Conductance - UPV/EHU · energy separation ∆ε between the quantum states in the reservoir, ∆ε...

Conductance

42

•Transport: quantization of the conductance.


Quasi-1D systems with width w comparable to the de Broglie wavelength

Diffusive transport regime = mean free path

Ballistic transport regime !

!

! << !, L !! >> !, L

43


Quantized conductance discovery

44


Quantized conductance discovery

45


!

k

r

!

k

r

eVbias

a

Quantization of conductanceBallistic regime

46

Landauer Formula47


DOS per unit

volume at Fermi level

More precise calculations.. (pdf-notes Bruus, Datta...)6.6. CONDUCTANCE AND SCATTERING MATRIX FORMALISM 73

Figure 6.8: Electrical currents through a nanostructure at zero temperature with injectionof electrons from a left and a right electron reservoir. Without an applied bias voltage Vwe have µL = µR, and the left- and right-going currents cancel each other yielding a zeronet current. For V > 0 we have µL = µR + eV and the current carried by right-goingelectrons with energies larger than µL ! eV and smaller than µL is not compensated, thusresulting in a non-zero net current.

6.6.1 Electron channels

For a given quantum number ! there are many possible quantum states, namely two (spinup and spin down) for each k. One therefore talks about the electron channel !. The totalenergy of a state in a given channel ! is denoted "k!. It consists of two contributions: onedenoted "! is the quantum energy of the transverse wavefunction #!(y, z), while the otheris the usual kinetic energy "k = !2k2/2m associated with the motion in the x direction.From a simple energy consideration we derive an expression for k:

"k! = "k + "! " k =1!

!2m("k! ! "!). (6.49)

Note that the lowest possible energy for electrons in channel ! is "! obtained when fillingthe state $0!.

6.6.2 Current, reservoirs, and electron channels

We now analyze the current flow sketched in Fig. 6.8 starting with the current Ik! carriedby an electron in state $k! in channel !. It is given by Eq. (6.10), with a factor of 2 forspin inserted,

Ik! = !2e!Lm

Im"e!ikx %

%x

#eikx

$%= !2e!k

Lm= !2

e

Lvk!, (6.50)

where vk! is the velocity of an electron in the state $k!.It is now easy to derive an expression for the current IL"R running from the left

reservoir L into the right reservoir R through the nanostructure. We consider the energy

The chemical potential can be changed in a controlled manner by applying a voltage V :

62 CHAPTER 6. TRANSPORT IN NANOSTRUCTURES

structure. The electrons that flow through the nanostructure is supplied from macroscopicmetal contacts. The energy and electron number of the large reservoirs are not a!ectedby the presence of the nanostructure. It is thus natural to denote the metal contacts aselectron reservoirs.

In the reservoirs the electrons behave more or less as classical particles. This is mainlydue to the large volume V of the reservoirs, since a large V leads to a vanishingly smallenergy separation "! between the quantum states in the reservoir, "! ! !2/2mV

23 . Thus

given even the slightest perturbation the electrons will easily and frequently jump backand forth between quantum states whereby the distinct quantum nature is washed out.A given reservoir is characterized by its temperature T and its chemical potential µ, i.e.the free energy of the lastly added electron. The chemical potential can be changed in acontrolled manner by applying a voltage V :

µ = µ0 " eV. (6.1)

The probability nF that a given quantum state with energy ! is occupied by an electronis determined by the Fermi-Dirac distribution as described in Secs. 2.9.3 and 3.6:

nF(!, µ0, V, T ) =1

exp!

!!µ0+eVkBT

"+ 1

. (6.2)

In the nanostructure between the reservoirs the electrons must however be describedusing quantum physics. This is due to the small transverse size w of the nanostructurewhich leads to appreciable energy gaps "! given by "! ! h2/2mw2. It now requires asubstantial perturbation to force an electron to jump from one quantum state to the next.It is therefore very probable that a given electron stays in the same quantum state whilepassing through the nanostructure. The quantum properties are therefore not washed outand it can therefore be concluded that although the electrons are fed into the nanostructureas classical particles from a large reservoir the transport through the nanostructure mustnevertheless be described in terms of the electron wave picture of quantum theory.

6.2 Current density and transmission of electron waves

We begin our analysis of electric conductance of nanostructures by studying simple 1Dwaves in piecewise constant potentials. We discuss the electron waves, then their currentdensity, and finally the concept of transmission and reflection coe#cients T and R.

