review article landauer-datta-lundstrom generalized...

Review ArticleLandauer-Datta-Lundstrom Generalized TransportModel for Nanoelectronics

Yuriy Kruglyak

Department of Information Technologies, Odessa State Environmental University, Odessa 65016, Ukraine

Correspondence should be addressed to Yuriy Kruglyak; [email protected]

Received 20 June 2014; Accepted 14 August 2014; Published 17 September 2014

Academic Editor: Xizhang Wang

Copyright © 2014 Yuriy Kruglyak. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Landauer-Datta-Lundstrom electron transport model is briefly summarized. If a band structure is given, the number ofconduction modes can be evaluated and if a model for a mean-free-path for backscattering can be established, then the near-equilibrium thermoelectric transport coefficients can be calculated using the final expressions listed below for 1D, 2D, and 3Dresistors in ballistic, quasiballistic, and diffusive linear response regimes when there are differences in both voltage and temperatureacross the device. The final expressions of thermoelectric transport coefficients through the Fermi-Dirac integrals are collected for1D, 2D, and 3D semiconductors with parabolic band structure and for 2D graphene linear dispersion in ballistic and diffusiveregimes with the power law scattering.

1. Introduction

Theobjectives of this short review is to give a condensed sum-mary of Landauer-Datta-Lundstrom (LDL) electron trans-port model [1–5] which works at the nanoscale as well asat the macroscale for 1D, 2D, and 3D resistors in ballistic,quasiballistic, and diffusive linear response regimes whenthere are differences in both voltage and temperature acrossthe device.

Appendices list final expressions of thermoelectric trans-port coefficients through the Fermi-Dirac integrals for 1D,2D, and 3D semiconductors with parabolic band structureand for 2D graphene linear dispersion in ballistic and diffu-sive regimes with the power law scattering.

2. Generalized Model for Current

The generalized model for current can be written in twoequivalent forms:

𝐼 =2𝑞

ℎ∫ 𝛾 (𝐸) 𝜋

𝐷 (𝐸)

2(𝑓1− 𝑓2) 𝑑𝐸, (1a)

𝐼 =2𝑞

ℎ∫𝑇 (𝐸)𝑀 (𝐸) (𝑓

1− 𝑓2) 𝑑𝐸, (1b)

where “broadening” 𝛾(𝐸) relates to transit time for electronsto cross the resistor channel:

𝛾 (𝐸) ≡ℏ

𝜏 (𝐸); (2)

density of states 𝐷(𝐸) with the spin degeneracy factor 𝑔𝑠=

2 is included; 𝑀(𝐸) is the integer number of modes ofconductivity at energy 𝐸; the transmission

𝑇 (𝐸) =𝜆 (𝐸)

𝜆 (𝐸) + 𝐿, (3)

where 𝜆(𝐸) is the mean-free-path for backscattering and 𝐿 isthe length of the conductor; Fermi function

𝑓 (𝐸) =1

𝑒(𝐸−𝐸𝐹)/𝑘𝑇 + 1(4)

is indexed with the resistor contact numbers 1 and 2; 𝐸𝐹is the

Fermi energy which as well as temperature𝑇may be differentat both contacts.

Hindawi Publishing CorporationJournal of NanoscienceVolume 2014, Article ID 725420, 15 pageshttp://dx.doi.org/10.1155/2014/725420

2 Journal of Nanoscience

Equation (3) can be derived with relatively few assump-tions and it is valid not only in the ballistic and diffusionlimits, but in between as well:

Diffusive: 𝐿 ≫ 𝜆; 𝑇 = 𝜆𝐿≪ 1,

Ballistic: 𝐿 ≪ 𝜆; 𝑇 → 1,

Quasi-ballistic: 𝐿 ≈ 𝜆; 𝑇 < 1.

(5)

The LDL transport model can be used to describe all threeregimes.

It is now clearly established that the resistance of a ballisticconductor can be written in the form

𝑅ball

=ℎ

𝑞2

1

𝑀 (𝐸), (6)

where ℎ/𝑞2 is fundamental Klitzing constant and numberof modes 𝑀(𝐸) represents the number of effective parallelchannels available for conduction.

This result is now fairly well known, but the commonbelief is that it applies only to short resistors and belongsto a course on special topics like mesoscopic physics ornanoelectronics.What is not well known is that the resistancefor both long and short conductors can be written in the form

𝑅 (𝐸) =ℎ

𝑞2

1

𝑀 (𝐸)(1 +

𝐿

𝜆 (𝐸)) . (7)

Ballistic and diffusive conductors are not two differentworlds, but rather a continuum as the length 𝐿 is increasing.Ballistic limit is obvious for 𝐿 ≪ 𝜆, while for 𝐿 ≫ 𝜆 it reducesinto standard Ohm’s law:

𝑅 ≡𝑉

𝐼= 𝜌

𝐿

𝐴. (8)

Indeed we could rewrite 𝑅(𝐸) above as

𝑅 (𝐸) =𝜌 (𝐸)

𝐴[𝐿 + 𝜆 (𝐸)] (9)

with a new expression for specific resistivity:

𝜌 (𝐸) =ℎ

𝑞2(

1

𝑀 (𝐸)/𝐴)

1

𝜆 (𝐸), (10)

which provides a different view of resistivity in terms of thenumber of modes per unit area and the mean-free-path.

Number of modes

𝑀(𝐸) = 𝑀1𝐷

(𝐸) =ℎ

4⟨V+𝑥(𝐸)⟩𝐷

1𝐷(𝐸) , (11a)

𝑀(𝐸) = 𝑊𝑀2𝐷

(𝐸) = 𝑊ℎ


2𝐷(𝐸) , (11b)

𝑀(𝐸) = 𝐴𝑀3𝐷

(𝐸) = 𝐴ℎ


3𝐷(𝐸) (11c)

is proportional to the width𝑊 of the resistor in 2D and to thecross-sectional area𝐴 in 3D; ⟨V+

𝑥(𝐸)⟩ is the average velocity in

the +𝑥 direction from contact 1 to contact 2.

For parabolic energy bands

𝐸 (𝑘) = 𝐸𝐶+ℏ2𝑘2

2𝑚∗(12)

the 1D, 2D, and 3D densities of states are given by

𝐷 (𝐸) = 𝐷1𝐷

(𝐸) 𝐿 =𝐿

𝜋ℏ√

2𝑚∗

(𝐸 − 𝐸𝐶)𝐻 (𝐸 − 𝐸

𝐶) , (13a)

𝐷 (𝐸) = 𝐷2𝐷

(𝐸)𝐴 = 𝐴𝑚∗

𝜋ℏ2𝐻(𝐸 − 𝐸

𝐶) , (13b)

𝐷 (𝐸) = 𝐷3𝐷

(𝐸)Ω = Ω

𝑚∗√2𝑚∗ (𝐸 − 𝐸

𝐶)

𝜋2ℏ3𝐻(𝐸 − 𝐸

𝐶) ,

(13c)

where 𝐴 is the area of the 2D resistor, Ω is the volume of the3D resistor, and 𝐻(𝐸 − 𝐸

𝐶) is the Heaviside step function.

Then number of modes is

𝑀(𝐸) = 𝑀1𝐷

(𝐸) = 𝐻 (𝐸 − 𝐸𝐶) , (14a)

𝑀(𝐸) = 𝑊𝑀2𝐷

(𝐸) = 𝑊𝑔V

√2𝑚∗ (𝐸 − 𝐸𝐶)

𝜋ℏ𝐻 (𝐸 − 𝐸

𝐶) ,

(14b)

𝑀(𝐸) = 𝐴𝑀3𝐷

(𝐸) = 𝐴𝑔V𝑚∗(𝐸 − 𝐸

𝐶)

2𝜋ℏ2𝐻(𝐸 − 𝐸

𝐶) , (14c)

where 𝑔V is the valley degeneracy.Figure 1 shows qualitative behavior of the density of states

and number of modes for resistors with parabolic bandstructure.

