Noise of hot electrons in anisotropic semiconductors

in the presence of a magnetic field

Francesco Ciccarello in collaboration with:

S. Zammito and M. Zarcone

CNISM & Dipartimento di Fisica e Tecnologie Relative, University of Palermo (Italy)

 UPON 2008, École Normale Supérieure de Lyon (France), June 2–6, 2008

a sketch of this work

in an anisotropic seminconductor:how hot-electron velocity fluctuations are affected by

the presence of a static magnetic field?

1. the relative importance of noise can be reduced

2. partition noise can be strongly attenuated

3. a simultaneous increase of conductivity can take place

2

problem

our outcomes

t

t

l

l

m

k

m

k

22)(

2222 k

conduction band of Si

3

3 pairs of energetically-equivalent

ellipsoidal valleys along:

<100> (“valleys 1” )

<010> (“valleys 2” )

<001> (“valleys 3” )

t

yz

l

x

m

k

m

k

22)(

2222 k

electrons’ inertia does depend on the direction and valleys

ml=0.916 m0 > mt=0.19 m0

t

xz

l

y

m

k

m

k

22)(

2222 k

we apply an intense electric field (~ kV/cm)

4

valleys 1 cold & less populated

valleys 2,3 hot & most populated

velocity fluctuations

5

v1x< v2x,v3x

heavy valleys 1 slow electrons

light valleys 2,3 fast electrons

v1x < vd < v2x=v3x

partition noise

in the <100> case

longitudinal velocity autocorrelation function

let us add a magnetic field in a Hall-geometry

6

B=0 B≠0

valleys1 cold & most populated hot & less populated

valleys 2,3 hot & less populated cold & most populated

Asche M and Sarbei O G, Phys. Stat. Sol. 37, 439 (1970)Asche M et al., Phys. Stat. Sol. (b) 60, 497 (1973)

also: conductivity is enhanced

EA

EH

B<001>

<010>

how are velocity fluctuations affected?

7

Ciccarello F and Zarcone M, J. Appl. Phys. 99,113702 (2006)

Ciccarello F and Zarcone, AIP Conf. Proc. 800, 492 (2005 ) Ciccarello F and Zarcone, AIP Conf. Proc. 780, 159 (2005 )

n-GaAs (isotropic)

T=77 K; stationary conditions; maximum magnetic-field strengths

allowing magnetic-field effects competing with electric-field ones

our Monte Carlo approach

8

free-flight Newtonian equation

BkvEk )(q

t

t

l

l

m

k

m

k

22)1(

2222 zyxi

m

k

i

ii ,,

)21(v

scattering processes

3 f -type intervalley processes

3 g -type intravalley processes

acoustic scattering

Hall field

we iteratively look for EH such that Ad Ev ||Raguotis R A, Repsas K K and Tauras V K, Lith. J. Phys. 31, 213 (1991)

Raguotis R, Phys. Stat. Sol. (b) 174 K67 (1992)

Brunetti R, et al., J. Appl. Phys. 52, 6713 (1981)Jacoboni C, Minder R and Majni ,J. Chem. Phys. Solids 36,1129 (1975)

regime

(valley i j)

(valley i i)

B (T)

0 2 4 6 8

rela

tive

sta

nd

ard

dev

iati

on

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

B (T)

0 2 4 6 8

ave

rag

e e

ne

rgy

(eV

)

0.040

0.042

0.044

0.046

0.048

0.050

0.052

0.054

0.056

0.058

0.060

B (T)

0 2 4 6 8

vari

an

ce

(1

01

4c

m2

s-2

)

1.4

1.6

1.8

2.0

2.2

2.4

B (T)

0 2 4 6 8

dri

ft v

elo

city

(10

6c

ms

-1)

8

9

10

11

12

13

central result

9

drift velocity

average energy

velocity variance

standard deviation/drift velocity

the magnetic-field action yieldsa more intense & cleaner current

~20

~%

EA=6 kV/cm

EA=10 kV/cm

EA=6 kV/cm

EA=10 kV/cm

EA=6 kV/cm

EA=10 kV/cm

EA=6 kV/cm

EA=10 kV/cm

B (T)

0 1 2 3 4 5 6 7 8 9

ave

rag

e v

elo

cit

y (1

06

cm

s-1

)

4

6

8

10

12

14

B (T)

