Landauer Buttiker Formalism

Frank Elsholz

December 17, 2002

Abstract

This is intended to be a very elementary introduction to the LandauerButtiker Formalism. At first, basic concepts of electronic transport in meso-scopic structures are introduced, like transverse modes, reflectionless contactsand the ballistic conductor. The current per mode per energy is calculatedand the value for the contact resistance derived. Then Landauer’s formula isproposed, including residual scatterer’s resistances. After investigating thequestion, where the voltage drop comes from, multiterminal devices are con-sidered, proposing Buttiker’s multi-terminal formula, wich is then applied toa simple three terminal device. The whole article is heavily based on [1].

Contents

1 Symbols 3

2 Ohmic Resistance Measurement 3

3 Concepts 4

3.1 Transmission probability . . . . . . . . . . . . . . . . . . . . . 43.2 Ballistic conductor . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Reflectionless contacts . . . . . . . . . . . . . . . . . . . . . . 43.4 Transverse modes . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . 53.6 Number of transverse modes . . . . . . . . . . . . . . . . . . . 63.7 Contact Resistance . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Landauer Formula 8

5 Residual scatterer’s resistance on a microscopic scale 9

6 Multiterminal Devices 10

7 Three Terminal Device 12

2

1 Symbols

Quantity Große Symbol SI-Unit

(2-D)Conductivity1 (spezifische) Leitfahigkeit σ Ω−1· m−1

(2-D)Resistivity1 (spezifischer) Widerstand ρ Ω · m−1

Conductance1 Leitwert G Ω−1

Resistance1 Widerstand R ΩBand edge energy (bulk) (Leitungs-)bandunterkante EC eV

Cutoff energy ? εc eV

Transmission function Transmissionsfunktion T

Heavyside function Stufenfunktion ϑ

Effective mass Effektive Masse m me

Length Lange L mWidth Breite w m

Number of transverse modes Anzahl transversaler Moden M

2 Ohmic Resistance Measurement

To start with, we consider a simple classical ohmic re-

w

L

I

UR

UDC

+ -

RUDC

R0

RI

RUR

RLRU

Figure 1:Measuringthe value of aresistance R0 isinfluenced byseveral sourcesof praticalerrors.

sistance measurement (fig. 1). The total resistance com-prises of of several parts: The actual resistor, the wires,the instruments, the internal resistance of the battery. . .

Rtot = RU +RI +R0 +RL (1)

But we won’t worry ’bout all these details, so usually wecalculate the resistors resistance by calculating

R0 =UR

I. (2)

On the other hand, we know, that the resistance can beexpressed by a specific, material dependend but geometryindependent (2-D)resistivity ρ, or, equivalently, it’s (2-D)conductivity σ, as:

R0 = G−10 =

L

σw, (3)

1Temparture dependent

3

Contact 1 Contact 2

L

w

Conductor

Figure 2: A conductor sandwiched between two contacts.

with σ,w dimensions of the resistor. So what happens, if we tend this geom-etry towards zero? We would expect the resistance to become zero too:

limw,L→0

R0

!= 0 (wrong)

which is not observed experimentally. For the length L going to zero undfor small width w, we find a limiting value limL→0R0 → RC(w), which doesdepend on the width. To find an explanation, we introduce several concepts.

3 Concepts

We treat the resistor as a conductor sandwiched between two contacts (fig.2).

3.1 Transmission probability

Conductance should be related to the ease, with wich electrons can pass aconductor, so we introduce the transmission probability T as the probabilityfor an electron to transmit through the conductor. Certainly, the reflectionprobability will be given by 1 − T .

3.2 Ballistic conductor

A ballistic conductor is an ideal transmitting conductor without scatterers,having a transmission probability of T = 1.

3.3 Reflectionless contacts

An electron inside the conductor can exit “into a wide contact with negligibleprobability of reflection”. This is a quite good approximation, for the givencase of a narrow conductor and almost infinitely wide contact, as we will see

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later. This assumption set us in the position to note, that the +k statesinside a ballistic conductor are populated by electron originating in the leftcontact only and vice versa.

3.4 Transverse modes

As we will see, electronic transport happens in discrete channels through anarrow conductor, which we call transverse modes. The electron dynam-ics in effective mass approximation inside the conductor is described bySchrodinger’s eigenvalue equation

[

EC +p2

2m∗+ V (x, y)

]

ψ(x, y, z) = Eψ(x, y, z) (4)

Here, EC is the conduction band edge ofV(x,y)

x,yw

Figure 3: Lateral potentialconfining the width of a con-ductor.

the (bulk) conductor material and V (x, y) is aconfining potential (fig. 3). Since the systemis translational invariant in the z direction, wechoose a separating ansatz which yields:ψ(x, y, z) = χ(x, y)exp(ikzz)

⇒

[p2

x+p2y

2m∗+ V (x, y)

]

χn(x, y) = εnχn(x, y)

En(kz) = EC + εn + h2k2z

2m∗

The χn(x, y) are called transverse modes andn is an index for the discrete spectrum. Withthis we can understand, why we can assumethe contact to be reflectionless: An electroninside the conductor most probably will findan empty state in the contact when exiting,for we have almost infinitely many modes in awide contact. For an electron in the contact, however, we have a differentsituation: To enter the conductor it must have exactly the correct energycorresponding to an empty transverse mode. Fig. 4 illustrates this matter.From now on we simply say k := kz.