For linear dispersion in graphene,

𝐸 (𝑘) = ±ℏV𝐹𝑘, (15)

where +sign corresponds to conductivity band with 𝐸𝐹> 0

(𝑛-type graphene) and −sign corresponds to valence bandwith 𝐸

𝐹< 0 (𝑝-type graphene),

V (𝑘) =1

ℏ

𝜕𝐸

𝜕𝑘≡ V𝐹≈ 1 × 10

8 cm/s. (16)

Density of states in graphene is

𝐷 (𝐸) =2 |𝐸|

𝜋ℏ2V2𝐹

(17)

and number of modes is

𝑀(𝐸) = 𝑊2 |𝐸|

𝜋ℏV𝐹

. (18)

Two equivalent expressions for specific conductivitydeserve attention, one as a product of𝐷(𝐸) and the diffusioncoefficient𝐷(𝐸):

𝜎 (𝐸) = 𝑞2𝐷 (𝐸)

𝐷 (𝐸)

𝐿{1,

1

𝑊,1

𝐴} , (19a)

Journal of Nanoscience 3

D1D

D2D

D3D

M1D

M2D

M3D

ECE

ECE

ECE

ECE

EC

EC

E

E

1

Figure 1: Comparison of the density of states 𝐷(𝐸) and number ofmodes𝑀(𝐸) for 1D, 2D, and 3D resistors with parabolic dispersion.

where

𝐷 (𝐸) = ⟨]2𝑥𝜏⟩ = ]2𝜏 (𝐸) {1,

1

2,1

3} (∗)

with 𝜏(𝐸) being the mean free time after which an electrongets scattered, and the other as a product of𝑀(𝐸) and 𝜆(𝐸):

𝜎 (𝐸) =𝑞2

ℎ𝑀 (𝐸) 𝜆 (𝐸) {1,

1

𝑊,1

𝐴} , (19b)

where the three items in parenthesis correspond to 1D, 2D,and 3D resistors.

Although (19b) is not well known, the equivalent versionin (19a) is a standard result that is derived in textbooks. Both(19a) and (19b) are far more generally applicable comparedwith traditional Drude model. For example, these equationsgive sensible answers even for materials like graphene whosenonparabolic bands make the meaning of electron masssomewhat unclear, causing considerable confusion whenusing Drude model. In general we must really use (19a) and(19b) and not Drude model to shape our thinking aboutconductivity.

These conceptual equations are generally applicable evento amorphous materials and molecular resistors. Irrespectiveof the specific𝐸(𝑝) relation at any energy, the density of states𝐷(𝐸), velocity ](𝐸), and momentum 𝑝(𝐸) are related to thetotal number of states 𝑁(𝐸) with energy less than 𝐸 by thefundamental relation

𝐷 (𝐸) ] (𝐸) 𝑝 (𝐸) = 𝑁 (𝐸) ⋅ 𝑑, (∗∗)

where 𝑑 is the number of dimensions. Being combinedwith (19a), it gives one more fundamental equation forconductivity:

𝜎 (𝐸) =𝑞2𝜏 (𝐸)

𝑚 (𝐸){𝑁 (𝐸)

𝐿,𝑁 (𝐸)

𝐿𝑊,𝑁 (𝐸)

𝐿𝐴} , (19c)

where electron mass is defined as

𝑚(𝐸) =𝑝 (𝐸)

V (𝐸). (20)

For parabolic 𝐸(𝑝) relations, the mass is independent ofenergy, but in general it could be energy-dependent as, forexample, in graphene the effective mass

𝑚∗=𝐸𝐹

V2𝐹

. (21)

2.1. Linear Response Regime. Near-equilibrium transport orlow field linear response regime corresponds to lim(𝑑𝐼/𝑑𝑉)𝑉→0

. There are several reasons to develop low fieldtransport model. First, near-equilibrium transport is thefoundation for understanding transport in general. Conceptsintroduced in the study of near-equilibrium regime areoften extended to treat more complicated situations, andnear-equilibrium regime provides a reference point whenwe analyze transport in more complex conditions. Second,near-equilibrium transport measurements are widely usedto characterize electronic materials and to understand theproperties of new materials. And finally, near-equilibriumtransport strongly influences and controls the performanceof most electronic devices.

Under the low field condition let

𝑓0(𝐸) ≈ 𝑓

1(𝐸) > 𝑓

2(𝐸) , (22)

where𝑓0(𝐸) is the equilibriumFermi function, and an applied

bias

𝑉 =Δ𝐸𝐹

𝑞=(𝐸𝐹1− 𝐸𝐹2)

𝑞(23)

is small enough. Using Taylor expansion under constanttemperature condition

𝑓2= 𝑓1+𝜕𝑓1

𝜕𝐸𝐹

Δ𝐸𝐹= 𝑓1+𝜕𝑓1

𝜕𝐸𝐹

𝑞𝑉 (24)

and property of the Fermi function

+𝜕𝑓

𝜕𝐸𝐹

= −𝜕𝑓

𝜕𝐸, (25)

one finds

𝑓1− 𝑓2= (−

𝜕𝑓0

𝜕𝐸) 𝑞𝑉. (26)

The derivative of the Fermi function multiplied by 𝑘𝑇 tomake it dimensionless

𝐹𝑇(𝐸, 𝐸𝐹) = 𝑘𝑇(−

𝜕𝑓

𝜕𝐸) (27)

is known as thermal broadening function and shown inFigure 2.


1010

1

5

00

0 0−10−10

−5

0.5 0.20.1 0.3

→ f(E) → kT (− 𝜕f𝜕E)

E − EFkT

↑

Figure 2: Fermi function and the dimensionless normalized ther-mal broadening function.

If one integrates 𝐹𝑇over all energy range, the total area

∫

+∞

−∞

𝑑𝐸𝐹𝑇(𝐸, 𝐸𝐹) = 𝑘𝑇, (28)

so that we can approximately visualize 𝐹𝑇as a rectangular

pulse centered around 𝐸 = 𝐸𝐹with a peak value of 1/4 and a

width of ∼4𝑘𝑇.The derivative (−𝜕𝑓

0/𝜕𝐸) is known as the Fermi conduc-

tion window function.Whether a conductor is good or bad isdetermined by the availability of the conductor energy statesin an energy window ∼±2𝑘𝑇 around the electrochemicalpotential 𝐸

𝐹0, which can vary widely from one material

to another. Current is driven by the difference 𝑓1− 𝑓2

in the “agenda” of the two contacts which for low biasis proportional to the derivative of the equilibrium Fermifunction (26). With this near-equilibrium assumption forcurrent (1b), we have

𝐼 = [2𝑞2

ℎ∫𝑇 (𝐸)𝑀 (𝐸) (−

𝜕𝑓0

𝜕𝐸)𝑑𝐸]𝑉 = 𝐺𝑉, (29)

with conductivity

𝐺 =2𝑞2

ℎ∫𝑇 (𝐸)𝑀 (𝐸) (−

𝜕𝑓0

𝜕𝐸)𝑑𝐸, (30)

known as the Landauer expression which is valid in 1D, 2D,and 3D resistors, if we use the appropriate expressions for𝑀(𝐸).

For ballistic limit 𝑇(𝐸) = 1. For diffusive transport 𝑇(𝐸)is given by (3). For a conductor much longer than a mean-free-path the current density equation for diffusive transportis

𝐽𝑥=𝜎

𝑞

𝑑 (𝐸𝐹)

𝑑𝑥, (31)

where the electrochemical potential 𝐸𝐹is also known as the

quasi-Fermi level.For a 2D conductor the surface specific conductivity is

𝜎𝑆=2𝑞2

ℎ∫𝑀2𝐷

(𝐸) 𝜆 (𝐸) (−𝜕𝑓0

𝜕𝐸)𝑑𝐸 (32)

or in a different form

𝜎𝑆= ∫𝜎

𝑆(𝐸) 𝑑𝐸, (33a)

where differential specific conductivity

𝜎

𝑆(𝐸) =

2𝑞2

ℎ𝑀2𝐷

(𝐸) 𝜆 (𝐸) (−𝜕𝑓0

𝜕𝐸) . (33b)

Similar expressions can be written for 1D and 3D resistors.Another way to write the conductance is the product of

the quantum of conductance, times the average transmission,times the number of modes in the Fermi windows:

𝐺 =2𝑞2

ℎ⟨⟨𝑇⟩⟩ ⟨𝑀⟩ , (34a)

⟨𝑀⟩ = ∫𝑀(𝐸) (−𝜕𝑓0

𝜕𝐸)𝑑𝐸, (34b)

⟨⟨𝑇⟩⟩ =

∫𝑇 (𝐸)𝑀 (𝐸) (−𝜕𝑓0/𝜕𝐸) 𝑑𝐸

∫𝑀(𝐸) (−𝜕𝑓0/𝜕𝐸) 𝑑𝐸

=⟨𝑀𝑇⟩

⟨𝑀⟩. (34c)

Yet another way to write the conductance is in terms ofthe differential conductance 𝐺(𝐸) as

𝐺 = ∫𝐺(𝐸) 𝑑𝐸, [S] (35a)

𝐺(𝐸) =

2𝑞2

ℎ𝑀 (𝐸) 𝑇 (𝐸) (−

𝜕𝑓0

𝜕𝐸) . (35b)

2.2. Thermocurrent and Thermoelectric Coefficients. Elec-trons carry both charge and heat. The charge current is givenby (1a) and (1b). To get the equation for the heat current,one notes that electrons in the contacts flow at an energy𝐸 ≈ 𝐸

𝐹. To enter a mode𝑀(𝐸) in the resistor electrons must

absorb (if 𝐸 > 𝐸𝐹) or emit (if 𝐸 < 𝐸

𝐹) a thermal energy

𝐸 − 𝐸𝐹. We conclude that to get the heat current equation

we should insert (𝐸 − 𝐸𝐹)/𝑞 inside the integral. The resulting

thermocurrent

𝐼𝑄=2

ℎ∫ (𝐸 − 𝐸

𝐹) 𝑇 (𝐸)𝑀 (𝐸) (𝑓

1− 𝑓2) 𝑑𝐸. (36)

It is important from practical point of view that bothexpressions—for the electric current (1a) and (1b) and ther-mocurrent (36)—are suitable for analysis of conductivity ofany materials from metals to semiconductors up to modernnanocomposites.