0 1 2 3 4 5 6 7 8 9

ave

rag

e e

ne

rgy

(eV

)

0.03

0.04

0.05

0.06

0.07

B (T)

0 1 2 3 4 5 6 7 8 9

occu

patio

n pr

obab

ility

0.1

0.2

0.3

0.4

0.5

0.6

insight into the mechanism behind

10

the magnetic field increases population of high-velocity valleys

valley occupation probabilitiesvalley average

energiesvalley average velocities

valleys 1

valleys 2, 3valleys 1

valleys 2, 3

mean

EA=6 kV/cm EA=6 kV/cm valleys 2, 3

EA=6 kV/cm

valleys 1

drift velocity

isotropic case: free-flight motion

BkvEk )(q *

22

2)(

m

kk

*m

kv

0

00

00

)(

sincos)(

cossin )(

zz

CxyCy

xyCx

kk

kkkkk

kkkk

B

EmkC

*

*m

qB

A. D. Boardman et al., Phys. Stat. Sol. (a), 4, 133 (1971)

cyclotron frequencycenter magnitude

11

measures themaximum amount of gainable energy

K*-space

12

Herring-Vogt transformation

3/12*** tldzz

dzy

y

dyx

x

dx mmmk

m

mkk

m

mkk

m

mk

dispersion law in each valley

nonparabolicity effects

)(21 *

**

k

kv

dm

dm

k

2)1(

2*2

dm

** k

v

velocity vs. wavevector

Herring C and Vogt E, Phys. Rev. 101, 944 (1956)

t

t

l

l

m

k

m

k

22)(

2222 k

dm

kk

2)(

2*2*

fields*

13

in k *-space fields are valley-dependent

kjiB

kjiE

BkvEk

)(

*

*

*****

yx

d

zx

d

zy

d

z

d

y

d

x

d

mm

m

mm

m

mm

m

m

m

m

m

m

m

q

free-flight equation in each valley

isotropic

*

2*2***

B

EEmk HAC

as B grows theeffect of EH becomes

more & more significantmeasures the

maximum energy that can be gained

fields*

14

kkB

kjE

iE

*

*

*

tl

d

tl

d

t

d

t

dH

l

dA

mm

m

mm

m

m

m

m

m

m

m

kjB

kjE

iE

*

*

*

tl

d

tt

d

t

d

l

dH

t

dA

mm

m

mm

m

m

m

m

m

m

m

valleys 1 valleys 2 valleys 3

kjiBkjiE **

yx

d

zx

d

zy

d

z

d

y

d

x

d

mm

m

mm

m

mm

m

m

m

m

m

m

m

kjB

kjE

iE

*

*

*

tt

d

tl

d

l

d

t

dH

t

dA

mm

m

mm

m

m

m

m

m

m

m

t (ps)

0 1 2 3 4 5 6au

toc

orr

ela

tio

n f

un

cti

on

(1

01

4c

m2

s-2

)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

t (ps)

0 1 2 3 4 5 6

auto

corr

elat

ion

fu

nct

ion

(10

14cm

2 s-2

)

-0.1

0.0

0.1

0.2

0.3

autocorrelation functions

15

longitudinal B=0 vs. B=4.5 T

partition-noise attenuation

EA=5 kV/cm

longitudinal vs. transversal

EA=5 kV/cmB=4.5 T

partition-noise is shifted towards the transversal direction

valleys 1

valleys 3

valleys 2

is partition noise lost?

different valleys have different

transversal velocitiesB (T)

0 1 2 3 4 5

tran

sver

sal a

vera

ge v

eloc

ity (

106 c

m s

-1)

-8

-6

-4

-2

0

2

4

6

8

10

transversal average velocity

conclusions

16

∙ B can give rise to a more ordered conductive state with:

- higher mobility

- reduced importance of fluctuations

- attenuated partition noise

- moderate electron heating

open questions & outlook

17

∙ what about different Hall geometries such asEA ll <011> & B ll <100> or EA ll <111> & B ll <110>??∙ what’s the optimal geometry in order to

minimize noise?

∙ is there a statistical procedure able to filter out mere fluctuations from cyclotronic oscillations?

∙ is there a better system able to exhibit such phenomena (if possible, at room temperature)?

∙ noise spectrum analysis

thanks for your attention!