3.5 Distribution Function

We will assume the contacts to be in thermodynamical equilibrium, so theelectrons simply are Fermi-distributed with some electrochemical potentials

5

E

ContactConductor

(a)

E(kz)

kz

EC

EC+e0

EC+e1

EC+e2

EC+e3

EC+e4

(b)

Figure 4: (4(a)): States in a conductor and contact. (4(b)): Schematicdispersionrelations for some transverse modes.

µ1 and µ2:

Left contact: f1(E)T=0K= ϑ(µ1 − E) Fermi distribution

Right contact: f2(E)T=0K= ϑ(µ2 − E) Fermi distribution

Conductor:+k states: f+(E) = f1(E)

T=0K= ϑ(µ1 − E)

-k states: f−(E) = f2(E)T=0K= ϑ(µ2 − E)

3.6 Number of transverse modes

The effectively current carrying states are

Em1m200

1f+-(E)

Figure 5: Current carryingstates are between µ1 andµ2.

the states between µ1 and µ2 (fig. 5), so weonly have to count the number of them andto calculate, which current is carried by eachstate. With cut-off energy εn = En(k = 0) foreach transverse mode n, the number of statesthat can be reached at an energy E is given byM(E) :=

∑

n

ϑ(E − εn). Now we consider the

+k states at first. Each mode n is occupiedaccording to the left contact distriution func-

tion f1(E) = f+(E) and carries a current I+n = Neveff, where N = 1

Lis the

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electron density for an electron inside a conductor of length L and veff is theeffective velocity of the electrons. So we have:

I+n =

e

L

∑

k

∂E

h∂kf+(E(k)). (5)

By using the formal transition∑

k

→ 2 ×L2π

∫dk this yields:

I+n =

2e

h

∞∫

εn

f+(E)dE. (6)

All modes together sum up to:

I+ =2e

h

∞∫

−∞

f+(E)M(E)dE. (7)

Here 2eh

= 80 nA

meVis the current per mode per energy. The same holds for

the −k states.

3.7 Contact Resistance

Apply a low voltage U = (µ1 − µ2) /e to a ballistic conductor, such thatM(E)=const=M for µ2 < E < µ1, which is referred to as transport at the

Fermi edge. Then the current will be

I = I+− I− =

2e

hM(µ1 − µ2) =

2e2

hM

(µ1 − µ2)

e

The conductance will be

GC =I

U=

2e2

hM

and the resistance (contact resistance)is

G−1

C =h

2e2M≈

12.9 kΩ

M

These results have been confirmed experimentally (fig. 6).

7

Figure 6: Discrete conductance steps in a narrow conductor (atopted from:[1]).

4 Landauer Formula

A fully analoguous treatment including a resident scatterer inside theconductor with transmission probability T yield Landauer’s formula for theconductance of a mesoscopic conductor:

Gtot = 2e2

hMT Landauer 1957 (8)

This formula includes:Contact resistance

Discrete modes

Ohm’s law

Ohm’s law is obtained considerering the limiting case of a long conductorincluding many scatterers, which will not be derived here. The interestedreader may be suggested to have a look in [1]. Finally we want to devide theresistance into two parts: The resistance originating in the transistion to thecontacts and the residual scatterer’s resistance:

G−1 =h

2e2MT=

h

2e2M︸ ︷︷ ︸

G−1

C

+h

2e2M

1 − T

T︸ ︷︷ ︸

G−1s

(9)

8

5 Residual scatterer’s resistance on a micro-

scopic scale

Sm1m2

XL XRRLDistributionfunctions (T=0 K):

Em100

1

m2

f+(E)=J(m1-E)

Em100

1

m2

f+(E)=J(m1-E)

Em100

1

m2

f+(E)=J(m2-E)+T[J(m1-E)-J(m2-E)]

T

F'' E00

1f+(E)=J(F''-E)

F''

E00

1f-(E)=J(F'-E)

F' Em100

1

m2

f-(E)=J(m2-E)+(1-T)[J(m1-E)-J(m2-E)]

T

F'

1-T

Em100

1

m2

f-(E)=J(m2-E)

T

F'

1-T

Em100

1

m2

f-(E)=J(m2-E)

T

F'

1-T

To have a look at the distribution function for the electrons inside the con-ductor for temperature 0K, we first consider the +k states. Coming in fromthe left contact (XL), they are Fermi distributed according to the left contactelectrochemical potential µ1 and move on to the scatterer (L). Here a frac-tion T transmits the scatterer, the remaining part is reflected back to the leftcontact, so these electrons turn into −k states. Directly after the scatterer(R) the +k states are highly nonequilibrium distributed. On their way tothe right contact, however they relaxate and form a new equilibrium Fermidistribution with some quasi-potential F”. The same holds for the −k statesoriginating in the right contact: First they are Fermi distributed accordingto the right contact electrochemical potential µ2, move on to the scatterer.Here, in pricipal a fraction T is transmitted and the rest reflected, howeverto simplify the matter, we assume the scatterer to act only on the +k states,so all −k states can transmit, which definitely is not quite correct. Afterpassing the scatterer, the transmitted −k states unify with the reflected +kstates, that turned into −k states and we again have a highly nonequilibrium

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distribution, which relexates on it’s way to the left contact. A quasi Fermi-potential F’ emerges.In that simplified model the quasi-Fermi Niveaus are given by:

F ′ = µ2 + (1 − T )(µ1 − µ2) (10)

F ′′ = µ2 + T (µ1 − µ2) (11)

Fig. 8 shows the electrochemical potentials for the two species across the

F'

E

F''

m1

m2

R XRLXLnonequilibriumdistributions

equilibriumdistributions

S

equilibriumdistributions

Figure 8: Electrochemical potentials for the +k states (red) and the −kstates (blue).

conductor. Clearly we can see, that the voltage drop at the scatterer is:+k states eV +

s = µ1 − F ′′ = (1 − T )∆µ = eG−1s I

-k states eV −s = F ′ − µ2 = (1 − T )∆µ = eG−1s I

whereas the voltage drop at the contacts is:

eVc = T (µ1 − µ2) = eG−1c I

according to eqn. 9.

6 Multiterminal Devices

Now we want to extend our investigations to multi-terminal devices, havingmore than 2 probes (or electrodes or contacts, generally terminals). Fig. 9

10

S T1

1-Tm2m1

mp1 mp2

Figure 9: Conceptual idea of a multiterminal device with 4 terminals (con-tacts).

schematically shows a 4 terminal device with a scatterer inside the conduc-tor. When treating such devices, we have to note, that there exist differentproblems, that may arise, some of which are sketched in fig. 10 So how do we

S m2m1

m p1m

p2

+k -k-k +k

(a)

m2m1

mp1mp2

SSS

(b)

m2m1

mp1mp2

S

(c)

Figure 10: Different problems with multiterminal devices arise: (10(a)):The terminals may couple differently to different species of states (e.g.+ − kstates). (10(b)): Since the terminal are invasive by themselves, theymay produce additional sources of scattering. (10(c)): A propagating wavemay interfer with it’s own from a scatterer reflected part. This is a purequantum-mechanical effect and the results of a measurement may depend onthe exact location of the terminals.

have to treat such multi-terminal devices? It was Buttiker, who realized, thatthere is no principal difference between voltage probes and current probes, sowe can simply extend the two terminal Landauer formula by summing overall probes:

Buttiker: Ip =2e

h

∑

q

(T q←pµp − T p←qµq

)(12)

11

Here T q←p := Mq←pTq←p is the product of transmission probability T fromcontact p to contact q and the number of transverse modes M between them,and is called transmission function. Just let us rewrite this a little:

with Gpq :=2e2

hT pq

Vq :=µq

e

∑

q

Gqp =∑

q

Gpq

Ip =∑

q

Gpq (Vp − Vq)

7 Three Terminal Device

V1 V3

V2

I1 I3I2

+ -I

V

Figure 11: Conceptual idea of a 3 terminal-device.

For a voltage contact p, we know that there is almost no current flowing,so we can write:

Ip = 0 ⇒ Vp =

∑

q 6=p

GpqVq

∑

q 6=p

Gpq

(13)

As an example we will apply this result to a three terminal device as shownin fig. 11. Here the probe at potential V2 may be the voltage probe and wejust want to measure the resistance of that device. From eqn. 12 we can

12

write:

I1I2I3

=

G11 (V1 − V1) +G12 (V1 − V2) +G13 (V1 − V3)G21 (V2 − V1) +G22 (V2 − V2) +G23 (V2 − V3)G31 (V3 − V1) +G32 (V3 − V2) +G33 (V3 − V3)

=

G12 +G13 −G12 −G13

−G21 G21 +G23 −G23

−G31 −G32 G31 +G32

V1

V2

V3

This can be reduced further. From Kirchhoff’s knot rule, we know, thatI1 + I2 + I3 = 0, so these three equations are not independent and we canonly solve for I1 and I2. I3 then follows immediately. Secondly we can choosea reference potential without changing the physics behind it, so we chooseV3 = 0 to simplify the matter. This yields:

⇒

(I1I2

)

=

[G12 +G13 −G12

−G21 G21 +G23

]

︸ ︷︷ ︸

2−1

(V1

V2

)

⇔

(V1

V2

)

=

[Raa Rab

Rba Rbb

] (I1I2

)

and the resistance is given as

R =V2

I1

∣∣∣∣I2=0

=RbaI1 +RbbI2

I1

∣∣∣∣I2=0

= Rba

R can be obtained from the conductance coefficients Gij and these can beobtained from the scattering matrix Slm, for which we have to solve thethreedimensional problem quantummechanically, e.g. using Green’s function.

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References

[1] S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge Uni-versity Press, Cambridge, 1995).

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