When there are differences in both voltage and temper-ature across the resistor, then we must the Fermi difference(𝑓1− 𝑓2) expands to Taylor series in both voltage and

temperature and get

𝑓1− 𝑓2≈ (−

𝜕𝑓0

𝜕𝐸) 𝑞Δ𝑉 − (−

𝜕𝑓0

𝜕𝐸)𝐸 − 𝐸

𝐹

𝑇Δ𝑇, (37)

where Δ𝑉 = 𝑉2− 𝑉1, Δ𝑇 = 𝑇

2− 𝑇1, and 𝑇 = (𝑇

1+ 𝑇2)/2.


Deriving a general near-equilibrium current equation isnow straightforward. The total current is the sum of thecontributions from each energy mode:

𝐼 = ∫ 𝐼(𝐸) 𝑑𝐸, (38a)

where the differential current is

𝐼(𝐸) =

2𝑞

ℎ𝑇 (𝐸)𝑀 (𝐸) (𝑓

1− 𝑓2) . (38b)

Using (37) we obtain

𝐼(𝐸) = 𝐺

(𝐸) Δ𝑉 + 𝑆

𝑇(𝐸) Δ𝑇, (39a)

where

𝐺(𝐸) =

2𝑞2

ℎ𝑇 (𝐸)𝑀 (𝐸) (−

𝜕𝑓0

𝜕𝐸) (39b)

is the differential conductance and

𝑆

𝑇(𝐸) = −

2𝑞2

ℎ𝑇 (𝐸)𝑀 (𝐸) (

𝐸 − 𝐸𝐹

𝑞𝑇)(−

𝜕𝑓0

𝜕𝐸)

= −𝑘

𝑞(𝐸 − 𝐸

𝐹

𝑘𝑇)𝐺(𝐸)

(39c)

is the Soret coefficient for electrothermal diffusion in differ-ential form.Note that 𝑆

𝑇(𝐸) is negative formodeswith energy

above 𝐸𝐹(𝑛-resistors) and positive for modes with energy

below 𝐸𝐹(𝑝-resistors).

Now we integrate (39a) over all energy modes and find

𝐼 = 𝐺Δ𝑉 + 𝑆𝑇Δ𝑇, [A] , (40a)

𝐼𝑄= − 𝑇𝑆

𝑇Δ𝑉 − 𝐾

0Δ𝑇, [W] (40b)

with three transport coefficients, namely: conductivity givenby (35a) and (35b); the Soret electrothermal diffusion coeffi-cient

𝑆𝑇= ∫ 𝑆

𝑇(𝐸) 𝑑𝐸 = −

𝑘

𝑞∫(

𝐸 − 𝐸𝐹

𝑘𝑇)𝐺(𝐸) 𝑑𝐸, [A/K]

(40c)

and the electronic heat conductance under the short circuitconditions (Δ𝑉 = 0)

𝐾0= 𝑇(

𝑘

𝑞)

2

∫(𝐸 − 𝐸

𝐹

𝑘𝑇)

2

𝐺(𝐸) 𝑑𝐸, [W/K] , (40d)

where current 𝐼 is defined to be positive when it flows inconductor from contact 2 to contact 1 with electrons flowingin opposite direction. The heat current 𝐼

𝑄is positive when it

flows in the +𝑥 direction out of contact 2.Equations (40a), (40b), (40c), and (40d) for long diffusive

resistors can be written in the common form used to describebulk transport as

𝐽𝑥= 𝜎

𝑑 (𝐸𝐹/𝑞)

𝑑𝑥− 𝑠𝑇

𝑑𝑇

𝑑𝑥, [0/m2] , (41a)

𝐽𝑄𝑥

= 𝑇𝑠𝑇


𝑑𝑥− 𝜅0

𝑑𝑇

𝑑𝑥[W/m2] (41b)

with three specific transport coefficients

𝜎 = ∫𝜎(𝐸) 𝑑𝐸,

𝜎(𝐸) =

2𝑞2

ℎ𝑀3𝐷

(𝐸) 𝜆 (𝐸) (−𝜕𝑓0

𝜕𝐸) , [1/Ω ⋅m ⋅ J] ,

(41c)

𝑠𝑇= −

𝑘

𝑞∫(

𝐸 − 𝐸𝐹

𝑘𝑇)𝜎(𝐸) 𝑑𝐸, [0/m ⋅ K] , (41d)

𝜅0= (

𝑘

𝑞)

2

∫(𝐸 − 𝐸

𝐹

𝑘𝑇)

2

𝜎(𝐸) 𝑑𝐸, [W/m ⋅ K] .

(41e)

These equations have the same form for 1D and 2D resistors,but the units of the various terms differ.

The inverted form of (40a), (40b), (40c), and (40d) isoften preferred in practice, namely:

Δ𝑉 = 𝑅𝐼 − 𝑆Δ𝑇, (42a)

𝐼𝑄= − Π𝐼 − 𝐾Δ𝑇, (42b)

where

𝑆 =𝑆𝑇

𝐺, (42c)

Π = 𝑇𝑆, (42d)

𝐾 = 𝐾0− Π𝑆𝐺. (42e)

In this form of the equations, the contributions from eachenergy mode are not added; for example, 𝑅 ̸= ∫𝑅(𝐸)𝑑𝐸.

Similarly, the inverted form of the bulk transport equa-tions (41a), (41b), (41c), (41d), and (41e) becomes


𝑑𝑥= 𝜌𝐽𝑥+ 𝑆

𝑑𝑇

𝑑𝑥, (43a)

𝐽𝑄𝑥

= 𝑇𝑆𝐽𝑥− 𝜅

𝑑𝑇

𝑑𝑥(43b)

with transport coefficients

𝜌 =1

𝜎, (43c)

𝑆 =𝑠𝑇

𝜎, (43d)

𝜅 = 𝜅0− 𝑆2𝜎𝑇. (43e)

In summary, when a band structure is given, number ofmodes can be evaluated from (11a), (11b), and (11c) and if amodel for the mean-free-path for backscattering 𝜆(𝐸) can bechosen, then the near-equilibrium transport coefficients canbe evaluated using the expressions listed above.

2.3. Bipolar Conduction. Let us consider a 3D semiconductorwith parabolic dispersion. For the conduction band

𝑀3𝐷

(𝐸) = 𝑔V𝑚∗

2𝜋ℏ2(𝐸 − 𝐸

𝐶) (𝐸 ≥ 𝐸

𝐶) (44a)


and for the valence band

𝑀(V)3𝐷

(𝐸) = 𝑔V

𝑚∗

𝑝

2𝜋ℏ2(𝐸𝑉− 𝐸) (𝐸 ≤ 𝐸

𝑉) . (44b)

The conductivity is provided with two contributions: forthe conduction band

𝜎 =𝑞2

ℎ∫

∞

𝐸𝐶

𝑀3𝐷

(𝐸) 𝜆 (𝐸) (−𝜕𝑓0

𝜕𝐸)𝑑𝐸 (45a)

and for the valence band

𝜎𝑝=𝑞2

ℎ∫

𝐸𝑉

−∞

𝑀V3𝐷

(𝐸) 𝜆𝑝(𝐸) (−

𝜕𝑓0

𝜕𝐸)𝑑𝐸. (45b)

The Seebeck coefficient for electrons in the conductionband follows from (41a), (41b), (41c), (41d), and (41e):

𝜎 = ∫

∞

𝐸𝐶

𝜎(𝐸) 𝑑𝐸, (46a)

𝜎(𝐸) =

2𝑞2

ℎ𝑀3𝐷

(𝐸 − 𝐸𝐶) 𝜆 (𝐸) (−

𝜕𝑓0

𝜕𝐸) , (46b)

𝑠𝑇= −

𝑘

𝑞∫

∞

𝐸𝐶

(𝐸 − 𝐸

𝐹

𝑘𝑇)𝜎(𝐸) 𝑑𝐸, (46c)

𝑆 =𝑠𝑇

𝜎. (46d)

Similarly, for electrons in the valence band we have

𝜎𝑝= ∫

𝐸𝑉

−∞

𝜎

𝑝(𝐸) 𝑑𝐸, (47a)

𝜎

𝑝(𝐸) =

2𝑞2

ℎ𝑀

V3𝐷

(𝐸𝑉− 𝐸) 𝜆

𝑝(𝐸) (−

𝜕𝑓0

𝜕𝐸) , (47b)

𝑠(V)𝑇

= −𝑘

𝑞∫

𝐸𝑉

−∞

(𝐸 − 𝐸

𝐹

𝑘𝑇)𝜎

𝑝(𝐸) 𝑑𝐸, (47c)

𝑆𝑝=𝑠(V)𝑇

𝜎𝑝

, (47d)

but the sign of 𝑆𝑝will be positive.

