a review on conduction mechanisms in dielectric films

Review ArticleA Review on Conduction Mechanisms in Dielectric Films

Fu-Chien Chiu

Department of Electronic Engineering, Ming Chuan University, Taoyuan 333, Taiwan

Correspondence should be addressed to Fu-Chien Chiu; [email protected]

Received 29 August 2013; Revised 5 December 2013; Accepted 11 December 2013; Published 18 February 2014

Academic Editor: Chun-Hsing Shih

Copyright © 2014 Fu-Chien Chiu. 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 conduction mechanisms in dielectric films are crucial to the successful applications of dielectric materials. There are twotypes of conduction mechanisms in dielectric films, that is, electrode-limited conduction mechanism and bulk-limited conductionmechanism.The electrode-limited conductionmechanism depends on the electrical properties at the electrode-dielectric interface.Based on this type of conduction mechanism, the physical properties of the barrier height at the electrode-dielectric interfaceand the effective mass of the conduction carriers in dielectric films can be extracted. The bulk-limited conduction mechanismdepends on the electrical properties of the dielectric itself. According to the analyses of bulk-limited conduction mechanisms,several important physical parameters in the dielectric films can be obtained, including the trap level, the trap spacing, the trapdensity, the carrier drift mobility, the dielectric relaxation time, and the density of states in the conduction band. In this paper, theanalytical methods of conduction mechanisms in dielectric films are discussed in detail.

1. Introduction

The application of dielectric films has always been a veryimportant subject for the semiconductor industry and thescientific community. This is especially true for metal-oxide-semiconductor field effect transistor (MOSFET) technologyin integrated circuits (ICs). The concept of MOSFET is basedon themodulation of channel carriers by an applied gate volt-age across a thin dielectric. Dielectric is a material in whichthe electrons are very tightly bonded. The electric charges indielectrics will respond to an applied electric field throughthe change of dielectric polarization. Dielectric materials arenearly insulators in which the electrical conductivity is verylow and the energy band gap is large. In general, the value ofenergy band gap of insulators is set to be larger than 3 eV or5 eV. Although not all dielectrics are insulators, all insulatorsare typical dielectrics. At 0K, the valence band is completelyfilled and the conduction band is completely empty. Thus,there is no carrier for electrical conduction. When thetemperature is larger than 0K, there will be some electronsthermally excited from the valence band and also from thedonor impurity level to the conduction band.These electronswill contribute to the current transport of the dielectric mate-rial. Similarly, holes will be generated by acceptor impurities

and vacancies will be left by excited electrons in the valenceband.The conduction current of insulators at normal appliedelectric field will be very small because their conductivitiesare inherently low, on the order of 10−20∼10−8Ω−1 cm−1.However, the conduction current through the dielectric filmis noticeable when a relatively large electric field is applied.These noticeable conduction currents are owing to manydifferent conduction mechanisms, which is critical to theapplications of the dielectric films. For example, the gatedielectric of MOSFETs, the capacitor dielectric of dynamicrandom access memories, and the tunneling dielectric ofFlash memories are of top importance to the IC applications.In these cases, the conduction current must be lower thana certain level to meet the specific reliability criteria undernormal operation of the devices. Consequently, the study ofthe various conduction mechanisms through dielectric filmsis of great importance to the success of the integrated circuits.

Tomeasure the conduction current through the dielectricfilm, one must prepare some kind of sample devices for test-ing. In general, there are two types of device structures used insample testing. One is metal-insulator-metal structure whichis called the MIM capacitor or the MIM diode. The consid-ered issue in MIM capacitors is the possible asymmetry ofthe electrical properties when the top and bottom electrodes

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2014, Article ID 578168, 18 pageshttp://dx.doi.org/10.1155/2014/578168

2 Advances in Materials Science and Engineering

Conduction mechanism

Bulk-limited conduction mechanism

Schottky emission Fowler-Nordheim tunneling Direct tunneling Thermionic-field emission

Poole-Frenkele missionHopping conduction Ohmic conduction Space-charge-limited conduction Ionic conduction Grain-boundary-limited conduction

Electrode-limited conduction mechanism

Figure 1: Classification of conduction mechanisms in dielectric films.

are made of different metals. Different metals generally leadto different work functions, and therefore result in differentmetal-dielectric interface barriers. The main parameters inthis type of measurement are the barrier height of the metal-dielectric interface and the effective mass of the conductioncarriers. The second type used to characterize a dielectricfilm is the metal-insulator-semiconductor (MIS) capacitor.Since MIS capacitor is the most useful device in the studyof semiconductor surfaces, it is of interest to characterizethe electrical properties of the device. However, the structureof the MIS capacitor is inherently asymmetric and oneshould be careful about the voltage drop across each layer.If the MIS capacitor can be biased in such a way that thesemiconductor surface is in accumulation, the voltage dropacross the semiconductor is minimal and most of the voltagewill be applied across the dielectric film. If the semiconductorsurface is in depletion or inversion, some voltage drop acrossthe semiconductor will take place and then the voltage dropneeds to be considered in calculation of the electric fieldacross the dielectric film.

Among the conduction mechanisms being investigated,some depend on the electrical properties at the electrode-dielectric contact. These conduction mechanisms are calledelectrode-limited conduction mechanisms or injection-lim-ited conduction mechanisms. There are other conductionmechanisms which depend only on the properties of thedielectric itself. These conduction mechanisms are calledbulk-limited conduction mechanisms or transport-limitedconduction mechanisms [1–10]. The methods to distinguishthese conduction mechanisms are essential because there area number of conduction mechanisms that may all contributeto the conduction current through the dielectric film at thesame time. Since several conduction mechanisms dependon the temperature in different ways, measuring the tem-perature dependent conduction currents may afford us ahelpful way to know the constitution of the conduction cur-rents.The electrode-limited conduction mechanisms include(1) Schottky or thermionic emission (2) Fowler-Nordheim

tunneling, (3) direct tunneling, and (4) thermionic-fieldemission. The bulk-limited conduction mechanisms include(1) Poole-Frenkel emission, (2) hopping conduction, (3)

ohmic conduction, (4) space-charge-limited conduction, (5)ionic conduction, and (6) grain-boundary-limited conduc-tion. Figure 1 shows the classification of conduction mech-anisms in dielectric films.

2. Electrode-Limited Conduction Mechanisms

The electrode-limited conduction mechanisms depend onthe electrical properties at the electrode-dielectric contact.The most important parameter in this type of conduc-tion mechanism is the barrier height at the electrode-dielectric interface. The electrode-limited conduction mech-anisms include (1) Schottky or thermionic emission, (2)

Fowler-Nordheim tunneling, (3) direct tunneling, and (4)

thermionic-field emission. The current due to thermionicemission is highly dependent on the temperature, whereas thetunneling current is nearly temperature independent. Asidefrom the barrier height at the electrode-dielectric interface,the effective mass of the conduction carriers in dielectricfilms is also a key factor in the electrode-limited conductionmechanisms.

2.1. Schottky or Thermionic Emission. Schottky emission isa conduction mechanism that if the electrons can obtainenough energy provided by thermal activation, the electronsin the metal will overcome the energy barrier at the metal-dielectric interface to go to the dielectric. Figure 2 shows theMIS energy band diagram when the metal electrode is undernegative bias with respect to the dielectric and the semi-conductor substrate. The energy barrier height at the metal-dielectric interface may be lowered by the image force. Thebarrier-lowering effect due to the image force is called Schot-tky effect. Such a conduction mechanism due to electronemission from the metal to the dielectric is called thermionicemission or Schottky emission.Thermionic emission is one of

Advances in Materials Science and Engineering 3

Metal Insulator Semiconductor

Schottky emission

EC

EFEV

q𝜙B

Figure 2: Schematic energy band diagram of Schottky emission inmetal-insulator-semiconductor structure.

the most often observed conduction mechanism in dielectricfilms, especially at relatively high temperature.The expressionof Schottky emission is

𝐽 = 𝐴

∗

𝑇

2 exp[

−𝑞 (𝜙

𝐵

− √𝑞𝐸/4𝜋𝜀

𝑟

𝜀

0

)

𝑘𝑇

] ,

𝐴

∗

=

4𝜋𝑞𝑘

2

𝑚

∗

ℎ

3

=

120𝑚

∗

𝑚

0

,

(1)

where 𝐽 is the current density, 𝐴∗ is the effective Richardsonconstant, 𝑚

0

is the free electron mass, 𝑚∗ is the effectiveelectron mass in dielectric, 𝑇 is the absolute temperature, 𝑞is the electronic charge, 𝑞𝜙

𝐵

is the Schottky barrier height(i.e., conduction band offset), 𝐸 is the electric field acrossthe dielectric, 𝑘 is the Boltzmann’s constant, ℎ is the Planck’sconstant, 𝜀

0

is the permittivity in vacuum, and 𝜀

𝑟

is the opticaldielectric constant (i.e., the dynamic dielectric constant). Inview of the classical relation between dielectric and opticalcoefficients, the dynamic dielectric constant should be closeto the square of the optical refractive index (i.e., 𝜀

𝑟

=

𝑛

2) [11]. It is worthy of note that the dielectric constant isgenerally a function of frequency, why the optical dielectricconstant is used in this case. During the emission process, ifthe electron transit time from themetal-dielectric interface tothe barrier maximum position is shorter than the dielectricrelaxation time, the dielectric does not have enough time tobe polarized; consequently, the dielectric constant at highfrequency or optical dielectric constant should be chosen.This optical dielectric constant is smaller than the staticdielectric constant or the value at low frequency wheremore polarization mechanisms can contribute to the totalpolarization [10].

Figure 3 shows the current density-electric field (𝐽-𝐸)characteristics of Al/CeO

2

/𝑝-Si MIS capacitors biased in

Schottky emission simulation

0 0.5 1 1.5 2 2.5 3 3.5 4Electric field (MV/cm)

1 × 103

1 × 101

1 × 10−1

1 × 10−3

1 × 10−5

1 × 10−7

1 × 10−9

1 × 10−11

Curr

ent d

ensit

y (A

/cm

2) n = 2.33

ΦB(Al/CeO2) = 0.62 ± 0.01eV

300 K350 K400 K

450 K500 K

Figure 3: Experimental 𝐽-𝐸 curves (symbols) and simulation ofSchottky emission (lines) for the Al/CeO

2

/𝑝-Si MIS capacitor.

accumulation mode at temperatures ranging from 300K to500K. According to the optical characterization of CeO

2

films, the refractive index (𝑛) at 632.8 nm is about 2.33.Therefore, the optical dielectric constant of CeO

2

films isabout 5.43. The measured 𝐽-𝐸 curves in [11] and the sim-ulations of Schottky emission are shown in Figure 3. Theexperimental data match the Schottky emission theory verywell at high temperature (≥400K) in a medium electric field(0.5∼1.6MV/cm).The corresponding conduction band offsetbetween Al and CeO

2

is then determined to be 0.62±0.01 eV.Besides the simulation method, Schottky plot is the most

popular way to identify the barrier height at the interface. Fora standard Schottky emission, the plot of log(𝐽/𝑇2) versus𝐸

1/2 should be linear. The barrier height can be obtainedfrom the intercept of Schottky plot. For example, the 𝐽-𝐸data measured at high temperatures (>425K) and in highfields (>1MV/cm) correlate well with the Schottky emissiontheory under the gate injection in an Al/ZrO

2

(17.4 nm)/𝑝-SiMIS capacitor [13], as shown in Figure 4. Moreover, the fittedoptical dielectric constant in the plot of standard Schottkyemission is extremely close to the square of optical refractiveindex (i.e., 6.25 = 2.5

2), and the extracted Schottky barrierheight between Al and ZrO

2

is about 0.92 eV.Simmons indicated that if the electronic mean free path

in the insulator is less than the thickness of dielectric film,the equation of standard Schottky emissionmust bemodified[14]. When excited electrons pass through dielectric films,the thermal electrons are affected by traps and interfacestates, which are generated from oxygen vacancies andthermal instability between dielectric and Si, respectively.The thickness of dielectric films also affects the behaviorof Schottky emission. A trap-limited mechanism governsthe carrier transportation in dielectric films [13]. Therefore,


0 0.5 1 1.5 2

Schottky plots

1 1.2 1.4 1.6

Si

Al

Si

1 × 10−5

1 × 10−5

1 × 10−6

1 × 10−6

1 × 10−7

1 × 10−7

1 × 10−8

1 × 10−9

1 × 10−10

1 × 10−11

1 × 10−12

1 × 10−13

1 × 10−14

J/T2

(A/c

m2

K2 )

E1/2 (MV/cm)1/2

E1/2 (MV/cm)1/2

425K

0.92 eV

450K475K

ZrO2

Al/ZrO2 (17.4nm)/p-Si

ZrO2 (RTA 500∘C)

Gate injection

300K: 400

425

450

475K: K:

K: K:

J/T2

(A/c

m2

K2 )𝜀r = 20.2

𝜀r = 17.9

𝜀r = 6.23

𝜀r = 6.30

𝜀r = 6.40

Figure 4: Characteristics of standard Schottky emission at various temperatures. Inset graphs present standard Schottky emission constrainedby 𝑛 = 𝜀

1/2

𝑟

at high temperatures; the band diagram is also shown.

(1) should bemodified into (2) when the electronic mean freepath (𝑙) is less than the dielectric thickness (𝑡

𝑑

):

𝐽 = 𝛼𝑇

3/2

𝐸𝜇(

𝑚

∗

𝑚

0

)

3/2

exp[

−𝑞 (𝜙

𝑏

− √𝑞𝐸/4𝜋𝜀

𝑟

𝜀

0

)

𝐾𝑇

]

(𝑙 < 𝑡

𝑑

) ,

(2)

where 𝛼 = 3×10

−4 A s/cm3K3/2 and 𝜇 is the electronicmobil-ity in the insulator; the other notations are the same as definedbefore. Notably, no clear distinction can be made betweenthe bulk- and electrode-limited conduction mechanisms, asindicated by (2), because each is involved in the conductionprocess [15]. By generating modified Schottky emission plotsfor ZrO

2

films of various thicknesses, the electronic meanfree path in ZrO

2

films can be determined to be between 16.2and 17.4 nm at high temperature (>425K) [13]. In addition,the electronic mobility (𝜇) in ZrO

2

films can be determinedfrom the intercept of the plot of modified Schottky emission.The obtained electronic mobility in ZrO

2

is 12-13 cm2/V-s ina medium field at high temperatures [13].

2.2. Fowler-Nordheim Tunneling. According to the classicalphysics, when the energy of the incident electrons is less thanthe potential barrier, the electrons will be reflected. However,quantummechanismpredicts that the electronwave function

will penetrate through the potential barrier when the barrieris thin enough (<100 A). Hence, the probability of electronsexisting at the other side of the potential barrier is not zerobecause of the tunneling effect. Figure 5 shows the schematicenergy band diagram of Fowler-Nordheim (F-N) tunneling.F-N tunneling occurs when the applied electric field is largeenough so that the electron wave function may penetratethrough the triangular potential barrier into the conductionband of the dielectric. The expression of the F-N tunnelingcurrent is

𝐽 =

𝑞

3

𝐸

2

8𝜋ℎ𝑞𝜙

𝐵

exp[

−8𝜋(2𝑞𝑚

∗

𝑇

)

1/2

3ℎ𝐸

𝜙

3/2

𝐵

] , (3)

where 𝑚

∗

𝑇

is the tunneling effective mass in dielectric; theother notations are the same as defined before. To extract thetunneling current, one canmeasure the current-voltage (𝐼-𝑉)characteristics of the devices at very low temperature. At sucha low temperature, the thermionic emission is suppressed andthe tunneling current is dominant.

For F-N tunneling, a plot of ln(𝐽/𝐸2) versus 1/𝐸 should belinear. Figure 6 shows the 𝐼-𝑉 data measured at 77 K for anHfO2

MIS capacitor biased in the accumulation mode [16].The inset graph indicates that the fitting of F-N tunneling



Fowler-Nordheim tunneling

EC

EF

EV

q𝜙B

Figure 5: Schematic energy band diagram of Fowler-Nordheimtunneling in metal-insulator-semiconductor structure.

theory in high electric fields is very good. The slope of F-N plot can be expressed in (4) [17] and is also a function ofelectron effective mass and barrier height:

slope = −6.83 × 10

7

√(

𝑚

∗

𝑇

𝑚

0

)𝜙

3

𝐵

.(4)

To identify electron effective mass and barrier height, it isuseful to measure the thermionic emission current at hightemperature and the tunneling current at low temperature.Chiu [16] reported that the electron effective mass in HfO

2

and barrier height at theAl/HfO2

interface can be determinedusing the intercept of Schottky plot at high temperatures andthe slope of F-N plot at 77 K, as shown in Figure 7. In general,𝑚

∗

𝑇

= 𝑚

∗ is assumed. Using a mathematical iterationmethodfor a 23.2 nmHfO

2

MIS capacitor, the electron effective massand barrier height at the Al/HfO

2

interface were extractedto be about 0.4𝑚

0

and 0.94 eV, respectively [16]. These twoparameters are self-consistent with the intercept of Schottkyplot and the slope of F-N plot. For the case of a 12.2 nmHfO2

MIS capacitor, the electron effective mass and Al/HfO2

barrier height were determined to be 0.09𝑚

0

and 0.94 eV,respectively.

For HfO2

films, the correlation between electron effectivemass, barrier height, equivalent oxide thickness, and currentconduction mechanism was summarized in [16]. In addition,reports showed that the electron effective mass in SiO

2

tendsto increase with decreasing oxide thickness in ultrathin sili-con dioxide layers [18, 19]. Both parabolic and nonparabolicenergy-momentum dispersion were used for measuring thetunneling effective mass in SiO

2

. The latter has been found towork better in many cases [18–21]. As the SiO

2

thickness isincreased beyond 4 nm, the nonparabolic tunneling effectivemass ultimately converges to electron effective mass in SiO

2

[19]. Also, the electron effectivemass in SiO2

is approximatelyconstant over the voltage bias range studied [19, 20]. Based on

these results, the tunneling effective mass can be assumed tobe equal to the electron effectivemass for dielectric films withwide enough thickness.

2.3. Direct Tunneling. There are two main gate currentconduction mechanisms in SiO

2

films. If the voltage acrossthe SiO

2

is large enough, the electrons see a triangular barrierand the gate current is due to F-N tunneling. On the otherhand, if the voltage across the SiO

2

is small, the electrons seethe full oxide thickness and the gate current is due to directingtunneling. The driving oxide voltage between the two mech-anisms is approximately 3.1 V for the SiO

2

-Si interface. ForSiO2

thicknesses of 4-5 nm and above, F-N tunneling domi-nates and for SiO

2

thickness less than about 3.5 nm, directingtunneling becomes dominant. The schematic energy banddiagramof direct tunneling is shown in Figure 8. Based on theresult of Lee and Hu on a polysilicon-SiO

2

-silicon structure,the expression of the direct tunneling current density is [12]

𝐽 =

𝑞

2

8𝜋ℎ𝜀𝜙

𝐵

𝐶 (𝑉

𝐺

, 𝑉, 𝑡, 𝜙

𝐵

)

× exp{−

8𝜋√2𝑚

∗

(𝑞𝜙

𝐵

)

3/2

3ℎ𝑞 |𝐸|

⋅ [1 − (1 −

|𝑉|

𝜙

𝐵

)

3/2

]} ,

(5)

where 𝑡 is the thickness of the dielectric, 𝑉 is the voltageacross the dielectric, and the other notations are the sameas defined before.The tunneling current components includeelectron tunneling from the conduction band (ECB), electrontunneling from the valence band (EVB), and hole tunnelingform the valence band (HVB).The correction function𝐶 canbe expressed as [12]

𝐶 (𝑉

𝐺

, 𝑉, 𝑡, 𝜙

𝐵

)

= exp [

20

𝜙

𝐵

(

|𝑉| − 𝜙

𝐵

𝜙

0

+ 1)

𝛼

⋅ (1 −

|𝑉|

𝜙

𝐵

)] ⋅

𝑉

𝐺

𝑡

⋅ 𝑁,

(6)

where 𝛼 is a fitting parameter depending on the tunnelingprocess, 𝑞𝜙

0

is the Si/SiO2

barrier height (e.g., 3.1 eV forelectron and 4.5 eV for hole), and 𝑞𝜙

𝐵

is the actual barrierheight (e.g., 3.1 eV for ECB, 4.2 eV for EVB, and 4.5 eV forHVB).𝑁 is an auxiliary functionwhich is used as an indicatorof carrier population for ECB andHVB cases or transmissionprobability for EVB case. For ECB and HVB tunnelingprocesses in both the inversion or accumulation regimes, 𝑁is given by

𝑁 =

𝜀

𝑡

{𝑛invV𝑇 ⋅ ln [1 + exp(

𝑉

𝐺,eff − 𝑉TH

𝑛invV𝑇)]

+ V𝑇

⋅ ln [1 + exp(−

𝑉

𝐺

− 𝑉FBV𝑇

)]} ,

(7)

where V𝑇

(= 𝑘𝑇/𝑞) is the thermal voltage, 𝑉TH is thresholdvoltage, 𝑉FB is flatband voltage, and 𝑉

𝐺,eff = 𝑉

𝐺

− 𝑉poly isthe effective gate voltage after accounting for the voltage dropacross the polysilicon depletion region. The rate of increase


Electric field (V/cm)

FN plot

0.24 0.28 0.32 0.36 0.41/E (cm/MV)

ln(J

/E2)

−30

−32

−34

−36

−38

−40

−42

m∗ = m00.4

ΦB = 0.94 eV

Al/HfO2/p-Si

RTA @ 700∘C

T = 77K

EOT = 6nm

0 1 × 106 2 × 106 3 × 106 4 × 106

1 × 100

1 × 10−1

1 × 10−2

1 × 10−3

1 × 10−4

1 × 10−5

1 × 10−6

Curr

ent d

ensit

y (A

/cm

2 )

Figure 6: Characteristics of 𝐽-𝐸 plots for HfO2

MIS capacitor at 77K. The inset graph presents the Fowler-Nordheim tunneling.

0.75

0.8

0.85

0.9

0.95

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Barr

ier h

eigh

t (eV

) SC

Al

IL

Al

p-Sip-Si

ILIL

FN

Effective mass (m∗

m∗

m∗

)

ΦB

ΦB

HfO2HfO2

( m00.4 , 0.94 eV)

Figure 7: The extracted relationships between electron effectivemass and Al/HfO

2

barrier height from the intercept of the Schottkyplot at a high temperature (465K) and the slope of the F-N plotat a low temperature (77K). The band diagrams for the Schottkyemission and F-N tunneling are also shown.

of the subthreshold carrier density with 𝑉

𝐺

is indicated bythe swing parameter 𝑛inv, where 𝑛inv = 𝑆/V

𝑇

and 𝑆 are thesubthreshold swing which is positive for NMOS and negativefor PMOS. For EVB tunneling process,𝑁 can be written as

𝑁 =

𝜀

𝑡

⋅ {3V𝑇

⋅ ln[1 + exp(

𝑞 |𝑉| − 𝐸

𝑔

3𝑘𝑇

)]} . (8)


Direct tunneling

q𝜙B

EC

EFEV

Figure 8: Schematic energy band diagram of direct tunneling inmetal-insulator-semiconductor structure.

Equation (5) can be simplifiedusing a binomial expansionand neglecting higher order terms, which leads to

𝐽 ∼ exp{−

8𝜋(𝑞𝜙

𝐵

)

3/2

√2𝑚eff

3ℎ𝑞 |𝐸|

[

3 |𝑉|

2𝜙

𝐵

]}

∼ exp{−

8𝜋√2𝑞

3ℎ

(𝑚eff𝜙𝐵

)

1/2

𝜅 ⋅ 𝑡ox,eq} .

(9)

Yeo et al. indicated the scaling limits of alternative gatedielectrics based on their direct tunneling characteristics and


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Inversion bias

RPN-SiON

ox,eqEquivalent SiO2 thickness t (nm)

103

100

10−3

10−6

Gat

e cur

rent

den

sityJ G

(A/c

m2 )

|VG | = 1.0V

SiO2

JVD-Si3 N

4

HfO

2

Al2 O

3

La2 O3

0 5 10 15 200

5

10

15

20

RPN-SiON

Scal

ing

limit

EOT

(A)

Figure of merit f = (meffΦB)1/2 𝜅·

SiO2

HfO2

La2O3

Al2O3JVD-Si3N4

JG,limit = 1A/cm2

Figure 9: Scaling limit for several gate dielectrics when 𝑉

𝑑𝑑

= 1.0V and 𝐽

𝐺,limit = 1A/cm2 [12].

gate leakage requirements for future CMOS technology [22].The tunneling leakage current for a given EOT (equivalentoxide thickness, 𝑡ox,eq) is not only dependent on the 𝜅 value ofa gate dielectric but also tunneling barrier height (𝑞𝜙

𝐵

= Φ

𝐵

)and tunneling effective mass (𝑚eff). They introduced a figureof merit to compare the relative advantages of gate dielectriccandidates.The figure of merit is given by 𝑓 = (𝑚effΦ𝐵)

1/2

⋅ 𝜅.Figure 9 shows the scaling limit for several gate dielectricswhen 𝑉

𝐺

or 𝑉

𝑑𝑑

is specified to be 1.0 V and the maximumtolerable gate current density 𝐽

𝐺,limit is 1 A/cm2. A dielectric

with a larger figure of merit possesses a lower scaling limitEOT, as shown in the inset of Figure 9.The EOT scaling limitsof La2

O3

and HfO2

are about 4 A and 9 A, respectively.

2.4. Thermionic-Field Emission. Thermionic-field emissiontakes place intermediately between field emission and Schot-tky emission. In this condition, the tunneling electronsshould have the energy between the Fermi level of metal andthe conduction band edge of dielectric.The schematic energyband diagram of thermionic-field emission is shown inFigure 10(a). The difference between thermionic emission,thermionic-field emission, and field emission is shown in Fig-ure 10(b). The current density due to thermionic-field emis-sion can be roughly expressed as [8]

𝐽 =

𝑞

2

√𝑚(𝑘𝑇)

1/2

𝐸

8ℎ

2

𝜋

5/2

exp(−

𝑞𝜙

𝐵

𝑘𝑇

) exp[

ℎ

2

𝑞

2

𝐸

2

24𝑚(𝑘𝑇)

3

] . (10)

3. Bulk-Limited Conduction Mechanisms

The bulk-limited conduction mechanisms depend on theelectrical properties of the dielectric itself. The most impor-tant parameter in this type of conduction mechanism is thetrap energy level in the dielectric films. The bulk-limitedconduction mechanisms include (1) Poole-Frenkel emission,(2) hopping conduction, (3) ohmic conduction, (4) space-charge-limited conduction, (5) ionic conduction, and (6)

grain-boundary-limited conduction. Based on the bulk-limited conduction mechanisms, some important electricalproperties in the dielectric films can be extracted, includingthe trap energy level, the trap spacing, the trap density, theelectronic drift mobility, and the dielectric relaxation time,the density of states in the conduction band.

3.1. Poole-Frenkel Emission. Poole-Frenkel (P-F) emissioninvolves amechanismwhich is very similar to Schottky emis-sion; namely, the thermal excitation of electrons may emitfrom traps into the conduction band of the dielectric. There-fore, P-F emission is sometimes called the internal Schottkyemission. Considering an electron in a trapping center, theCoulomb potential energy of the electron can be reducedby an applied electric field across the dielectric film. Thereduction in potential energy may increase the probabilityof an electron being thermally excited out of the trap intothe conduction band of the dielectric. The schematic energyband diagram of P-F emission is shown in Figure 11. For



q𝜙B

EC

EFEV

Thermionic-field emission

(a)


Thermionic emission

Field emission

q𝜙B

EC

EFEV

Thermionic-field emission

(b)

Figure 10: (a) Schematic energy band diagram of thermionic-field emission in metal-insulator-semiconductor structure. (b) Comparison ofthermionic-field emission, thermionic emission, and field emission.


Poole-Frenkel emission

EC

EFEV

q𝜙T

Figure 11: Schematic energy band diagram of Poole-Frenkel emis-sion in metal-insulator-semiconductor structure.

a Coulombic attraction potential between electrons and traps,the current density due to the P-F emission is

𝐽 = 𝑞𝜇𝑁

𝐶

𝐸 exp[

−𝑞 (𝜙

𝑇

− √𝑞𝐸/𝜋𝜀

𝑖

𝜀

0

)

𝑘𝑇

] , (11)

where 𝜇 is the electronic drift mobility, 𝑁𝐶

is the density ofstates in the conduction band, 𝑞𝜙

𝑇

(=Φ𝑇

) is the trap energylevel, and the other notations are the same as defined before.Since P-F emission is owing to the thermal activation underan electric field, this conductionmechanism is often observedat high temperature and high electric field. Chiu et al.reported that the dominant conduction mechanism throughPr2

O3

is the P-F emission at high temperature and highelectric field [23]. Figure 12 shows the measured 𝐽-𝐸 data and

0 0.5 1 1.5 2

Simulations of Poole-Frenkel emission

E (MV/cm)

10−2

10−3

10−4

10−5

10−6

10−7

10−8

10−9

J(A

/cm

2 )

400K, n = 1.6

K, n = 1.7

K, n = 1.8

K, n = 1.8

K, n = 1.8375

350

325

300

Figure 12: Characteristics of 𝐽-𝐸 plot and simulation of P-Femission for the laminated Pr

2

O3

/SiON MIS capacitors at highfields.

the modeled P-F emission curves for the laminated Pr2

O3

/SiON MIS capacitors. The experimental data match themodeled curves very well in high electric fields (>1MV/cm)from 300K to 400K. Therefore, the dominant conductionmechanism is the P-F emission in high electric fields attemperatures ranging from 300K to 400K. In addition, thetrap energy level in Pr

2

O3

can be extracted by using theArrhenius plot, as shown in Figure 13. The determined trapenergy level in Pr

2

O3

is about 0.56 ± 0.01 eV. Besides thesimulation method, P-F plot is also a popular way to identifythe trap energy level in dielectric films. For the P-F emission,a plot of ln(𝐽/𝐸) versus 𝐸1/2 is linear. The trap barrier heightcan be extracted from the intercept of P-F plot. Figure 14shows the P-F plot for a 17.4 nm ZrO

2

MIS capacitor in


2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4

Arrhenius plot

1000/T

ln(J/E

)

−22

−23

−24

−25

−26

E = 1.65 MV/cmE = 1.63MV/cmE = 1.60MV/cmE = 1.58MV/cm

E = 1.56MV/cmE = 1.54MV/cmE = 1.52MV/cm

ΦT = 0.56 ± 0.02 eV

Figure 13: Arrhenius plot of the P-F emission for the laminatedPr2

O3

/SiONMIS capacitors at high fields.

the accumulationmode at high temperatures (>425K) and inlow electric fields (<0.6MV/cm) [13]. Under the constraint of𝑛 = 𝜀

1/2

𝑟

, the trap barrier height in ZrO2

films was extractedto be about 1.1 eV according to the intercepts of the fitted linesin the P-F plot.

Aside from the intercept of P-F plot for the trap barrier,the slope of P-F plot is an important factor for determiningthe optical dielectric constant (𝜀

𝑟

) in dielectric films. For thecase of Pr

2

O3

MIS capacitors [23], the fitted 𝜀

𝑟

is 3 ± 0.3

and the fitted refractive index is 1.7 ± 0.1. The refractiveindex obtained by the electrical method is fairly close tothe one obtained by the optical method (𝑛 = 1.7-1.8). Thisimplies that during the emission process, the electron transittime from the trap site to the barrier maximum position isshorter than the dielectric relaxation time. Consequently, thedielectric film does not have enough time to be polarizedduring the emission process and the optical dielectric con-stant should be used in this case. Besides the trap energylevel and the optical dielectric constant, the electronic driftmobility in dielectric films can also be extracted by theanalyses of P-F emission. One can find the result of electronmobility in Pr

2

O3

films in [23]. Note that Angle and Talleyproposed that the free electrons in P-F emission are emittedfrom the donor centers [24]. If there are other defect statessuch as trap states or acceptor centers in dielectric films, thenumber of emitted free electrons from the donor center willdecrease. The existence of the trapping centers may influencethe relation of current density and electric field. Accordingly,the P-F emission is dependent on the concentration of trapcenters (𝑁

𝑡

) and donor centers (𝑁𝑑

). When 𝑁

𝑡

< 𝑁

𝑑

, theconduction mechanism is called the normal P-F emission[24]. When 𝑁

𝑡

≅ 𝑁

𝑑

, the conduction mechanism is calledthemodified P-F emission or anomalous Poole-Frenkel effect[24]. In such a case, the slope of P-F plot is reduced by half andequals the slope of Schottky plot. To distinguish the Schottkyemission and modified P-F emission, the effect of different

electrode materials on the conduction characteristics is avaluable means.

3.2. Hopping Conduction. Hopping conduction is due to thetunneling effect of trapped electrons “hopping” from onetrap site to another in dielectric films. Figure 15 shows theschematic energy band diagram of hopping conduction. Theexpression of hopping conduction is [5, 10, 23]

𝐽 = 𝑞𝑎𝑛V exp [

𝑞𝑎𝐸

𝑘𝑇

−

𝐸

𝑎

𝑘𝑇

] , (12)

where 𝑎 is the mean hopping distance (i.e., the mean spacingbetween trap sites), 𝑛 is the electron concentration in theconduction band of the dielectric, ] is the frequency ofthermal vibration of electrons at trap sites, and 𝐸

𝑎

is theactivation energy, namely, the energy level from the trap statesto the bottom of conduction band (𝐸

𝐶

); the other termsare as defined above. The P-F emission corresponds to thethermionic effect and the hopping conduction correspondsto the tunnel effect. In P-F emission, the carriers can over-come the trap barrier through the thermionic mechanism.However, in hopping conduction, the carrier energy is lowerthan the maximum energy of the potential barrier betweentwo trapping sites. In such case, the carriers can still transitusing the tunnel mechanism.

Chiu et al. [23] reported that the experimental 𝐽-𝐸 datamatch the simulated hopping conduction curves very wellfrom 300K to 400K in low electric fields (<0.6MV/cm)in a Pr

2

O3

MIS structure, as shown in Figure 16. Fromthe simulations of hopping conduction, the mean hoppingdistance in the Pr

2

O3

films was determined to be about 1.5 ±

0.1 nm. Using the slopes of Arrhenius plot in low fields, theactivation energy was determined to be about 50 ± 1meV,as shown in Figure 17. Furthermore, according to (12), thetrap spacing can be extracted by the slope of linear part oflog(𝐽) versus 𝐸. Figure 18 shows the 𝐽-𝐸 characteristics attemperatures ranging from 300K to 425K for a Pt/MgO/Ptmemory device in high resistance state [25]. Simulationresults show that the measured data match the theory ofhopping conduction very well when the electric field is largerthan about 0.25MV/cm. Based on the slope in Figure 18, thetrap spacing in MgO can be determined to be about 1.0 nm[25]. In Figure 18,we can observe an interesting characteristic,that is, a lower current density is observed at a highertemperature.This finding is far different from the normal 𝐽-𝐸characteristics in dielectric films in which the higher currentdensity can be achieved in a higher temperature. In the case ofPt/MgO/Pt, a simulation work can be adopted by varying thetrapping level [25]. Therefore, the temperature dependenceof the trap energy levels in MgO can be obtained, as shownin Figure 19. The trap energy level increases with increasingtemperature.This result indicates that the defects with deeperlevel are activated by the elevated temperature. Hence, thedeeper trap level activated at higher temperature leads to theexponential decrease in current density.

3.3. Ohmic Conduction. Ohmic conduction is caused by themovement of mobile electrons in the conduction band and


0 0.5 1 1.5 2

Poole-Frenkel emissions

0.2 0.4 0.6 0.8 1

1 × 10−6

1 × 10−7

1 × 10−8

1 × 10−9

1 × 10−10

1 × 10−11

1 × 10−12

1 × 10−13

−8

−9

−10

−11

E1/2 (MV/cm)1/2

E1/2 (MV/cm)1/2

Al/ZrO2 (17.4nm)/p-Si

Log(J/E)

300K: 350

425

450

475K: K:

K: K:

K, n = 2.50

K, n = 2.52

425

450

K, n = 2.49475

Φt = 1.1 eV

𝜀r < 5

𝜀r < 5

𝜀r = 6.26

𝜀r = 6.36

𝜀r = 6.19

J/E

(A/c

m·V)

Figure 14: Characteristics of P-F emission at various temperatures. Inset graph presents Poole-Frenkel emission constrained by 𝑛 = 𝜀

1/2

𝑟

athigh temperatures.

Metal

Insulator

Semiconductor

Hopping conduction

EC

EFEV

q𝜙T

a

Figure 15: Energy band diagram of hopping conduction in metal-insulator-semiconductor structure.

holes in the valence band. In this conduction mechanism,a linear relationship exists between the current density andthe electric field. Figure 20 shows a schematic energy banddiagram of the Ohmic conduction due to electrons. Althoughthe energy band gap of dielectrics is by definition large, therewill still be a small number of carriers that may be generateddue to the thermal excitation. For example, the electrons maybe excited to the conduction band, either from the valence

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7E (MV/cm)

400K375K350K

325K300K

3.0 × 10−7

2.5 × 10−7

2.0 × 10−7

1.5 × 10−7

1.0 × 10−7

5.0 × 10−8

1.0 × 10−9

a = 1.50 ± 0.1nm

Al/Pr2O3 (23 nm)/SiON (1nm)/p-SiRTA at 600∘C in N2 for 30min

J(A

/cm

2)

Figure 16: 𝐽-𝐸 characteristics and simulation of hopping conduc-tion for a laminated Pr

2

O3

/SiON MIS capacitors at low electricfields.

band or from the impurity level. The carrier numbers will bevery small but they are not zero.The current density of ohmicconduction can be expressed as

𝐽 = 𝜎𝐸 = 𝑛𝑞𝜇𝐸, 𝑛 = 𝑁

𝐶

exp[

− (𝐸

𝐶

− 𝐸

𝐹

)

𝑘𝑇

] , (13)

where 𝜎 is electrical conductivity, 𝑛 is the number of electronsin the conduction band, 𝜇 is electron mobility, and 𝑁

𝐶

is


2.4 2.6 2.8 3 3.2 3.4

−15

−15.5

−16

−17

−18

−18.5

−17.5

−16.5

lnJ

= 50 ± 1meV

5nma = 1.

1000/T

E =E = 0.40MV/cmE = 0.38MV/cmE = 0.35MV/cmE = 0.33MV/cm

E = 0.31MV/cmE = 0.29MV/cmE = 0.27MV/cm

Ea

Figure 17: Arrhenius plot of the hopping conduction at low fieldsfor the laminated Pr

2

O3

/SiONMIS capacitors.

0 300 600 900 1200 1500

Hopping conduction

Pt/MgO/Pt MIM diode

Electric field (kV/cm)

Curr

ent d

ensit

y (A

/cm

2 )

10−1

10−3

10−5

10−7

10−9

300K, Φt = 0.70 eV325K, Φt = 0.80 eV350K, Φt = 0.90 eV

375K, Φt = 1.00 eV400K, Φt = 1.15 eV425K, Φt = 1.29 eV

Figure 18: Experimental data and simulation curves of hoppingconduction in high resistance state in Pt/MgO/Pt memory device.

the effective density of states of the conduction band; theother terms are as defined above.

Because the energy band gap of a dielectric is very large,we can assume that the Fermi level𝐸

𝐹

is close to themiddle ofthe energy band gap; that is, 𝐸

𝐶

− 𝐸

𝐹

∼𝐸

𝑔

/2. In this case, theohmic conduction current is 𝐽 = 𝑞𝜇𝐸𝑁

𝐶

exp(−𝐸𝑔

/2𝑘𝑇). Themagnitude of this current is very small. This current mecha-nism may be observed if there is no significant contributionfrom other conduction mechanisms of current transport indielectrics [10]. The ohmic conduction current due to mobileelectrons in the conduction band or similarly holes in thevalence band is linearly dependent on the electric field. Thiscurrent usually may be observed at very low voltage in thecurrent-voltage (𝐼-𝑉) characteristics of the dielectric films.

300 320 340 360 380 400 420 4400.6

0.8

1.0

1.2

1.4

1.6

3.1 eV

Pt

Pt

MgO

Pt/MgO/Pt Hopping conduction

Temperature (K)

Φt

Trap

ener

gy le

vel,Φ

t(e

V)

a3.1 eV

Hopping distance (a) = 1nm

Figure 19: Temperature dependence of the trap energy levels in highresistance state. Inset graph shows the band diagram of hoppingconduction in Pt/MgO/Pt memory cells.


Ohmic conduction

q𝜙B

EC

EFEV

Figure 20: Energy band diagram of ohmic conduction in metal-insulator-semiconductor structure.

Recently, the resistance switching behaviors in dielectricfilms have been extensively studied [26–29]. The ohmic con-duction can be often observed in low resistance state in theresistance switchingmemory devices. Figure 21 shows the 𝐽-𝐸curves in a double-logarithmic plot in low resistance state inPt/ZnO/Pt memory devices [29].The linear relation betweencurrent density and electric field is observed, which matchesthe ohmic conduction very well because the slopes are veryclose to 1. In Figure 21, the current density increases withincreasing temperature. Hence, the linear relation betweenelectrical conductivity (𝜎) and inverse temperature can bederived from Figure 21, as shown in Figure 22. Using theArrhenius plot, the Fermi level (𝐸

𝐹

) of ZnO is determinedto be about 0.4 eV below the conduction band edge in ZnO(𝐸𝐶

), as shown in the inset of Figure 22. Consequently, theproduct of 𝜇 and𝑁

𝐶

at each temperature can be extracted by


Ohmic plot

Electric field (MV/cm)10−3

10−3

10−2

10−2

10−1

10−1

100

101

100

Pt/ZnO (25nm)/Pt

Curr

ent d

ensit

y (A

/cm

2 )

150∘C125∘C100∘C

75∘C50∘C25∘C

Figure 21: Linear relation between current density and electric fieldat temperature ranging from 25 to 150∘C in low resistance state inPt/ZnO/Pt memory devices.

2.2 2.4 2.6 2.8 3.0 3.2 3.4

10−3

10−4

10−5

10−6

10−7

10−8

Con

duct

ivity

𝜎(S

/cm

)

Ea

ECEF

EV

𝜎 = 𝜎0 exp(−Ea/kT)

Ea = 0.40 eV

1000/T (K−1)

Figure 22: Temperature dependence of the electrical conductivity inlow resistance state in Pt/ZnO/Pt memory devices. The inset graphshows the location of Fermi level in ohmic conduction.

the combination of 𝐸𝐹

and 𝜎. In addition, 𝑁𝐶

is a functionof temperature, which is proportional to 𝛽𝑇

3/2, where 𝛽 is aconstant [30].The effective density of states of the conductionband (𝑁

𝐶

) in ZnO at room temperature is 4.8 × 10

18 cm−3[31].Therefore, the temperature-dependent electronmobility(𝜇) and effective density of states of the conduction band(𝑁𝐶

) in ZnO can be obtained, as shown in Figure 23. Atroom temperature the electronmobility in ZnOfilms is about4.6 cm2/V-s.

3.4. Space-Charge-Limited Conduction. The mechanism ofspace-charge-limited conduction (SCLC) is similar to thetransport conduction of electrons in a vacuum diode.

The thermionic cathode of a vacuumdiode can emit electronswith a Maxwellian distribution of initial velocities (V). Thecorresponding charge distribution can be written by thePoisson’s equation:

𝜕

2

𝑉

𝜕𝑥

2

= −

𝜌 (𝑥)

𝜀

0

. (14)

Moreover, in the steady state, with the condition V(𝑥) =

[2𝑞𝑉(𝑥)/𝑚]

1/2, the continuity equation is

𝑗

𝑥

= 𝑞𝑛 (𝑥) V (𝑥) . (15)

The current density-voltage (𝐽-𝑉) characteristic of a vacuumdiode is governed by the Child’s law:

𝐽 =

4𝜀

0

9

(

2𝑒

𝑚

)

1/2

𝑉

3/2

𝑑

2

.(16)

In a solid material, the space-charge-limited current iscaused by the injection of electrons at an ohmic contact. Thecontinuity equation should include the diffusion componentand can be written as

𝑗

𝑥

= 𝑒𝑛 (𝑥) V (𝑥) + 𝑒𝐷

𝑑𝑛

𝑑𝑥

. (17)

A typical 𝐽-𝑉 characteristic plotted in a log-log curve forspace-charge-limited current is shown in Figure 24. The 𝐽-𝑉characteristic in the log 𝐽-log𝑉 plane is bounded by the threelimited curves, namely, ohm’s law (𝐽Ohm ∝ 𝑉), traps-filledlimit (TFL) current (𝐽TFL ∝ 𝑉

2), and Child’s law (𝐽Child ∝

𝑉

2). 𝑉tr and 𝑉TFL are the transition voltage at the departurefrom ohm’s law and TFL curve, respectively:

𝐽Ohm = 𝑞𝑛

0

𝜇

𝑉

𝑑

, (18)

𝐽TFL =

9

8

𝜇𝜀𝜃

𝑉

2

𝑑

3

,(19)

𝐽Child =

9

8

𝜇𝜀

𝑉

2

𝑑

3

,(20)

𝑉tr =8

9

×

𝑞𝑛

0

𝑑

2

𝜀𝜃

,(21)

𝜃 =

𝑁

𝐶

𝑔

𝑛

𝑁

𝑡

exp (

𝐸

𝑡

− 𝐸

𝐶

𝑘𝑇

) , (22)

𝑉TFL =

𝑞𝑁

𝑡

𝑑

2

2𝜀

,(23)

𝜏

𝑐

=

𝑑

2

𝜇𝜃𝑉tr, (24)

𝜏

𝑑

=

𝜀

𝑞𝑛𝜇𝜃

, (25)

where 𝑛

𝑜

is the concentration of the free charge carriers inthermal equilibrium, 𝑉 is the applied voltage, 𝑑 is the thick-ness of thin films, 𝜀 is the static dielectric constant, 𝜃 is the


300 320 340 360 380 400 420 440Temperature (K)

Ohmic conduction

Pt/ZnO/Pt

8

10

6

4

2

×1018N

C(c

m−3)

2

4

6

810

EC − EF = 0.40 eV

Mob

ility

(cm

2 /V·s)

Figure 23: Temperature dependence of the electron mobility andthe effective density of states of the conduction band in lowresistance state in Pt/ZnO/Pt memory devices.

Square region

Linear region

Square region

(J ∼ V)

(J ∼ V2)

(J ∼ V2)

Vtr VTFL Log V

Log J

Figure 24: A typical current density-voltage (𝐼-𝑉) characteristic ofspace-charge-limited conduction current. 𝑉tr is transition voltage.𝑉TFL is trap-filled limit voltage.

ratio of the free carrier density to total carrier (free andtrapped) density, 𝑔

𝑛

is the degeneracy of the energy statein the conduction band, 𝐸

𝑡

is the trap energy level, 𝑁𝑡

isthe trap density, 𝑛 is the concentration of the free carrier inthe insulator, and the other terms are as defined above. It isnoticed that (20) is theMott-Gurney relation which indicatesthe space-charge-limited current under the condition ofsingle type of carriers and no traps. If there are traps in thedielectric, the SCL conduction can be described by (19) basedon the assumption of monoenergetical trapping levels in thedielectric [5–7, 10].

At low applied voltages (𝑉<𝑉tr), 𝐽-𝑉 characteristicsfollowed the ohm’s law, which implies that the density ofthermally generated free carriers (𝑛

0

) inside the films is largerthan the injected carriers [32]. This ohmic mode takes place

in the electrically quasineutral state corresponding to thesituationwhen partial trap centers are filled at weak injection.When the transition from the ohmic to the space-charge-limited region, the carrier transit time (𝜏

𝑐

) at 𝑉tr (the min-imum voltage required for the transition) becomes equalto the dielectric relaxation time (𝜏

𝑑

) [2]. The onset of thedeparture from ohm’s law or the onset of the SCL conductiontakes place when the applied voltage reaches the value of𝑉tr. Accordingly, 𝜏𝑐 ≅ 𝜏

𝑑

can be extracted at the transitionpoint𝑉tr. If the applied voltage𝑉 is smaller than𝑉tr, then thecarrier transit time 𝜏

𝑐

is larger than the dielectric relaxationtime 𝜏

𝑑

. This implies that the injected carrier density 𝑛 issmall in comparison with 𝑛

0

and that the injected carrierswill redistribute themselves with a tendency to maintainelectric charge neutrality internally in a time comparable to𝜏

𝑑

. Consequently, the injected carriers have no chance totravel across the insulator. The redistribution of the chargeis known as dielectric relaxation. The ohmic behavior canbe observed only after these space charge carriers becometrapped. Figure 25 shows the schematic diagrams of carrierdistributions in dielectric film in SCLC mechanism underthe conditions of (a) very weak injection (𝑉 < 𝑉tr, 𝑛 < 𝑛

0

,𝜏

𝑐

> 𝜏

𝑑

), (b) dielectric relaxation and carriers redistribution,and (c) weak injection at 𝑉tr (𝑉 = 𝑉tr, 𝑛 = 𝑛

0

, 𝜏𝑐

= 𝜏

𝑑

).In the case of strong injection, the traps are filled up and

a space charge appears. When 𝑉 > 𝑉tr and 𝜏

𝑐

∼ 𝜏

𝑑

or 𝜏𝑐

< 𝜏

𝑑

,the injected excess carriers dominate the thermally generatedcarrier since the injected carrier transit time is too short fortheir charge to be relaxed by the thermally generated carriers.It is noted that for𝑉 < 𝑉tr, 𝜏𝑐 increases with decreasing𝑉 but𝜏

𝑑

remains almost constant, while for 𝑉 > 𝑉tr, 𝜏𝑐 decreaseswith increasing 𝑉 and 𝜏

𝑑

also decreases with increasing 𝑉

since the increase in 𝑉 causes an increase in free carrierdensity in the dielectric. The increase of applied voltage mayincrease the density of free carriers resulting from injectionto such a value that the Fermi level (𝐸

𝐹𝑛

) moves up abovethe electron trapping level (𝐸

𝑡

). The trap-filled limit (TFL)is the condition for the transition from the trapped 𝐽-𝑉characteristics to the trap-free 𝐽-𝑉 characteristics. It can beimagined that after all traps are filled up, the subsequentlyinjected carriers will be free to move in the dielectric films,so that at the subthreshold voltage (𝑉TFL) to set on thistransition, the current will rapidly jump from its low trap-limited value to a high trap-free SCL current. 𝑉TFL is definedas the voltage required to fill the traps or, in other words, asthe voltage at which Fermi level (𝐸

𝐹𝑛

) passes through 𝐸

𝑡

.In the case of very strong injection, all traps are filled and

the conduction becomes the space-charge-limited (Child’slaw).Thus a space charge layer in the dielectric builds up andthe electric field cannot be regarded as constant any longer.While the bias voltage reaches 𝑉TFL in the strong injectionmode, the traps get gradually saturated, which means thatthe Fermi-level gets closer to the bottom of the conductionband. This results in a strong increase of the number of freeelectrons, thus explaining the increase of the current for 𝑉 =

𝑉TFL. For the voltage >𝑉TFL, the current is fully controlledby the space charge, which limits the further injection offree carriers in the dielectric. Square law dependence of


Ohm’s law

(a)

(b)

(c)

Injected carriers (n) Thermally generated free carriers (n0)

Dielectric relaxation within 𝜏d

Carriers redistributed by n0

V < Vtr

V < Vtr

V = Vtr

𝜏c > 𝜏d

𝜏c = 𝜏d

n < n0

n < n0

n = n0

Figure 25: Carriers distribution in dielectric film under carrier weak injection (𝑉 ≤ 𝑉tr) in space-charge-limited conduction. (a) Very weakinjection (𝑉 < 𝑉tr, 𝑛 < 𝑛

0

, 𝜏𝑐

> 𝜏

𝑑

), (b) dielectric relaxation and carriers redistribution, and (c) weak injection at𝑉tr (𝑉 = 𝑉tr, 𝑛 = 𝑛

0

, 𝜏𝑐

= 𝜏

𝑑

).𝑉tr is the transition voltage at the onset of departure from ohm’s law; 𝑛 is the concentration of injected carriers; 𝑛

0

is the concentration ofthermally generated free carriers in dielectric film; 𝜏

𝑐

is the carrier transit time; 𝜏𝑑

is the dielectric relaxation time.

the current (𝐽 ∼ 𝑉

2, Child’s law) is the consequenceof the space charge controlled current. Figure 26 showsthe schematic diagrams of carrier and trap distributions indielectric film in SCLC mechanism under the conditions of(a) strong injection (𝑉 > 𝑉tr, 𝑛 < 𝑛

0

< 𝑁

𝑡

, 𝜏𝑐

< 𝜏

𝑑

), (b)trap-filled-limited conduction (𝑉tr < 𝑉 < 𝑉TFL, 𝑛 < 𝑛

0

< 𝑁

𝑡

,𝐸

𝐹𝑛

< 𝐸

𝑡

): trapped behavior, parts of traps are filled up, and(c) space-charge-limited conduction (𝑉 > 𝑉TFL, 𝑛0 > 𝑁

𝑡

,𝐸

𝐹𝑛

> 𝐸

𝑡

): trap-free behavior, all traps are filled up.A report [32] showed that the dominant conduction

mechanism through the polycrystalline La2

O3

films is space-charge-limited current, as shown in Figure 27.Three differentregions, ohm’s law region, trap-filled-limited region, andChild’s law region are observed in the current density-voltage(𝐽-𝑉) characteristics at room temperature. Based on theSCLC study in [32], some valuable electrical properties inpolycrystalline La

2

O3

were obtained. For example, at roomtemperature, the trap density (𝑁

𝑡

) is about 9.2 × 10

17 cm−3,the trap energy level (𝐸

𝑡

) is about 0.21 eV, the trap capturecross-section (𝜎

𝑡

) is about 1.2 × 10

−21 cm2, the effectivedensity of states in the conduction band (𝑁

𝐶

) is about

5.5 × 10

18 cm−3, the maximum of dielectric relaxation time(𝜏𝑑,max) is about 8.8 × 10

−5 s, and the electron mobility (𝜇)is about 8.2 × 10

−7 cm2/V-s. In addition, some importantelectrical properties in polycrystalline Dy

2

O3

were alsoobtained according to the SCLC mechanism [33]. At 350K,the trap density (𝑁

𝑡

) is about 1.5 × 10

19 cm−3, the trap energylevel (𝐸

𝑡

) is about 0.20 eV, the trap capture cross-section(𝜎𝑡

) is about 3.2 × 10

−21 cm2, the effective density of statesin the conduction band (𝑁

𝐶

) is about 4.5 × 10

21 cm−3,the maximum of dielectric relaxation time (𝜏

𝑑,max) is about8.2 × 10

−6 s, and the electron mobility (𝜇) is about 1.2 ×

10

−6 cm2/V-s.