6.2.1 Electron waves in constant potentials in 1D

For an electron with energy ! moving in a given constant potential V the solution "!(x) ofthe time-independent Schrodinger equation

" !2

2m#2"!(x)

#x2+ V "!(x) = E "!(x) (6.3)

The probability n F that a given quantum state with energy ! is occupied by an electron

is determined by the Fermi-Dirac distribution




23 . Thus


µ = µ0 " eV. (6.1)


nF(!, µ0, V, T ) =1

exp!

!!µ0+eVkBT

"+ 1

. (6.2)






" !2

2m#2"!(x)

#x2+ V "!(x) = E "!(x) (6.3)

3.6. THE ELECTRON GAS AT FINITE TEMPERATURE 35

Figure 3.6: The Fermi-Dirac distribution nF(!k), its derivative !!nF!"k

, and its product withthe density of states, nF(!k)d(!k), shown at the temperature kT = 0.03 !F, correspondingto T = 2400 K in metals. This rather high value is chosen to have a clearly observabledeviation from the T = 0 case, which is indicated by the dashed lines.

3.6 The electron gas at finite temperature

In the previous section temperature was tacitly taken to be zero. As temperature is raisedfrom zero the occupation number is given by the Fermi-Dirac distribution nF(!k), seeEq. (2.55). The main characteristics of this function is shown in Fig. 3.6. Note that to beable to see any e!ects of the temperature in Fig. 3.6, kT is set to 0.03 !F correspondingto T " 2400 K. Room temperature yields kT/!F " 0.003, thus the low temperature limitof nF(!k) is of importance:

nF(!k) =1

e#("k!µ) + 1!#T"0

"(µ ! !k), (3.28a)

!#nF

#!k

=$

41

cosh2[#2 (!k ! µ)]!#T"0

%(µ ! !k). (3.28b)

At T = 0 the chemical potential µ is identical to !F. But in fact µ varies slightly withtemperature. A careful analysis based on the so-called Sommerfeld expansion combinedwith the fact that the number of electrons does not change with temperature yields

n(T = 0) = n(T ) =! #

0d! d(!)f(!) $ µ(T ) = !F

"1 ! &2

12

#kT

!F

$2+ . . .

%(3.29)

Because !F according to Eq. (3.13) is around 80000 K for metals, we find that even at themelting temperature of metals only a very limited number "N of electrons are a!ectedby thermal fluctuations. Indeed, only the states within 2kT of !F are actually a!ected,and more precisely we have "N/N = 6kT/!F (" 10!3 at room temperature). The Fermisphere is not destroyed by heating, it is only slightly smeared. Now we have at hand anexplanation of the old paradox in thermodynamics, as to why only the ionic vibrationaldegrees of freedom contribute significantly to the specific heat of solids. The electronic

48


Current density and transmission of electron waves

•Electron waves in constant potentials in 1D




23 . Thus


µ = µ0 " eV. (6.1)


nF(!, µ0, V, T ) =1

exp!

!!µ0+eVkBT

"+ 1

. (6.2)






" !2

2m#2"!(x)

#x2+ V "!(x) = E "!(x) (6.3)

6.2. CURRENT DENSITY AND TRANSMISSION OF ELECTRON WAVES 63

is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2

2m +V . The two possible signs ’+’ and ’!’ correspond to waves travellingto the right and to the left, respectively. This can be seen by writing out the full time-dependent solutions !+k(x, t) and !!k(x, t) as given in Eq. (2.30):

!+k(x, t) = ei(+kx! !! t), (right-moving wave), (6.5a)

!!k(x, t) = ei(!kx! !! t), (left-moving wave). (6.5b)

In the following we suppress the trivial time-dependence exp(!i !!t) and work only with

the time-independent Schrodinger eqaution and the time-independent solutions !±k(x).But of course the concept of left- and right-moving waves remains the same:

!+k(x) = e+ikx, (right-moving wave), (6.6a)

!!k(x) = e!ikx, (left-moving wave). (6.6b)

6.2.2 The current density J

The classical expression for current density is J = n v = n p/m, n being the density ofparticles. In quantum theory the density of a single particle is n = !"!, and a naıve guessfor the quantum expression for J would thus be J = !"! p/m. However, as argued inSec. 2.8.2 the true result is slightly more complicated owing to the fact that in quantumtheory p is a di!erential operator and not a number. Moreover, we need to ensure that Jis real, so as stated in Eq. (2.45) we end up with

J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)

The electric current I passing through the yz plane in the x direction is given by

I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)

In the case of simple product wavefunctions described in terms of some quantum numbersk in the x direction and $ in the yz directions,

!k"(r) = %k(x) &"(y, z), (6.9)

we find that the current Ik" carried by the quantum state !k" is

Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)

• Work with time-independent solutions


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)

49


The current density J


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)

The electric current I passing through the yz plane in the x direction


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)

But


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)


is given by

!!(x) = e±ikx, k =1!!