What is going on when both the conduction and valencebands contribute to conduction?This can happen for narrowbandgap conductors or at high temperatures. In such a case,we have to simply integrate over all the modes and will find

𝜎tot

≡ 𝜎 + 𝜎𝑝=𝑞2

ℎ∫

𝐸2

𝐸1

𝑀tot3𝐷

(𝐸) 𝜆 (𝐸) (−𝜕𝑓0

𝜕𝐸)𝑑𝐸, (48a)

𝑀tot3𝐷

(𝐸) = 𝑀3𝐷

(𝐸) + 𝑀V3𝐷

(𝐸) ; (48b)

moreover, we do not have to be worried about integratingto the top of the conduction band or from the bottom ofthe valence band because the Fermi function ensures thatthe integrand falls exponentially to zero away from the bandedge. What is important is that in both cases we integratethe same expression with the appropriate 𝑀

3𝐷(𝐸) and 𝜆(𝐸)

over the relevant energy difference 𝐸2− 𝐸1. Electrons carry

current in both bands. Our general expression is the same forthe conduction and valence bands.There is no need to changesigns for the valence band or to replace 𝑓

0(𝐸) with 1 − 𝑓

0(𝐸).

To calculate the Seebeck coefficient when both bandscontribute let us be reminded that in the first direct form ofthe transport coefficients (41a), (41b), (41c), (41d), and (41e)the contributions from eachmode are added in parallel so thetotal specific Soret coefficient

𝑠tot𝑇

= −𝑘

𝑞∫

+∞

−∞

(𝐸 − 𝐸

𝐹

𝑘𝑇)𝜎(𝐸) 𝑑𝐸 = 𝑆𝜎 + 𝑆

𝑝𝜎𝑝; (49a)

then, the Seebeck coefficient for bipolar conduction

𝑆tot

=

𝑆𝜎 + 𝑆𝑝𝜎𝑝

𝜎 + 𝜎𝑝

. (49b)

Since the Seebeck coefficients for the conduction and valencebands have opposite signs, the total Seebeck coefficientjust drops for high temperatures and the performance of athermoelectrical device falls down.

In summary, given a band structure dispersion, thenumber of modes can be evaluated and if a model for amean-free-path for backscattering can be established, thenthe near-equilibrium transport coefficients can be calculatedusing final expressions listed above.

3. Heat Transfer by Phonons

Electrons transfer both charge and heat. Electrons carry mostof the heat in metals. In semiconductors electrons carry onlya part of the heat but most of the heat is carried by phonons.

The phonon heat flux is proportional to the temperaturegradient

𝐽ph𝑄𝑥

= −𝜅𝐿

𝑑𝑇

𝑑𝑥[W/m2] (50)

with coefficient 𝜅𝐿known as the specific lattice thermal

conductivity. Such an exceptional thermal conductor likediamond has 𝜅

𝐿≈ 2 ⋅ 10

3W/m ⋅K while such a poor thermalconductor like glass has 𝜅

𝐿≈ 1W/m ⋅ K. Note that electrical

conductivities of solids vary over more than 20 orders ofmagnitude, but thermal conductivities of solids vary over arange of only 3-4 orders of magnitude. We will see that thesame methodology used to describe electron transport canbe also used for phonon transport. We will also discuss thedifferences between electron and phonon transport. For athorough introduction to phonons use classical books [6–9].

To describe the phonon current we need an expressionlike for the electron current (1b) written now as

𝐼 =2𝑞

ℎ∫𝑇el (𝐸)𝑀el (𝐸) (𝑓1 − 𝑓2) 𝑑𝐸. (51)

For electrons the states in the contacts were filled accord-ing to the equilibrium Fermi functions, but phonons obeyBose statistics; thus, the phonon states in the contacts arefilled according to the equilibriumBose-Einstein distribution

𝑛0(ℏ𝜔) =

1

𝑒ℏ𝜔/𝑘𝑇 − 1. (52)


Let temperature for the left and the right contacts be𝑇1and 𝑇

2. As for the electrons, both contacts are assumed

ideal. Thus the phonons that enter a contact are not able toreflect back, and transmission coefficient𝑇ph(𝐸) describes thephonon transmission across the entire channel.

It is easy now to rewrite (51) to the phonon heat current.Electron energy 𝐸 we replace by the phonon energy ℏ𝜔. Inthe electron current we have charge 𝑞moving in the channel;in case of the phonon current the quantum of energy ℏ𝜔is moving instead; thus, we replace 𝑞 in (51) with ℏ𝜔 andmove it inside the integral. The coefficient 2 in (51) reflectsthe spin degeneracy of an electron. In case of the phononswe remove this coefficient, and instead the number of thephonon polarization states that contribute to the heat flowlets us include to the number of the phononmodes𝑀ph(ℏ𝜔).Finally, the heat current due to phonons is

𝑄 =1

ℎ∫ (ℏ𝜔) 𝑇ph (ℏ𝜔)𝑀ph (ℏ𝜔) (𝑛1 − 𝑛2) 𝑑 (ℏ𝜔) , [W] .

(53)

In the linear response regime by analogy with (26),

𝑛1− 𝑛2≈ −

𝜕𝑛0

𝜕𝑇Δ𝑇, (54)

where the derivative according to (52) is

𝜕𝑛0

𝜕𝑇=ℏ𝜔

𝑇(−

𝜕𝑛0

𝜕 (ℏ𝜔)) , (55)

with

𝜕𝑛0

𝜕 (ℏ𝜔)= (−

1

𝑘𝑇)

𝑒ℏ𝜔/𝑘𝑇

(𝑒ℏ𝜔/𝑘𝑇 − 1)2. (56)

Now (53) for small differences in temperature becomes

𝑄 = −𝐾𝐿Δ𝑇, (57)

where the thermal conductance

𝐾𝐿=𝑘2𝑇

ℎ∫𝑇ph (ℏ𝜔)𝑀ph (ℏ𝜔)

× [(ℏ𝜔

𝑘𝑇)

2

(−𝜕𝑛0

𝜕 (ℏ𝜔))] 𝑑 (ℏ𝜔) , [W/K] .

(58)

Equation (57) is simply Fourier’s law stating that heatflows down to a temperature gradient. It is also useful tonote that the thermal conductance (58) displays certainsimilarities with the electrical conductance

𝐺 =2𝑞2

ℎ∫𝑇el (𝐸)𝑀el (𝐸) (−

𝜕𝑓0

𝜕𝐸)𝑑𝐸. (59)

The derivative

𝑊el (𝐸) ≡ (−𝜕𝑓0

𝜕𝐸) (60)

known as the Fermi window function that picks out thoseconduction modes which only contribute to the electriccurrent. The electron windows function is normalized:

∫

+∞

−∞

(−𝜕𝑓0

𝜕𝐸)𝑑𝐸 = 1. (61)

In case of phonons the term in square brackets of (58) actsas a window function to specify which modes carry the heatcurrent. After normalization,

𝑊ph (ℏ𝜔) =3

𝜋2(ℏ𝜔

𝑘𝑇)(

𝜕𝑛0

𝜕 (ℏ𝜔)) ; (62)

thus finally

𝐾𝐿=𝜋2𝑘2𝑇

3ℎ∫𝑇ph (ℏ𝜔)𝑀ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔) (63)

with

𝑔0≡𝜋2𝑘2𝑇

3ℎ≈ (9.456 × 10

−13W/K2) 𝑇, (64)

known as the quantum of thermal conductance experimen-tally observed first in 2000 [10].

Comparing (59) and (63) one can see that the electricaland thermal conductances are similar in structure: bothare proportional to corresponding quantum of conductancetimes an integral over the transmission times the number ofmodes times a window function.

The thermal broadening functions for electrons andphonons have similar shapes and each has a width of a few𝑘𝑇. In case of electrons this function is given by (27) or

𝐹𝑇(𝑥) ≡

𝑒𝑥

(𝑒𝑥 + 1)

2(65)

with 𝑥 ≡ (𝐸−𝐸𝐹)/𝑘𝑇 and is shown on Figure 2.This function

for phonons is given by (62) or

𝐹ph𝑇(𝑥) ≡

3

𝜋2

𝑥2𝑒𝑥

(𝑒𝑥 − 1)

2(66)

with 𝑥 ≡ ℏ𝜔/𝑘𝑇. Both functions are normalized to a unityand shown together on Figure 3.

Along with the number of modes determined by thedispersion relation, these two window functions play a keyrole in determining the electrical and thermal conductances.

3.1. Thermal Conductivity of the Bulk Conductors. The ther-mal conductivity of a large diffusive resistor is a key materialproperty that controls performance of any electronic devices.By analogy with the transmission coefficient (3) for electrontransport, the phonon transmission

𝑇ph (ℏ𝜔) =𝜆ph (ℏ𝜔)

𝜆ph (ℏ𝜔) + 𝐿

𝐿≫𝜆ph

→

𝜆ph (ℏ𝜔)

𝐿. (67)

It is also obvious that for large 3D conductors the numberof phonon modes is proportional to the cross-sectional areaof the sample:

𝑀ph (ℏ𝜔) ∝ 𝐴. (68)


10

5

0

−5

−100 0.1 0.2 0.3

FT(x)

FT(x)

FT(x)

phx

Figure 3: Broadening function for phonons compared to that ofelectrons.

Now let us return to (57) dividing and multiplying it by𝐴/𝐿, which immediately gives (50) for the phonon heat fluxpostulated above:

𝑄

𝐴≡ 𝐽

ph𝑄𝑥

= −𝜅𝐿

𝑑𝑇

𝑑𝑥(69)

with specific lattice thermal conductivity

𝜅𝐿= 𝐾𝐿

𝐿

𝐴, (70)

or substituting (67) into (63), one for the lattice thermalconductivity finally obtains

𝜅𝐿=𝜋2𝑘2𝑇

3ℎ∫

𝑀ph (ℏ𝜔)

𝐴𝜆ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔) . (71)

It is useful now to define the average number of phononmodes per cross-sectional area of the conductor that partici-pate in the heat transport:

⟨

𝑀ph

𝐴⟩ ≡ ∫

𝑀ph (ℏ𝜔)

𝐴𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔) . (72)

Then


3ℎ⟨

𝑀ph

𝐴⟩⟨⟨𝜆ph⟩⟩ , (73)

where the average mean-free-path is defined now as

⟨⟨𝜆ph⟩⟩ =∫ (𝑀ph (ℏ𝜔) /𝐴) 𝜆ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔)

∫ (𝑀ph (ℏ𝜔) /𝐴)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔).

(74)

Thus, the couple of the phonon transport equations (69)and (73) corresponds to similar electron transport equations:

𝐽𝑥=𝜎

𝑞

𝑑 (𝐸𝐹)

𝑑𝑥, (75)

𝜎 =2𝑞2

ℎ⟨𝑀el𝐴

⟩⟨⟨𝜆el⟩⟩ . (76)

The thermal conductivity (73) and the electrical conduc-tivity (76) have the same structure. It is always a product of thecorresponding quantum of conductance times the numberof modes that participate in transport, times the averagemean-free-path. These three quantities for phonons will bediscussed later.

3.2. Specific Heat versus Thermal Conductivity. The connec-tion between the lattice specific thermal conductivity andthe lattice specific heat at constant volume is well known[6–9]. We will show now that corresponding proportionalitycoefficient is a product of an appropriately-defined mean-free-path ⟨⟨Λ ph⟩⟩ and an average phonon velocity ⟨Vph⟩,namely:

𝜅𝐿=1

3⟨⟨Λ ph⟩⟩ ⟨Vph⟩𝐶𝑉. (77)

The total phonon energy per unit volume

𝐸ph = ∫∞

0

(ℏ𝜔)𝐷ph (ℏ𝜔) 𝑛0 (ℏ𝜔) 𝑑 (ℏ𝜔) , (78)

where𝐷ph(ℏ𝜔) is the phonon density of states. By definition,

𝐶𝑉≡

𝜕𝐸ph

𝜕𝑇=

𝜕

𝜕𝑇∫

∞

0

(ℏ𝜔)𝐷ph (ℏ𝜔) 𝑛0 (ℏ𝜔) 𝑑 (ℏ𝜔)

= ∫

∞

0

(ℏ𝜔)𝐷ph (ℏ𝜔) (𝜕𝑛0(ℏ𝜔)

𝜕𝑇)𝑑 (ℏ𝜔)

=𝜋2𝑘2𝑇

3∫

∞

0

𝐷ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔) ,

(79)

where (55) and (62) were used. Next, multiply and divide (71)by (79) and obtain the proportionality we are looking for:𝜅𝐿

= [

[

(1/ℎ) ∫∞

0(𝑀ph (ℏ𝜔) /𝐴) 𝜆ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔)

∫∞

0𝐷ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔)

]

]

𝐶𝑉.

(80)To obtain final expression (77) and correct interpretation

of the proportionality coefficient we need to return to (67).This expression can be easily derived for 1D conductorwith several simplifying assumptions. Nevertheless it worksvery well in practice for a conductor of any dimension.Derivation of (67) is based on the interpretation of themean-free-path 𝜆(𝐸) or 𝜆(ℏ𝜔) as that its inverse value is theprobability per unit length that a positive flux is convertedto a negative flux. This is why 𝜆 is often called a mean-free-path for backscattering. Let us relate it to the scattering time𝜏. The distinction between mean-free-path and mean-free-path for backscattering is easiest to see for 1D conductor.Let an electron undergo a scattering event. For isotropicscattering the electron can forward scatter or backscatter.Only backscattering is relevant for the mean-free-path forscattering, so the time between backscattering events is 2𝜏.Thus the mean-free-path for backscattering is twice themean-free-path for scattering:

𝜆1𝐷

(𝐸) = 2Λ (𝐸) = 2V (𝐸) 𝜏 (𝐸) . (81a)

It was shown that the proper definition of the mean-free-path for backscattering for a conductor of any dimension [11]is

𝜆 (𝐸) = 2

⟨V2𝑥𝜏⟩

⟨V𝑥

⟩, (∗ ∗ ∗)


where averaging is performed over angles. For isotropicbands,

𝜆2𝐷

(𝐸) =𝜋

2V (𝐸) 𝜏 (𝐸) , (81b)

𝜆3𝐷

(𝐸) =4

3V (𝐸) 𝜏 (𝐸) . (81c)

The scattering time is often approximately written as thepower law scattering:

𝜏 (𝐸) = 𝜏0(𝐸 − 𝐸

𝐶

𝑘𝑇)

𝑠

, (82)

where exponent 𝑠 describes the specific scattering mecha-nism: for acoustic phonon scattering in 3D conductor withparabolic dispersion 𝑠 = −1/2 and for ionized impurityscattering 𝑠 = +3/2 [12].

Analogous power law is often used for mean-free-path:

𝜆 (𝐸) = 𝜆0(𝐸 − 𝐸

𝐶

𝑘𝑇)

𝑟

. (83)

For parabolic zone structure V(𝐸) ∝ 𝐸1/2; thus 𝑟 = 𝑠 + 1/2with 𝑟 = 0 for acoustic phonon scattering, and 𝑟 = 2 forionized impurity scattering.

Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have

𝜆ph (ℏ𝜔) =4

3Vph (ℏ𝜔) 𝜏ph (ℏ𝜔) , (84)

where, according to (81a),

Vph (ℏ𝜔) 𝜏ph (ℏ𝜔) = Λ ph (ℏ𝜔) , (85)

and finally

𝜆ph (ℏ𝜔) =4

3Λ ph (ℏ𝜔) . (86)

It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are

𝑀el (𝐸) = 𝐴𝑀3𝐷 (𝐸) = 𝐴ℎ


3𝐷(𝐸) . (87)

Let us rewrite this formula for phonons. Note that the spindegeneracy for electrons 𝑔

𝑆= 2 is included with the density

of states:

𝐷3𝐷

(𝐸) = 2𝐷

3𝐷(𝐸) , (88)

and for spherical bands in 3D conductor,

⟨V+𝑥(𝐸)⟩ =

Vel (𝐸)2

. (89)

Collecting (87) up to (89) all together in case of phonons, wehave

𝑀ph (ℏ𝜔) = 𝐴ℎ

2(

Vph (ℏ𝜔)2

) 2𝐷ph (ℏ𝜔)

= 𝐴ℎ

4Vph (ℏ𝜔)𝐷ph (ℏ𝜔) .

(90)

Substituting (86) and (90) into (80), we obtain

𝜅𝐿

= [

[

(1/3) ∫∞

0Λ ph (ℏ𝜔) Vph (ℏ𝜔)𝐷ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔)

∫∞


]

]

𝐶𝑉.

(91)

Multiplying and dividing (91) by

∫

∞

0

Vph (ℏ𝜔)𝐷ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔) , (92)

we finally get (77) with proportionality coefficient between 𝜅𝐿

and 𝐶𝑉as the product of an average mean-free-path as

⟨⟨Λ ph⟩⟩

≡

∫∞

0Λ ph (ℏ𝜔) Vph (ℏ𝜔)𝐷ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔)

∫∞

0Vph (ℏ𝜔)𝐷ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔)

(93)

and an average velocity as

⟨Vph⟩ ≡∫∞

0Vph (ℏ𝜔)𝐷ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔)

∫∞


, (94)

with the appropriate averaging.Equation (77) is often used to estimate the average mean-

free-path from the measured 𝜅𝐿and 𝐶

𝑉, if we know the

average velocity, which is frequently assumed to be the lon-gitudinal sound velocity. The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩. If a phonondispersion is chosen one can always compute the averagevelocity according to (94), and it is typically very differentfrom the longitudinal sound velocity. Thus, estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13].

3.3. Debye Model. For 3D conductors there are three polar-ization states for lattice vibrations: one for atoms displacedin the direction of propagation (longitudinal/𝐿) and two foratoms displaced orthogonally to the direction of propagation(transverse/𝑇). The low energy modes are called acousticmodes/𝐴: one 𝐿𝐴 mode analogous to sound waves propa-gating in air and two 𝑇𝐴 modes. Near 𝑞 → 0 dispersion ofacoustic modes is linear:

ℏ𝜔 = ℏV𝐷𝑞, (95)

and is known as Debye approximation. The Debye velocityV𝐷is an average velocity of the 𝐿 and 𝑇 acoustic modes. In

case of 𝐿𝐴mode V𝐷is simply the sound velocity V

𝑠∝ 𝑚−1/2

with 𝑚 being an effective mass of vibrating atom. Typically,V𝑠≈ 5 × 10

3m/s, about 20 times slower that the velocity of atypical electron.

The bandwidth of the electronic dispersion is typicallyBW ≫ 𝑘𝑇, so only states near the bottom of the conductionband where the effective mass model is reasonably accurate


are occupied. For phonons the situation ismuch different; thebandwidth BW ≈ 𝑘𝑇, so states across the entire Brillouinzone are occupied. The widely used Debye approximation(95) fits the acoustic branches as long as 𝑞 is not too far fromthe center of the Brillouin zone.

With the Debye approximation (95) it is easy to find thedensity of the phonon states:

𝐷ph (ℏ𝜔) =3(ℏ𝜔)

2

2𝜋2(ℏV𝐷)3, [J−1 ⋅m−3] , (96)

where the factor of three is for the three polarizations. Thenone can obtain the number of phonon modes per cross-sectional area from (90):

𝑀ph (ℏ𝜔) =3(ℏ𝜔)

2

4𝜋(ℏV𝐷)2. (97)

Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures, we are to be sure that we accountfor the correct number of states. For a crystal there are 3𝑁/Ωstates per unit volume. To find the total number of states wehave to integrate the density of states:

∫

ℏ𝜔𝐷

0

𝐷ph (ℏ𝜔) 𝑑 (ℏ𝜔) , (98)

with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states, namely:

ℏ𝜔𝐷= ℏV𝐷(6𝜋2𝑁

Ω)

1/3

≡ 𝑘𝑇𝐷. (99)

The Debye frequency defines a cutoff frequency abovewhich no states are accounted for. This restriction can alsobe expressed via a cutoff wave vector 𝑞

𝐷or as a Debye tem-

perature:

𝑇𝐷=ℏ𝜔

𝑘. (100)

For 𝑇 ≪ 𝑇𝐷, only states with 𝑞 → 0 for which the Debye

approximation is accurate are occupied.Now we can calculate the lattice thermal conductivity by

integrating (71) to the Debye cutoff energy


3ℎ∫

ℏ𝜔𝐷

0

𝑀ph (ℏ𝜔)

𝐴𝜆ph (ℏ𝜔)𝑊ph (ℏ𝜔) 𝑑 (ℏ𝜔) (101)

and estimate𝑀ph(ℏ𝜔) according to (71). The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used. This is how the lattice thermalconductivities were first calculated [14, 15]. The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11, 13, 16].

3.4. Phonon Scattering. Phonons can scatter from defects,impurity atoms, isotopes, surfaces and boundaries, and elec-trons and from other phonons. Phonon-phonon scattering

occurs because the potential energy of the bonds in the crystalis not exactly harmonic. All higher order terms are treatedas a scattering potential. Two types of phonon scattering areconsidered. In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved:

ℏ ⃗𝑞1+ ℏ ⃗𝑞2= ℏ ⃗𝑞3,

ℏ�⃗�1+ ℏ�⃗�2= ℏ�⃗�3.

(102)

The total momentum of the phonon ensemble is conserved;thus this type of scattering has little effect on the heat flux.

In a second type of scattering, umklapp/𝑈-scattering,the two initial phonons have larger momentum; thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions. The 𝑈-scatterings are thebasic processes in the heat transport especially at hightemperatures. Scattering on defects/𝐷 and on boundaries/𝐵is also important. Scattering rates are additive; thus the totalphonon scattering rate is

1

𝜏ph (ℏ𝜔)=

1

𝜏𝑈(ℏ𝜔)

+1

𝜏𝐷(ℏ𝜔)

+1

𝜏𝐵(ℏ𝜔)

(103)

or alternatively, in terms of the mean-free-path (84),

1

𝜆ph (ℏ𝜔)=

1

𝜆𝑈(ℏ𝜔)

+1

𝜆𝐷(ℏ𝜔)

+1

𝜆𝐵(ℏ𝜔)

. (104)

Expressions for each of the scattering rates are developed[17]. For scattering from point defects,

1

𝜏𝐷(ℏ𝜔)

∝ 𝜔4, (105)

known as the Rayleigh scattering which is like the scatteringof light from the dust.

For boundaries and surfaces,

1

𝜏𝐵(ℏ𝜔)

∝

Vph (ℏ𝜔)𝐿

, (106)

where 𝐿 is the shortest dimension of the sample.A commonly used expression for 𝑈-scattering is

1

𝜏𝑈(ℏ𝜔)

∝ 𝑇3𝜔2𝑒−𝑇𝐷/𝑏𝑇. (107)

With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity.

3.5. Lattice Thermal Conductivity versus Temperature. Thetemperature dependence of the lattice thermal conductivity𝜅𝐿is illustrated for bulk Si on Figure 4.According to (73) 𝜅

𝐿is proportional to the number of the

phononmodes that are occupied ⟨𝑀ph/𝐴⟩ and to the averagevalue of the phononmean-free-path ⟨⟨𝜆ph⟩⟩.The curve 𝜅𝐿(𝑇)can be explained by understanding how ⟨𝑀ph⟩ and ⟨⟨𝜆ph⟩⟩vary with temperature.


100 101101

102

102

103

103

104

𝜅ph

(Wm

−1K

−1)

T (K)

Figure 4: The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature.

It can be shown using (72) that, at low temperatures,

⟨𝑀ph⟩ ∝ 𝑇3, (𝑇 → 0) (108)

so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature. At low temperatures boundary scattering isimportant. As the temperature increases more short-wave-length phonons are produced. These phonons scatter frompoint defects, so defect scattering becomes more and moreimportant. As the temperature approaches the 𝑇

𝐷, all of

the phonon modes are populated and further increases intemperature do not change ⟨𝑀ph⟩. Instead, the higher tem-peratures increase the phonon scattering by𝑈-processes, andthe thermal conductivity drops with increasing temperatures.

3.6. Difference between Lattice Thermal and Electrical Con-ductivities. We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76). The average electron andphonon mean-free-paths are of the same order of magni-tude. Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few? The answer lies in thecorresponding windows functions (60) and (62). For bothelectrons and phonons, higher temperatures broaden thewindow function and increase the population of states. Forelectrons, however, the position of the Fermi level has adramatic effect on the magnitude of the window function.By controlling the position of the Fermi level, the electricalconductivity can be varied over many orders of magnitude.For the phonons the width of the window function isdetermined by temperature only.

