1 ,

0

0

TE

TEEEgn

iEE

r

rrr

ri

E1

E2

Previous work - 3

Trap noise, high T

behavior 1/f-quasi - )(

)(

ln2

explnexp)(

)/ln( ,)(

1

00

00

0000

PI

Ta

gvrTgvP

TgEgnconstgEg

dd

dd

d

rr

Trap noise, low T

Previous work - 4

1~)/(

)( ,)(

32

2

6

3

6

2

rr

ereg

e

EEEgn

e

EEg

One charge with energy e2/r per volume r3 (Efros, Shklovskii, 1975)r

e2/r

2e2/r

Previous work - 5

behavior 1/f-quasi -

lnexp

)(

lnexp)(

6

0

3

2

6

0

3

2

eaT

I

e

aTP

d

d

Exponent reaches 1 for the variable-range hopping rate

a

e

T

T 2

0

2/1

00 T ,exp

Problems of trap model

)(?!100exp

1ln21)dln(

dln(I)

lnexp

)(

5

3

00

5

0

3

2

6

0

3

2

HzT

T

e

aTd

eaT

I

I

1001010.1 (Hz)

Hypothesis: Involvement of multi-electron tunneling

(1) Simultaneous tunneling of multi-electron (N-electron) coupled clusters is characterized by tunneling amplitude V ~ exp(-aN), leads to 1/f noise if transition rates

(2) Clusters can be formed due to long-range interaction (Burin, Kagan, 1995, 1996)

(3) We exploit the most straightforward case of “random order”, i. e. Wigner crystal like configuration formed statistically

(4)External noise source (atomic tunneling, etc.) is less probable because of the correlation of noise with metal-insulator transition

Chessboard cluster

r

Probability to form chessboard cluster of N sites

rRr rii Structure close to that of the Wigner’s crystal

1~ ,),,(2

3322

6 N

N

ErNcr e

r

er

r

eg

r

erEP

rR

r

eE Eii

2

Site energy reproduces that of Wigner’s crystal

Transition of chessboard cluster: tunneling

Tunneling

a

rNtun

N 2exp0

Transition of chessboard cluster: thermal activation

Thermal activation of domain boundary

rT

eNact

N

23

2

0 2exp

Statistics of transition rates

rT

eN

a

rNact

NtunN

totN

23

2

0 2exp2exp

)(),(26

totn

N

n e

r

r

edEf

r

Statistics of transition rates - 2

Main contribution comes from the crossover regime N~Nc

0

2

3

2

~2exp~2exp

Tr

eN

a

rN

cc

cc

05/65/3

2

2/3

222

05/65/3

2c05/1

5/32

lnexp11

),(

;lnN ;ln

e

Ta

e

Ta

eaEf

e

Ta

aT

erc

r

rc

Deviations from 1/f statistics

325

0

05/15/3

2

05/65/3

2

2/3

222

6

5exp

ln5

61

ln

ln

lnexp11

),(

aT

e

e

Ta

d

fd

e

Ta

e

Ta

eaEf

Practically unlimited applicability at low temperature T<0.1e2/a

Conductivity noise

e2/T432

int

3int

32

2

2

,)0,(

1~

)(

/

str

str

aT

e~aR

TREfP

PT

e

V

05/6

5/324/1123

2

2

lnexp)(

aT

e

aT

e

V

a

Hooge constant, comparison with experiment

law) s(Hooge' )()(

2

2

donorsN

T

05/6

5/324/1123

2

2

lnexp)(

aT

e

aT

eand

Kak

e

an

B

d

11

,104.8

,395.2

2

33

Results, for higher temperature, lower dimension

Conclusions

1. Correlated transitions in coupled many-electron clusters account for the 1/f noise in a hopping conduction

2. Clusters are made of ordered “crystalline” configurations formed due to fluctuations of a random potential

Acknowledgements

Boris Shklovskii , special acknowledge for supporting my life and work in UMN in the Fall 2005 (where this work has been done) during the disaster in New Orleans

Coworkers:

Veniamin Kozub

Yuri Galperin

Valery Vinokur

Funding: