Tunneling transport

Courtesy Prof. S. Sawyer, RPIAlso Davies Ch. 5

Electron transport properties

le: electronic mean free path

lφ: phase coherence length

λF: Fermi wavelength

Lecture Outline• Important Concepts for Resonant Tunneling

Diodes (RTDs)• RTD Physics and Phenomena• RTD Equations and Parameters• RTDs vs. Tunnel Diodes

– Advantages and Disadvantages of RTDs

• Applications• Summary

RTD Concepts: Why Tunneling Devices?

• Advantage of this quantum effect device– Works at room temperature– High switching speed – Low power consumption

• Differing operating principles– Quantization– Quantum tunneling– Negative Differential Resistance

(NDR) http://www.cse.unsw.edu.au/~cs4211/projects/presentations/james-pp.ppt#266,9,Resonant Tunnelling Diodes

RTD Concepts: Tunneling• Tunneling

– Quantum mechanical phenomenon

• Calculate tunneling probability with Schrödinger’s equation

• Complex barrier shapes

– Requires finite barrier height and thin barrier width

RTD Concepts: Tunneling• Tunneling

– Majority carrier effect – Not governed by

conventional time transit concept

– Governed by quantum transition probability per unit time proportional to exp[-2<k(0)>W]

W

<k(0)> is the average value of momentum encountered in the tunneling path…..

Tunneling transport: single barrier

I L

Davies Ch. 5

Current in one-dimension

L

L

L

UL

ULL

ULL

dkTkfhe

hvdkTkvkfeI

dv

dddkdk

dkkTkvkfeI

)(]),([2

)()(]),([2

1

2)()(]),([2

T(k): transmission coefficient

LU

LL dTfheI )(],[2

RU

RR dTfheI )(],[2

LU

RLRL dTffheIII )()],(),([2

Total current in one-dimension

Low bias limit

),(),(),(),( feVfeVff RL

dTfheG

VIG

dTfhVeI

L

L

U

U

)(2

/

)(2

2

2

)(2

)(

2

TheG

f

LU

RLRL dTffheIII )()],(),([2

: conductance

at low temperatures

Tunneling probability

d2

dx2

2 m*

2E U x( )( ) 0

k2m* E U 0

• To determine tunneling probability

• Wavefunction for simple rectangular barrier height of U0 and width W isψ =exp(±ikx) where

Tunneling probability

• Solution to tunneling probability

• Using WKB for other barrier shapes where wavefunction is

T t B 2

A 21

U 02 sinh2 k W( )

4E U 0 E

1

~16E U 0 E

U 02

exp 22 m* U 0 E

2 W

x( ) exp xik x( )

d

T t B 2

A 2exp 2

x 1

x 2x

2 m*

2U x( ) E( )

d

16

Transmission coefficient for single barrier

Potential barrier of 0.3 eV and thickness of a = 10 nm in GaAs

Current in 2 and 3 dimensions

)]),,(()2(

2)[()(2

22),(

)()exp(

2

2

0

2222

,

Lzzzzz

L

zLz

kkk

kkfkdkTkvdkeI

mk

mkUkk

zurikzz

LULDL

zLL

BB

D

dEETEnheI

mkU

TkTmkn

)()(

2

))/exp(1ln()(

2

22

22

Dn2

:wave function

:energy

L

L

URDLD

ULDL

zL

dEETEnEnheJ

dEETEnheJ

mkUE

)()]()([

)()(

2

22

2

22

L

LUL dEETEm

heJ

)()(2

Large bias and low temperature limit

Total resonant tunneling current

Tunneling vs. Resonant Tunneling

Tunneling vs. Resonant Tunneling

http://w3.ualg.pt/~jlongras/OIC-NDRd.pdf

RTD concepts• RTD consists of

– Emitter region: source of electrons T(E)

– Double barrier structure: inside is the quantum well, with discrete energy levels

– Collector region: collect electrons tunneling through the barrier T(C)

RTD concepts• Double barriers formed• Quantum well quantizes

energy

– Assumes infinite barrier height

• Actual barrier height (ΔEc) ~0.2-0.5eV giving quantized levels of ~0.1eV

E n E Cwh2 n2

8 m* W2

A bound state vs. a resonant state

RTD concepts• Carriers tunnel from one

electrode to the other via energy states within the well

• Wavefunctions of Schrodinger equation must be solved for emitter, well, and collector

• Tunneling probability exhibits peaks where the energy of the incoming particle coincides with quantized levels

Profile through a three-dimensional resonant-tunnelling diode. The bias increases from (a) to (d), giving rise to the I(V) characteristic shown in (e). The shaded areas on the left and right are the Fermi seas of electrons.