Another key difference between electrons and phononsrelates to how the states are populated. For a thermoelectricdevice 𝐸

𝐹≈ 𝐸𝐶, and the electron and phonon window

functions are quite similar. However, for electrons the BWof the dispersion is very large, so only a few states near thebottom of the conduction band are populated: the effectivemass approximation works well for these states, and it is easyto obtain analytical solutions. For phonons, the BW of thedispersion is small. Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied: simple analyticalapproximations do not work, and it is hard to get analyticalsolutions for the lattice thermal conductivity.

3.7. Lattice Thermal Conductivity Quantization. By analogywith the electronic conduction quantization,

𝐺ball

=2𝑞2

ℎ𝑀(𝐸𝐹) , (109)

over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow. In fact, if𝑇 → 0 in (63), thenthe phonon window𝑊ph(ℏ𝜔) is sharply peaked near ℏ𝜔 = 0:


3ℎ𝑇ph (0)𝑀ph (0) . (110)

For a bulk conductor 𝑀ph(ℏ𝜔) → 0 as ℏ𝜔 → 0,but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes. For ballistic phonontransport, 𝑇ph = 1 and one can expect that


3ℎ𝑀ph. (111)

Exactly this result was proved experimentally using 4-mode resistor at 𝑇 < 0.8K [10]; thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20–22].

The quantum of thermal conductance

𝑔0≡𝜋2𝑘2𝑇

3ℎ

(112)

represents themaximumpossible value of energy transportedper phonon mode. Surprisingly, it does not depend onparticle statistics: the quantum of thermal conductance isuniversal for fermions, bosons, and anyons [23–25].

4. Conclusions

In summary, we see that the LDL concept used to describeelectron transport can be generalized for phonons. In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic, quasiballistic, and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device. We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity, but there are twoimportant differences.

The first difference between electrons and phonons is thedifference in bandwidths of their dispersions. For electrons,


the dispersion BW ≫ 𝑘𝑇 at room temperature, so onlylow energy states are occupied. For phonons, BW ≈ 𝑘𝑇,so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied. As a result, the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion.

The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature. For phonons, the window functiondepends only on the temperature. The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies, while lattice conductivitiesvary over only a few orders of magnitude.

Finally, we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5, 26].

Appendices

A. Thermoelectric Coefficients for1D, 2D, and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes

Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 𝑗 defined as

I𝑗(𝜂𝐹) =

1

Γ (𝑗 + 1)∫

∞

0

𝜂𝑗

exp (𝜂 − 𝜂𝐹) + 1

𝑑𝜂, (A.1)

where the location of the Fermi level 𝐸𝐹relative to the

conduction band edge 𝐸𝐶

is given by the dimensionlessparameter

𝜂𝐹=𝐸𝐹− 𝐸𝐶

𝑘𝑇. (A.2)

In expressions below thermoelectric coefficients (40a),(40b), (40c), (40d), (42a), (42b), (42c), (42d), and (42e)for diffusive regime were calculated with the power lawscattering:

𝜆 (𝐸) = 𝜆0(𝐸

𝑘𝑇)

𝑟

. (A.3)

Thermoelectric coefficients for 1D ballistic resistors are:

𝐺 =2𝑞2

ℎI−1(𝜂𝐹) ,

𝑆𝑇= −

𝑘

𝑞

2𝑞2

ℎ[I0(𝜂𝐹) − 𝜂𝐹I−1(𝜂𝐹)] ,

𝑆 = −𝑘

𝑞[I0(𝜂𝐹)

I−1(𝜂𝐹)− 𝜂𝐹] ,

𝐾0= 𝑇(

𝑘

𝑞)

22𝑞2

ℎ[2I1(𝜂𝐹) − 2𝜂

𝐹I0(𝜂𝐹) + 𝜂2

𝐹I−1(𝜂𝐹)] ,

𝐾 = 𝑇(𝑘

𝑞)

22𝑞2

ℎ[2I1(𝜂𝐹) −

I20(𝜂𝐹)

I−1(𝜂𝐹)] .

(A.4)

Thermoelectric coefficients for 1D diffusive resistors are:

𝐺 =2𝑞2

ℎ(𝜆0

𝐿) Γ (𝑟 + 1)I

𝑟−1(𝜂𝐹) ,

𝑆𝑇= −

𝑘

𝑞

2𝑞2

ℎ(𝜆0

𝐿) Γ (𝑟 + 1)

× [(𝑟 + 1)I𝑟(𝜂𝐹) − 𝜂𝐹I𝑟−1

(𝜂𝐹)] ,

𝑆 = −𝑘

𝑞[(𝑟 + 1)I

𝑟(𝜂𝐹)

I𝑟−1

(𝜂𝐹)

− 𝜂𝐹] ,

𝐾0= 𝑇(

𝑘

𝑞)

22𝑞2

ℎ(𝜆0

𝐿)

× [Γ (𝑟 + 3)I𝑟+1

(𝜂𝐹) − 2𝜂

𝐹Γ (𝑟 + 2)

×I𝑟(𝜂𝐹) + 𝜂2

𝐹Γ (𝑟 + 1)I

𝑟−1(𝜂𝐹)] ,

𝐾 = 𝑇(𝑘

𝑞)

22𝑞2

ℎ(𝜆0

𝐿) Γ (𝑟 + 2)

× [(𝑟 + 2)I𝑟+1

(𝜂𝐹) −

(𝑟 + 1)I2𝑟(𝜂𝐹)

I𝑟−1

(𝜂𝐹)

] .

(A.5)

Conductivity 𝐺 = 𝜎1𝐷/𝐿 is given in Siemens: [𝜎

1𝐷] = 1 S ⋅

m. Similarly for other specific coefficients: 𝑠𝑇= 𝑆𝑇𝐿; 𝜅0=

𝐾0𝐿; 𝜅 = 𝐾𝐿.Thermoelectric coefficients for 2D ballistic resistors are:

𝐺 = 𝑊2𝑞2

ℎ

√2𝜋𝑚∗𝑘𝑇

ℎI−1/2

(𝜂𝐹) ,

𝑆𝑇= −𝑊

𝑘

𝑞

2𝑞2

ℎ


ℎ

× [3

2I1/2

(𝜂𝐹) − 𝜂𝐹I−1/2

(𝜂𝐹)] ,

𝑆 = −𝑘

𝑞[3I1/2

(𝜂𝐹)

2I−1/2

(𝜂𝐹)− 𝜂𝐹] ,

𝐾0= 𝑊𝑇(

𝑘

𝑞)

22𝑞2

ℎ


ℎ

× [15

4I3/2

(𝜂𝐹) − 3𝜂

𝐹I1/2

(𝜂𝐹) + 𝜂2

𝐹I−1/2

(𝜂𝐹)] ,

𝐾 = 𝑊𝑇(𝑘

𝑞)

22𝑞2

ℎ


ℎ

× [15

4I3/2

(𝜂𝐹) −

9I21/2

(𝜂𝐹)

4I−1/2

(𝜂𝐹)] .

(A.6)



𝐺 = 𝑊2𝑞2

ℎ(𝜆0

𝐿)√2𝑚∗𝑘𝑇

𝜋ℎΓ (𝑟 +

3

2)I𝑟−1/2

(𝜂𝐹) ,

𝑆 = −𝑘

𝑞[(𝑟 + 3/2)I

𝑟+1/2(𝜂𝐹)

I𝑟−1/2

(𝜂𝐹)

− 𝜂𝐹] ,

𝑆𝑇= −𝑊

𝑘

𝑞

2𝑞2

ℎ(𝜆0


𝜋ℎ

× [Γ (𝑟 +5

2)I𝑟+1/2

(𝜂𝐹) − 𝜂𝐹Γ (𝑟 +

3

2)I𝑟−1/2

(𝜂𝐹)] ,

𝐾0= 𝑊𝑇(

𝑘

𝑞)

22𝑞2

ℎ(𝜆0


𝜋ℎ

× [Γ (𝑟 +7

2)I𝑟+3/2

(𝜂𝐹) − 2𝜂

𝐹Γ (𝑟 +

5

2)

× I𝑟+1/2

(𝜂𝐹) + 𝜂2

𝐹Γ (𝑟 +

3

2)I𝑟−1/2

(𝜂𝐹)] ,

𝐾 = 𝑊𝑇(𝑘

𝑞)

22𝑞2

ℎ(𝜆0


𝜋ℎΓ (𝑟 +

5

2)

× [(𝑟 +5

2)I𝑟+3/2

(𝜂𝐹) −

(𝑟 + 3/2)I2𝑟+1/2

(𝜂𝐹)

I𝑟−1/2

(𝜂𝐹)

] .