2m(" ! V ). (6.4)

Note that " = !2k2










J(r) =1m

Re"!"(p!)

#=

!m

Im"!"(!!)

#. (6.7)


I =$ #

!#dy

$ #

!#dz (!e)Jx = !e!

m

$ #

!#dy

$ #

!#dz Im

"!"

%#!

#x

&#. (6.8)


!k"(r) = %k(x) &"(y, z), (6.9)


Ik" = !e!m

Im"%k(x)"

#

#x

'%k(x)

(#. (6.10)

50


Electron channels

•For a given quantum number " there are many possible

quantum states, namely two (spin up and spin down) for

each k

6.6. CONDUCTANCE AND SCATTERING MATRIX FORMALISM 73




"k! = "k + "! " k =1!

!2m("k! ! "!). (6.49)




Ik! = !2e!Lm

Im"e!ikx %

%x

#eikx

$%= !2e!k

Lm= !2

e

Lvk!, (6.50)







"k! = "k + "! " k =1!

!2m("k! ! "!). (6.49)




Ik! = !2e!Lm

Im"e!ikx %

%x

#eikx

$%= !2e!k

Lm= !2

e

Lvk!, (6.50)



•Ik! carried by an electron in state "k! in channel !





"k! = "k + "! " k =1!

!2m("k! ! "!). (6.49)




Ik! = !2e!Lm

Im"e!ikx %

%x

#eikx

$%= !2e!k

Lm= !2

e

Lvk!, (6.50)



vk"is the velocity of an electron in the state #k"

•Current IL !R running from the left reservoir L into the right reservoir R through the nanostructure

51


Current IL !R running from the left reservoir L into the right reservoir R through

the nanostructure


! = !k! and multiply three factors: (1) the Fermi-Dirac probability nLF(!) that the right-

moving state "k! is occupied from the left reservoir; (2) the current Ik! carried by thatstate; and (3) the transmission probability T!(!) of an electron at energy ! making itthrough to the reservoir to the right. Then we sum over all possible electron channels #and wavenumbers k:

IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)

First we convert the k-sum into a k-integral by use of Eq. (3.10), which in 1D reads$k # (L/2$)

%dk. Then we convert the k-integral to an !-integral:

%dk #

%d! "k

"# . Butfrom Eq. (2.8b) we know that "k

"# = 1!v . So collecting all this yields

$k # (L/2$)

%d! 1

!vand Eq. (6.51) becomes

IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)

Note how all references to geometry, i.e. L, and velocity have vanished from the expression.The current IR!L flowing from the right to the left is obtained in a similar way,

IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)

The total current I flowing in the system is therefore

I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)

6.6.3 The conductance formula for nanostructures

We now have a relatively simple expression for the current, so to obtain the conductancewe need to find the voltage dependence. In our set-up in Fig. 6.1b the voltage V is appliedto the left reservoir. If we restrict ourselves to consider only small voltages (the linearresponse limit) it follows from Eq. (6.2) that

nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)

With this Eq. (6.54) becomes

I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)

Finally, we obtain the temperature dependent conductance G(T ),

G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)

Fermi-Dirac probability

sum for all channels

and wave vectors

transmission probability

of an electron at energy ! making it

through to the reservoir to the right




IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)



%dk #

%d! "k



$k # (L/2$)

%d! 1


IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)


IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)


I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)



nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)


I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)


G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)

(in 1D)




IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)



%dk #

%d! "k



$k # (L/2$)

%d! 1


IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)


IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)


I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)



nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)


I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)


G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)




IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)



%dk #

%d! "k



$k # (L/2$)

%d! 1


IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)


IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)


I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)



nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)


I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)


G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)

But,

Then,




IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)



%dk #

%d! "k



$k # (L/2$)

%d! 1


IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)


IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)


I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)



nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)


I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)


G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)




IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)



%dk #

%d! "k



$k # (L/2$)

%d! 1


IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)


IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)


I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)



nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)


I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)


G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)




IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)



%dk #

%d! "k



$k # (L/2$)

%d! 1


IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)


IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)


I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)



nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)


I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)


G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)

52


53





IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)



%dk #

%d! "k



$k # (L/2$)

%d! 1


IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)


IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)


I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)



nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)


I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)


G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)




IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)



%dk #

%d! "k



$k # (L/2$)

%d! 1


IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)


IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)


I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)



nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)


I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)


G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)




IL!R =!