Profile through a three-dimensional resonant tunneling diode

L

Resonant Tunneling Diode

Negative Differential Resistance

(c)

Animation courtesy of the group of Prof. G Klimeckand the NanoHub

RTD Concepts: NDR• Negative Differential

Resistance

• DC biasing in the NDR region can be used for– Oscillation– Amplification– High speed switching

rdVdI

http://www.answers.com/topic/gunn-diode?cat=technology

Transmission coefficient for resonant tunneling

)(2

)2/

(1)(

2

RL

pk

pk

TTav

EET

ET

2)(4

RL

RLpk TT

TTT

1)( pkET

If TL=TR

Transmission coefficient of a resonant-tunneling structure

RTD parameters• Probability of tunneling when electron energy

does not align with quantized state

• Probability of tunneling when electron energy does align with quantized state

• Resonant tunneling current is given by

Jq

2 EN E( ) T E( )

d N EkT m*

2ln 1 exp

E F E

kT

T E( ) T L T R

T E E n 4 T L T R

T L T R 2

Characteristics of real resonant tunneling diodes

RTD Research (2010)

Devices: 4 × 4 to 30 × 30 µm2

Structure: low Al-composition (18%) barriers RMS roughness of 8 Å

6×6

8×8

4×4

30×30

AlGaN/GaN resonant tunneling diodes

D. Li et al., Appl. Phys. Lett. 100, 252105 (2012).

RTDs vs. Tunnel Diodes• Tunnel Diodes were

discovered by Esaki in 1958– Studied heavily

(degenerately) doped germanium p-n junctions

– Depletion layer width is narrow

– Found NDR over part of forward characteristics

RTDs vs. Tunnel Diodes• (a) Fermi level is constant across

the junction– Net tunneling current zero applied

voltage is zero– Voltage applied: tunneling occurs

• Under what conditions?

• (b) Maximum tunneling current• (c) Tunneling current ceases

– No filled states opposite of unoccupied states

• (d) Normal diffusion and excess current dominates

High dopingLarge capacitanceDifficult device growth

RTDs vs. Tunnel Diodes• Tunneling Probability for

tunnel diodes (triangular barrier)

• Both effective mass and bandgap should be small

• Electric field should be large

T t exp4 2 m*

E g

3

2

3 q

RTDs vs. Tunnel Diodes• Comparison of typical current

voltage characteristics• Ip/Iv ratios

– 8:1 for Ge– 12:1 GaSb– 28:1 for GaAs– 4:1 for Si

• Limitation on ratio– Peak current (doping, effective

tunneling mass, bandgap)– Valley current (distribution of energy

levels in forbidden gap (defect densities)

RTDs vs. Tunnel Diodes• Advantages of RTDs

– Not transit time limited• No minority carrier charge storage • Maximum operational oscillation projected in the THz range (at

room temperature)• Better leakage current (can be used as a rectifier)

– Lower doping than p-n (reduced capacitance)– Easier to fabricate and design than tunnel diodes– Multiple NDR peaks (multivalue logic and memory)

• Disadvantages of RTDs– Does not supply enough current for high power oscillations

Applications• Nine-State Resonant

Tunneling Diode Memory– Eight double barriers

Al/In0.53Ga0.47As/InAs grown by MBE

A.C. Seabaugh et al., IEEE Electron Dev. Lett., EDL-13, 479, (1992)

Applications• High frequency, low

power dissipation– Trigger circuits

• AlAs/GaAs RTDs 110 GHz

– Pulse Generator • 1.7 ps switching transition

times with InAs/AlSb RTDs

– Oscillators• 712 GHz with InAs/Alsb

T.C. Sollner, GaAs IC Symposium, 15, (1990)

• Two paths to THz– Light/optics

(photonics)– Radio/microwave

(electronics)

Emerging technologies

Why Resonant Tunnelling Devices?

• Works at room temperature!• Extremely high switching speed (THz)• Low power consumption• Well demonstrated uses

– Logic gates, fast adders, ADC etc.• Can be integrated on existing processes• In one word: Feasible

Summary• Tunneling and negative differential resistance are key

characteristics of RTDs• These devices are used for amplification, oscillation, and

high speed switching• RTDs are not transit time limited (no minority carrier storage

charge)• Tunneling occurs when incoming energy of electrons

coincide with quantized states in quantum wells (resonance)• Diminished current due to lack of available electrons in line

with quantized states causes NDR• Thermionic emission dominates in the valley