(A.7)

Conductivity 𝐺 = 𝜎2𝐷𝑊/𝐿 is given in Siemens: [𝜎

2𝐷] =

1 S. Similarly for other specific coefficients: 𝑠𝑇= 𝑆𝑇𝐿/𝑊; 𝜅

0=

𝐾0𝐿/𝑊; 𝜅 = 𝐾𝐿/𝑊.Thermoelectric coefficients for 3D ballistic resistors are:

𝐺 = 𝐴2𝑞2

ℎ

𝑚∗𝑘𝑇

2𝜋ℏ2I0(𝜂𝐹) ,

𝑆𝑇= −𝐴

𝑘

𝑞

2𝑞2

ℎ

𝑚∗𝑘𝑇

2𝜋ℏ2[2I1(𝜂𝐹) − 𝜂𝐹I0(𝜂𝐹)] ,

𝑆 = −𝑘

𝑞[2I1(𝜂𝐹)

I0(𝜂𝐹)− 𝜂𝐹] ,

𝐾0= 𝐴𝑇(

𝑘

𝑞)

22𝑞2

ℎ

𝑚∗𝑘𝑇

2𝜋ℏ2

× [6I2(𝜂𝐹) − 4𝜂

𝐹I1(𝜂𝐹) + 𝜂2

𝐹I0(𝜂𝐹)] ,

𝐾 = 𝐴𝑇(𝑘

𝑞)

22𝑞2

ℎ

𝑚∗𝑘𝑇

2𝜋ℏ2[6I2(𝜂𝐹) −

4I21(𝜂𝐹)

I0(𝜂𝐹)] .

(A.8)


𝐺 = 𝐴2𝑞2

ℎ(𝜆0

𝐿)𝑚∗𝑘𝑇

2𝜋ℏ2Γ (𝑟 + 2)I

𝑟(𝜂𝐹) ,

𝑆 = −𝑘

𝑞[(𝑟 + 2)I

𝑟+1(𝜂𝐹)

I𝑟(𝜂𝐹)

− 𝜂𝐹] ,

𝑆𝑇= −𝐴

𝑘

𝑞

2𝑞2

ℎ(𝜆0


2𝜋ℏ2

× [Γ (𝑟 + 3)I𝑟+1

(𝜂𝐹) − 𝜂𝐹Γ (𝑟 + 2)I

𝑟(𝜂𝐹)] ,

𝐾0= 𝐴𝑇(

𝑘

𝑞)

22𝑞2

ℎ(𝜆0


2𝜋ℏ2

× [Γ (𝑟 + 4)I𝑟+2

(𝜂𝐹) − 2𝜂

𝐹Γ (𝑟 + 3)I

𝑟+1(𝜂𝐹)

+ 𝜂2

𝐹Γ (𝑟 + 2)I

𝑟(𝜂𝐹)] ,

𝐾 = 𝐴𝑇(𝑘

𝑞)

22𝑞2

ℎ(𝜆0


2𝜋ℏ2Γ (𝑟 + 3)

× [(𝑟 + 3)I𝑟+2

(𝜂𝐹) −

(𝑟 + 2)I2𝑟+1

(𝜂𝐹)

I𝑟(𝜂𝐹)

] .

(A.9)

Conductivity 𝐺 = 𝜎3𝐷𝐴/𝐿 is given in Siemens: [𝜎

3𝐷] =

1 S/m. Similarly for other specific coefficients: 𝑠𝑇= 𝑆𝑇𝐿/𝐴;

𝜅0= 𝐾0𝐿/𝐴; 𝜅 = 𝐾𝐿/𝐴.

B. Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes

Graphene is a 2D conductor with a unique linear bandstructure (15). Its transport coefficients are calculated from(40a), (40b), (40c), (40d), (42a), (42b), (42c), (42d), and (42e)with the number of modes given by (18). The power lawscattering for diffusive regime (A.3) is used.

Conductivity 𝐺 = 𝜎𝑊/𝐿 is given in Siemens: [𝜎] = 1 S.Similarly for other specific coefficients: 𝑠

𝑇= 𝑆𝑇𝐿/𝑊; 𝜅

0=

𝐾0𝐿/𝑊; 𝜅 = 𝐾𝐿/𝑊.Ballistic regime are:

𝐺ball

= 𝑊2𝑞2

ℎ(2𝑘𝑇

𝜋ℏV𝐹

) [I0(𝜂𝐹) +I0(−𝜂𝐹)] ,

𝑆ball

= −𝑘

𝑞{2 [I1(𝜂𝐹) −I1(−𝜂𝐹)]

I0(𝜂𝐹) +I0(−𝜂𝐹)

− 𝜂𝐹} ,

𝑆ball𝑇

= −𝑊2𝑞2

ℎ(2𝑘𝑇

𝜋ℏV𝐹

)(𝑘

𝑞)

× {2 [I1(𝜂𝐹) −I1(−𝜂𝐹)]

− 𝜂𝐹[I0(𝜂𝐹) +I0(−𝜂𝐹)]} ,


𝐾ball

= 𝑊𝑇2𝑞2

ℎ(2𝑘𝑇

𝜋ℏV𝐹

)(𝑘

𝑞)

2

× {6 [I2(𝜂𝐹) +I2(−𝜂𝐹)]

−4[I1(𝜂𝐹) −I1(−𝜂𝐹)]2

I0(𝜂𝐹) +I0(−𝜂𝐹)

} ,

𝐾ball0

= 𝑊𝑇2𝑞2

ℎ(2𝑘𝑇

𝜋ℏV𝐹

)(𝑘

𝑞)

2

× {6 [I2(𝜂𝐹) +I2(−𝜂𝐹)]

− 4𝜂𝐹[I1(𝜂𝐹) −I1(−𝜂𝐹)]

+ 𝜂2

𝐹[I0(𝜂𝐹) +I0(−𝜂𝐹)]} .

(B.1)

Diffusive regime are:

𝐺diff

= 𝑊2𝑞2

ℎ(2𝑘𝑇

𝜋ℏV𝐹

)(𝜆0

𝐿) Γ (𝑟 + 2)

× [I𝑟(𝜂𝐹) +I𝑟(−𝜂𝐹)] ,

𝑆diff

= −𝑘

𝑞{(𝑟 + 2) [I

𝑟+1(𝜂𝐹) −I𝑟+1

(−𝜂𝐹)]

I𝑟(𝜂𝐹) +I𝑟(−𝜂𝐹)

− 𝜂𝐹} ,

𝑆diff𝑇

= −𝑊2𝑞2

ℎ

𝑘

𝑞(2𝑘𝑇

𝜋ℏV𝐹

)(𝜆0

𝐿)

× {Γ (𝑟 + 3) [I𝑟+1

(𝜂𝐹) −I𝑟+1

(−𝜂𝐹)]

− 𝜂𝐹Γ (𝑟 + 2) [I

𝑟(𝜂𝐹) +I𝑟(−𝜂𝐹)]} ,

𝐾 = 𝑊𝑇2𝑞2

ℎ(𝑘

𝑞)

2

(2𝑘𝑇

𝜋ℏV𝐹

)(𝜆0

𝐿)

× Γ (𝑟 + 3)

× { (𝑟 + 3) [I𝑟+2

(𝜂𝐹) +I𝑟+2

(−𝜂𝐹)]

−(𝑟 + 2) [I

𝑟+1(𝜂𝐹) −I𝑟+1

(−𝜂𝐹)]2

I𝑟(𝜂𝐹) +I𝑟(−𝜂𝐹)

} ,

𝐾0= 𝑊𝑇

2𝑞2

ℎ(𝑘

𝑞)

2

(2𝑘𝑇

𝜋ℏV𝐹

)(𝜆0

𝐿)

× {Γ (𝑟 + 4) [I𝑟+2

(𝜂𝐹) +I𝑟+2

(−𝜂𝐹)]

− 2𝜂𝐹Γ (𝑟 + 3) [I

𝑟+1(𝜂𝐹) +I𝑟+1

(−𝜂𝐹)]

+ 𝜂2

𝐹Γ (𝑟 + 2) [I

𝑟(𝜂𝐹) +I𝑟(−𝜂𝐹)]} .

(B.2)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures “Fundamentalsof Nanoelectronics, Part I: Basic Concepts” and “Near-Equilibrium Transport: Fundamentals and Applications”online in 2011 and 2012 under initiative of Purdue Universi-ty/nanoHUB-U (https://nanohub.org/groups/u).This reviewis closely based on these lectures and books [4, 5].

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