!

!

k

"nL

F(!)#!

"" 2

e

Lvk!

#!

"T!(!)

#. (6.51)



%dk #

%d! "k



$k # (L/2$)

%d! 1


IL!R = " 2eh

!

!

& "

#"d! nL

F(!) T!(!). (6.52)


IR!L = " 2eh

!

!

& "

#"d! nR

F (!) T!(!). (6.53)


I = IL!R " IR!L = " 2eh

!

!

& "

#"d! T!(!)

"nL

F(!) " nRF (!)

#. (6.54)



nLF(!) " nR

F (!) = nF(!, µ0 " eV ) " nF(!, µ0) $nF

%µ

''''µ0

("eV ) =("

nF

%!

)("eV ). (6.55)


I =2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

)V. (6.56)


G(T ) =I

V=

2e2

h

!

!

& "

#"d! T!(!)

("

nF

%!

). (6.57)

Temperature dependent conductance G(T )6.7. QUANTIZED CONDUCTANCE 75

In the limit of very low temperature (temperatures around 1 K are routinely obtained inthe lab cooling with liquid helium) Eqs. (3.28a) and (3.29) state that ! nF

!" = !("F ! ").Inserting this in Eq. (6.56) yields the zero temperature conductance

G(T = 0) =2e2

h

!

#

T#("F). (6.58)

Eq. (6.57) is the conductance formula for nanostructures. We derived it by consid-ering scattering states, i.e. the scattering matrix formalism. We note that the integralis dimensionless, so that means that the prefactor G0 = e2/h, depending entirely onuniversal constants, is some kind of conductance quantum, a natural unit for measuringconductance,

G0 =e2

h= 3.87404614 " 10!5 S, R0 =

1G0

=h

e2= 2.58128056 " 104 k!. (6.59)

The zero temperature conductance Eq. (6.58) is simply the conductance quantum 2e2

htimes the sum of the transmission coe"cients for each channel # evaluated at the Fermienergy set by the metal reservoirs. All information about the nanostructure lies in thetransmission coe"cient T#(").

In the following section we shall see a spectacular consequence of the scattering wavenature of the conductance of nanostructures.

6.7 Quantized conductance

The conductance quantum is directly observable in the beautiful experiment shown inFig. 6.9 based on the quantum point contact depicted in Fig. 6.1a. In the following wegive a simple explanation of the observed quantization of the conductance.

The quantum point contact is fabricated on a GaAs-GaAlAs heterostructure. Asexplained in Exercise 3.4 all conduction electrons of this structure are bound to move atthe interface between GaAs and GaAlAs, all having the same wavefunction $0(z) in the zdirection (perpendicular to the interface). By etching techniques a narrow wire is createdin the x direction. As can be seen in Fig. 6.1a the width w of the wire in the y directionvaries with position, w(x), being of the order 100 nm at the narrowest point. The actualwidth can be controlled during the experiment by changing the gate voltage Vg on theside electrodes as sketched in Fig. 6.9a.

Let us model the potential Vx(y) of the wire at each position x as a simple potentialbox stretching w/2 on both sides of the x axis:

Vx(y) =

"0, for ! 1

2 w(x) < y < 12 w(x),

#, for |y| > 12 w(x).

(6.60)

We further imagine that the change of width as we move along the x axis is so slow that asimple product wavefunction fitting to the width at any given position is a good solution

6.7. QUANTIZED CONDUCTANCE 75



G(T = 0) =2e2

h

!

#

T#("F). (6.58)


G0 =e2

h= 3.87404614 " 10!5 S, R0 =

1G0

=h

e2= 2.58128056 " 104 k!. (6.59)








Vx(y) =

"0, for ! 1

2 w(x) < y < 12 w(x),

#, for |y| > 12 w(x).

(6.60)


The zero temperature conductance

Conductance quantum

6.7. QUANTIZED CONDUCTANCE 75



G(T = 0) =2e2

h

!

#

T#("F). (6.58)


G0 =e2

h= 3.87404614 " 10!5 S, R0 =

1G0

=h

e2= 2.58128056 " 104 k!. (6.59)








Vx(y) =

"0, for ! 1

2 w(x) < y < 12 w(x),

#, for |y| > 12 w(x).

(6.60)


All information about the nanostructure lies in the

transmission coefficient

Landauer Formula

! !

!

!

!