3.5. Ionic Conduction. Ionic conduction results from themovement of ions under an applied electric field. The move-ment of the ions may come from the existence of latticedefects in the dielectric films. Due to the influence of externalelectric field on defect energy level, the ions may jumpover a potential barrier from one defect site to another.Figure 28(a) shows a schematic energy band diagram of ionic


Injected carriers (n)

Trap-filled-limited

Space-charge-limited

Trapped behavior

Trap-free behavior

(a)

(b)

(c)

V>Vtr

Trap (Nt)

Within 𝜏d

𝜏c < 𝜏d

n < n0 <Nt

n < n0 <Nt

Vtr <V<VTFL

V>VTFL

n0 > Nt

EFn <Et

EFn >Et

Figure 26: Carriers distribution in dielectric film under carrier strong injection (𝑉 > 𝑉tr) in space-charge-limited conduction. (a) Stronginjection (𝑉 > 𝑉tr, 𝑛 < 𝑛

0

< 𝑁

𝑡

, 𝜏𝑐

< 𝜏

𝑑

), (b) trap-filled-limited (𝑉tr < 𝑉 < 𝑉TFL, 𝑛 < 𝑛

0

< 𝑁

𝑡

, 𝐸𝐹𝑛

< 𝐸

𝑡

): parts of traps are filled up(trapped behavior), and (c) space-charge-limited (𝑉 > 𝑉TFL, 𝑛0 > 𝑁

𝑡

, 𝐸𝐹𝑛

> 𝐸

𝑡

): all traps are filled up (trap-free behavior). 𝑉tr and 𝑉TFLare the transition voltage at the departure from ohm’s law and TFL curve, respectively; 𝑛 is the concentration of injected carriers; 𝑛

0

is theconcentration of thermally generated free carriers;𝑁

𝑡

is the trap density; 𝜏𝑐

is the carrier transit time; 𝜏𝑑

is the dielectric relaxation time; 𝐸𝐹𝑛

is the Fermi level; 𝐸𝑡

is the trapping level.

conduction without the applied electric field. Figure 28(b)shows the condition with the applied electric field [10]. Theionic conduction current can be expressed as

𝐽 = 𝐽

0

exp [−(

𝑞𝜙

𝐵

𝑘𝑇

−

𝐸𝑞𝑑

2𝑘𝑇

)] , (26)

where 𝐽

0

is the proportional constant, 𝑞𝜙𝐵

is the potentialbarrier height,𝐸 is the applied electric field, 𝑑 is the spacing oftwo nearby jumping sites, and the other terms are as definedabove. Because the ion mass is large, the mechanism of ionicconduction is usually not important for the applications ofdielectric films in CMOS technology.

3.6. Grain-Boundary-Limited Conduction. In a polycrys-talline dielectric material, the resistivity of the grain bound-aries may be much higher than that of the grains. Therefore,the conduction current could be limited by the electricalproperties of the grain boundaries. In this case, the con-duction mechanism is called the grain-boundary-limitedconduction [10, 34]. The grain boundary will build a grainboundary potential energy barrier (Φ

𝐵

) which is inversely

proportional to the relative dielectric constant of the dielec-tric material. The potential energy barrier can be written as

Φ

𝐵

= 𝑞𝜙

𝐵

=

𝑞

2

𝑛

2

𝑏

2𝜀𝑁

,

(27)

where 𝑛

𝑏

is the grain boundary trap density, 𝜀 is the rela-tive dielectric constant of the dielectric material, and 𝑁 isthe dopant concentration. From (27), it can be seen thatthe dielectric constant can significantly affect the potentialenergy barrier at the grain boundaries.

Figure 29(a) shows the charge distribution across anelectron-trapped grain boundary and the existence of deple-tion regions next to the grain boundary.The potential energybarrier at the grain boundary is shown in Figure 29(b) due tothe charge distribution close to the grain boundary. Figure 30indicates an energy band diagram of the grain-boundary-limited conduction in a metal-insulator-metal MIM diode.

4. Summary

The conduction mechanisms in dielectric films are discussedin detail in this review. There are two types of conduction


SCL conduction mechanism

0.001 0.01 0.1 1 10Applied voltage (V)

Region (I): ohm’s lawRegion (II): trap-filled-limitedRegion (IV): Child’s law

Slope = 1.04(I)

(II)

( III )

( IV )

10+2

10+1

10+0

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

10−9

Curr

ent d

ensit

y (A

/cm

2 )

Slope = 6.77

Slope = 2.01

Slope = 2.21

300K

Al/La2O3/p-SiRTN @ 700∘C

V = 0.3V

VTFL

V = 0.08on

Figure 27: Logarithm of the current density plotted as a function of the logarithm of the applied voltage under negative bias at roomtemperature.

q𝜙B

d

EE

= 0

x

(a)

d

E

x

E →q𝜙B + qEd/2

q𝜙B − qEd/2

(b)

Figure 28: Energy band diagram of ionic conduction (a) without the applied electric field and (b) with the applied electric field. 𝐸 is theapplied electric field, 𝑑 is the spacing between ionic sites, and 𝑞𝜙

𝐵

is the potential barrier height.

−x 0− 0+ x

(a)

Electron

W W

−x 0− 0+ x

q𝜙B

EC

EF

EV

(b)

Figure 29: (a) The charge distribution of an electron trapping grain boundary and (b) the resulting potential energy barrier at the grainboundary. 𝑞𝜙

𝐵

is the potential barrier and𝑊 is the depletion width.


Metal

Metal

Insulator

q𝜙B

qV

Figure 30: Schematic energy band diagram for grain-boundary-limited conduction in metal-insulator-metal structure.

mechanisms in dielectric films, that is, electrode-limitedconductionmechanism and bulk-limited conductionmecha-nism.Themost important parameter in the electrode-limitedconductionmechanisms is the barrier height at the electrode-dielectric interface; meanwhile, the most important param-eter in the bulk-limited conduction mechanisms is the trapenergy level in dielectric films. During the analysis of con-duction mechanisms, the dielectric constant is a key factor.For the case of thermionic emission process, the dielectricconstant should be equal to the optical dielectric constantif the electron transit time is shorter than the dielectricrelaxation time. Based on the electrode-limited conductionmechanisms, the physical properties of the barrier height atthe electrode-dielectric interface and the effective mass ofthe conduction carriers in dielectric films can be obtained.Similarly, based on the bulk-limited conductionmechanisms,the physical properties of the trap level, the trap spacing, thetrap density, the carrier driftmobility, the dielectric relaxationtime, and the density of states of the conduction band indielectric films can be determined. In general, the conductionmechanism in dielectric films may be influenced by thefollowing factors: temperature, electric field, stress condition,device structure (MIS or MIM), electrode material, filmspecies (SiO

2

, high-dielectric-constant material, or ferroelec-tric material), film thickness, deposition method, and so on.Because all the mentioned factors can affect the dielectric-electrode interface and/or the dielectric bulk property, all thefactors are important in the studies of dielectric conductionmechanisms.

Conflict of Interests

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

Acknowledgment

The author would like to thank the National Science Councilof the Republic of China, Taiwan, for supporting this workunder Contract no. NSC 102-2221-E-130-015-MY2.

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