High frequency hopping conductivity in semiconductors. Acoustical methods of

research.

I.L.Drichko

Ioffe Physicotechnical Institute RAS

Физико-технический институт им. А.Ф.Иоффе РАН, 194021, С.-Петербург, ул.Политехническая, 26

Outline

• 1.Two-site model of high frequency hopping conductivity

• 2. 3-dimensional high frequency hopping

• 3. 2-dimensional high frequency hopping

• 4. high frequency hopping in system with

• dense arrays of Ge –in- Si quantum dots

High- frequency hopping conductivity

• Two-site model

E

2 2E

,

=1-2 is the difference between initial

energies of impurity sites 1 and 2 (r)= 0e

-r/ is the overlap integral, where

0 EB, is the localization length

r

1.Resonant (phononless) absorption

2.Relaxation (nonresonant) absorption

kT E

E

Two-site model can be applied if ()>>(0). The hops between different pairs are absent..

12

Relaxation case

2

2 2~

1p

1 2 2( ) ( ( ) )E t e t r

0n n n

t

M.PollakV.GurevichYu.GalperinD.ParshinA.Efros

B.Shklovskii

n0 is the equilibrium value of n

The very important point is that it is necessary to take into account the Coulomb correlation (A.Efros, B.Shklovskii)

Two regimes

2

0

1 1( )

( , ) ( )E r E E

0 (E) is the minimum value of the population relaxation time for symmetrical pairs with =0

0<<1 ~hf ~T00>>1, ~hf~1/0(kT)~0Tn

~cos t

Effect of magnetic field

0 2

cH

e

c

eH

• An external magnetic field deforms the wave function of the impurity electrons and reduces the overlap integrals . This integral depends on the angle between the magnetic field Н and an arm of pair r.

Weak magnetic field Н<H0 ~H2 ~H2

High magnetic field Н>H0 ~H-4/3 ~H-2

-(H)=(0)- (H) (H)=- (0)+b/H2

Acoustic methods

Sample CABLE

piezotransducer

Setup for 3-dimensional systems

Setup for low dimensional systems

17-400 MHz150-1500 MHz

T=0.3-4.2 K, H=0-8 T

Dependences of (0) от Т; f=810(1),

630(2), 395(3),336(4), 268(5),207MHz(6) Dependences of оn Н;

1-0.58К, 2-2.15К, 3-4.2К f=810 MHz

Lightly doped strongly compensated (К=0.84) n-InSb, 3-dimensional case

21

2 22 1

(4 ( ) / )8.68

2 [1 4 ( ) / ] [4 ( ) / ]s

s s

K t q VqA

t q V t q V

A = 8b(q)(1+0) 02sexp[2q(a+d)],

VV

KA

t q V

t q V t q Vs

s s

22

22

122

1 4

1 4 4

[ ( ) / ]

[ ( ) / ] [ ( ) / ]

28.68

2 21 ( )M

M

K q

2- Dimensional case

14M

1 2hf i

3-dimensional case

1= Re hf ~ 2= Im hf ~ V/V

HF-hopping in 2D case

1 2hf i 2 1Im( ) Re( )hf hf

Re ~hf Im ~hf

A.L.Efros, Sov.Phys.JETP 62 (5),p.1057 (1985)

21

2 22 1

(4 ( ) / )8.68

2 [1 4 ( ) / ] [4 ( ) / ]s

s s

K t q VqA

t q V t q V

A = 8b(q)(1+0) 02sexp[2q(a+d)],

VV

KA

t q V

t q V t q Vs

s s

22

22

122

1 4

1 4 4

[ ( ) / ]

[ ( ) / ] [ ( ) / ]

28.68

2 21 ( )M

M

K q

2- Dimensional case

14M

1 2hf i

3-dimensional case

1= Re hf ~ 2= Im hf ~ V/V

The absorption coefficient Γ and the velocity shift V/V vs. magnetic

field (f=30 MHz)

The dependences of real 1 and imaginary 2 parts of high frequency conductivity , T=1.5 K, f=30 MHz; n-GaAs/AlGaAs

Dependences of 1, 2 on H near =2 at different T, n-GaAs/AlGaAs

Two-site model nonlinearity

0 50 100 150 200 250 300 350

1.0

1.5

2.0

2.5

3.0

3.5

E, a

.u

t, a .u.

kT

2 2( ) ( ( ) )E t e t r

The systems with a dense (4The systems with a dense (410101111 cm cm–2–2) array of Ge ) array of Ge quantum dots in silicon, doped with B.quantum dots in silicon, doped with B.

Quantum dots (QD) has a pyramidal shape with the square base 100×100 ÷ 150×150 Ǻ2 and the height of 10-15 Ǻ. The samples have been delta-doped with B with the concentration (1÷1.12)·1012 cm-2.

The boron concentration The boron concentration corresponds to the average corresponds to the average QD filling QD filling 2.85 2.852.5 per 2.5 per dotdot

Linear regime

In linear regime the high frequency hopping conductivity looks like hopping predicted by of "two-site model" provided >1 if holes hop between quantum dots. But 1> 2.

Left-Temperature dependence of in the sample 1 for f=30.1 and 307 MHz, a=510-5cm. Right-Frequency dependence of in the

sample 2 at T-4.2 K, a=410-5cm

Nonlinear regime

Results of numerical simulations for b (the distance between the dots) Galperin, Bergli

Conclusion

• Hopping relaxation conductivity• At R>, • where R is the distance between pairs of impurity site, is the localization length• 1. Hopping conductivity in 3-dimensional strongly compensated

lightly and heavily doped semiconductors (n-InSb) is successfully• explained by two-site model• In strongly compensated lightly doped n-InSb it was observed

crossover from <1 to >1.

• 2.In two-dimensional structures with quantum Hall effect there is hopping conductivity. This one is observed in minima of conductivity at small filling-factors and it is successfully explained by two-site model too. In this case Im >Re

• At R

• 3. The main mechanism of HF conduction in hopping systems with large localization length (dense arrays of Ge –in- Si quantum dots) is due to charge transfer within large clusters.

Acknowledgments

• I am very grateful to my numerous co-authors: • Yu.M.Galperin, L.B.Gorskaya, A.M.Diakonov,

I.Yu.Smirnov, A. V.Suslov, V.D.Kagan, D.Leadley, • V. A.Malysh, N.P.Stepina, E.S.Koptev, J.Bergli,

B.A.Aronzon, D. V.Shamshur

• and ours very good technologists: V.S.Ivleva, A.I.Toropov, A.I. Nikiforov