•More precise treatments: Non-equilibrium Green´s function method (see e.g. Datta’s book)

•Implemented in ab-initio calculation packages as TRANSIESTA

M. Bradbyge et al. PRB 65,165401 (2002)


•Conductance as a function of radius from cylindrical infinite wire calculations

55

The radius can be obtained from the conductance with this curve

G/G0 =!

!R

"F

"2 !1! 2#

!

"F

R

"

A. García et al. PRB 49, 16581 81994)

A.I. Yanson Thesis

Geometrical information can be obtained from the conductance histograms

56

•Quantum size effects (from conductance measurements)

•Conductance histograms


Conductance histograms for alkali metals

57


•Shell and supershell structure of nanowires from

conductance histograms

A.I. Yanson et al. PRL 84, 5834 (2000); Nature 400, 144

(1999)

Increasing the temperature the

atoms can move and

reorganize to the positions

which minimize the energy.

Surf

ace e

nerg

y

Radius58

Calculations with jellium model

M.J. Puska , E. Ogando, N. Zabala, PRB 64, 033401

(2001)

Quantum beat structure in stability

Two states whose quantum

numbers fulfill

belongs to the same orbit

L= 4 R L= 5.2 R L= 5.7 R L= 9.5 R L=10.9 R!= 4/"F != 5.2/"F != 5.7/"F !=10.9/"F!= 9.5/"F

(2,1) (3,1) (4,1) (5,2) (7,2) (P,Q)

n=1 R=

n=2 R=

n=3 R=

n=4 R=...

Semiclassical orbit closes

when its length L is an integer

times the Fermi wavelength

Semiclassical model

59

•Shell and supershell structure


explain existence of more stable radii

M.J. Puska , E. Ogando, N. Zabala, PRB 64, 033401

(2001)

FF

T

Frequency

The Fourier transform reveals the connection with the semiclassical approach

The pendular and triangular orbits are the most important ones

60


Interference of two waves of similar frequencies produce beat structure

Frequencies do not change

Radius

FF

T

Frequency

Al: beats are shifted

•Shell and supershell for other metals

61


The potential softness

changes the orbit shape

Distance to surface

No

rma

lize

d e

ffe

ctive

po

ten

tia

ls

•Potential softness effect

62


This model represents the ultimate limit in which the positive background is

completely deformed until its charge density is the same as the electron

density

•Ultimate jellium wire breaking simulations

Bulk density

Density (a

0-3

)

Radius

63


1.- Free parameter model

2.- Lowest energy by deformation

3.- Optimal shape from electrons

point of view

M. Koskinen et at. Z Phys. D 35, 285 (1995)

d

Liquid-like

shape

Cluster

derived

structure

Cluster

derived

structure

Straight

wire

structure

ba

c

64


•Ultimate jellium wire breaking simulations

Transport

Geometry

Mechanics

Experiements

•Summary of properties

65


(ultimate jellium wire

breaking simulations)

Atomic shell and supershell

A.I. Yanson et al. PRL 87, 216805 (2001)66


67

•Other hot topics

•Magnetism in nanowires and the 0.7 conductance anomaly

Materials which are non-magnetic can exhibit magnetism in small dimensions # spontaneosus magnetization in nanowires-Experiments: non-integer values of onductance quanta

K. J. Thomas, J.T. Nicholls, M. Y. Simmons, M. Pepper,

D. R. Mace and D.A. Ritchie, Phys. Rev. Lett. 77, 135 (1996).

;K. J. Thomas, J.T. Nicholls, M. Pepper, W.R. Tribe,

M.Y. Simmons and D.A. Ritchie, Phys. Rev. B 61, R13365 (2000).

Wires on surfaces can also exhibit magnetism

• Kondo effect in QPC and quantum dots

• Luttinger liquid effects, important for 1D systems

These effects belong to cutting edge of modern nanoscience and

nanotechnology.

68

•Other hot topics

•Summary

69

•We have studied some important systems with 1D behaviour, as 1D metals, polymers, semiconductor wires formed at heterostructures and nanowires formed by contact breaking.

•We have adressed some important phenomena taking place in these systems as the Peierls transition and the quantization of the conductance. For the later we have obtained the Landauer formula describing the conductance in ballistic regime.

•Using a cylindrical jellium model we have obtained some of these properties and described the oscillations occuring in the physical properties and in particular in the stability, giving rise to quantum size effects.

•Finally we have pointed out some hot topics related to 1D systems.

Conductance - UPV/EHU · energy separation ∆ε between the quantum states in the reservoir, ∆ε...

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Transcript of Conductance - UPV/EHU · energy separation ∆ε between the quantum states in the reservoir, ∆ε...