H. S. Leipner: Physics of materials
9. Dielectric properties
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1. Polarization mechanisms
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2. Dielectric constant
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5. Dielectric breakdown
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6. High and low-k dielectrics
3. Piezoelectricity 4. Ferroelectrics
Introduction
Electrical conductivity very small; insulators They are, however, affected by an electric field. Capacitance of a parallel plate capacitor
(vacuum/air)
(with dielectric)
Stored energy
Dielectric + − + − + − − + − + − + −
d
+ + + + + + +
− − − − − −
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Parallel plate capacitor with plates of area A and separation d. When a dielectric material is placed between the plates, the dielectric becomes polarized. 21:01
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Design of ceramic capacitors
Examples of ceramic capacitors. Single-layer ceramic capacitor (disk capacitors) and multilayer ceramic capacitor (stacked ceramic layers).
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[Askeland 1994/Kasap 1997]
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8.1 Polarization mechanisms Effect of an electric field on an isolated atom
E G−
G+ d
The electron orbits of an isolated atom in an electric field are distorted. The centroids of charge are separated by a distance d.
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Dipole moment
Dipole moment of one atom induced by the external electric field Pa = qd
Polarization of a dielectric crystal (dipole moment per unit volume) P = Nqd E + −
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Dielectric crystal as an ensemble of atomic dipoles. As a result of the application of the external electric field, a surface charge appears.
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Polarizability
Macroscopic polarization (surface charge density) P = ε 0 χE (χ = εr – 1, electric susceptibility)
Similarly, the atomic dipole is proportional to the field, Pa = αpEloc Local electric field
If the differences in the local field for surface atoms are not taken into account, we get P = NαpEloc
The polarizability αp can be expressed with the dielectric constant εr, taken into account the relation between the external and local electric field
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Other polarization mechanisms
In ionic crystals, the applied electric field pulls the kations and anions in different directions. The result is an ionic contribution to the polarization, Pi.
di
E
+
Na+
Cl−
Na+
Cl−
Na+
Cl−
Na+
Cl−
Ptot = P + Pi
−
Pi = Nqidi
Na+
Cl−
The effect of applying an electric field to the positions of the ions in a NaCl crystal
Na+ Cl−
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Polar molecular crystals
Materials with molecules of a permanent dipole moment Dipole moment much larger than the induced dipole moment in nonpolar molecules → much higher dielectric constant Normally in a solid the molecules are fixed and cannot follow the electric field, but there are exceptions, e. g. hydrogen chloride.
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Temperature variation of dielectric constant
εr
e− e− e−
e− e− e−
Cl− e−
H+
e−
The HCl molecule and variation of the dielectric constant εr with temperature T for HCl. The abrupt change corresponds to the temperature at which the molecules are no longer able to align themselves with the external electric field. [Turton 2000] hsl 2006 – Physics of materials 9 – Dielectric properties
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Dielectric constant HCl
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8.2 Dielectric constant
E P
Time For a slowly varying field E, the polarization P is expected to vary at the same frequency.
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Polarization vs time
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High-frequency dielectric constant
High frequencies (optical range, ν ≈ 5·1014 Hz) Dielectric constants from electromagnetic theory Speed of light
(vacuum)
Thus,
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(inside the dielectric)
or
εopt = nˆ2
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Dielectric constant at optical frequencies
Material Diamond NaCl LiCl TiO2 Quartz
εs 5.68 5.90 11.95 94.00 3.85
εopt 5.66 2.34 2.78 6.80 2.13
Values of the dielectric constant in a static electric field, εr = εs, and at optical frequencies, εr = εopt [Turton 2000]
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Dielectric constant
Usually, the static dielectric constant is higher. Electrons can quickly follow the alternating electric field. However, the ionic contribution to the polarization becomes much smaller. The ions cannot follow the quick change of the field. At high frequencies, only the induced polarization remains.
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Survey of polarization mechanisms E d
E
E
– – – – –
+ + + + +
E
(c)
E
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(f)
Polarization mechanisms in materials: (a) electronic, (b) ionic, (c) high-frequency dipolar or orientation (present in ferroelectrics), (d) low-frequency dipolar (present in linear dielectrics and glasses), (e) interfacial space charge at electrodes, and (f) interfacial space charge at heterogeneities such as grain boundaries. [Askeland 1994/Hench, West 1990] 14
Polarization
Frequency dependence
Molecular Ionic Electronic 104
108
1012
1016
1020
Frequency (Hz) Frequency dependence of polarization mechanisms
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8.3 Piezoelectricity
In certain dielectric materials, the application of an external stress produces electrical charges on the surface.
F
F
Converse piezoelectric effect: A piezoelectric material becomes strained when placed in an electric field. Δl
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Dielectric effects
When any material undergoes polarization, its ions and electronic clouds are displaced, causing the development of a mechanical strain in the material. This effect is seen in all materials subjected to an electric field and is known as the electrostriction. Piezoelectrics – materials that develop voltage upon the application of a stress and develop strain when an electric field is applied. Pyroelectric – the ability of a material to spontaneously polarize and produce a voltage due to changes in temperature. Ferroelectric – a material that shows spontaneous and reversible dielectric polarization.
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Piezoelectric crystals Inversion center − − 3+ −
3+ −
−
−
When a stress is applied to a crystal structure with three-fold symmetry and zero polarization, the symmetry is altered and the material acquires a non-zero polarization even without an electric field.
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Piezoelectric coefficients
Material
~ d (pC/N)
~g (mV m/N)
Quartz BaTiO3 PZT PbNb2O6 PbTiO3 LiNbO3 LiTaO3
2.3 190 268 to 480 80 47 6 5.7
50 12 12 to 35
[Askeland 1996]
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Design of a spark igniter Example:
Solution:
A PZT spark igniter is made using a disk that has a 5 mm diameter and 20 mm height. Calculate the voltage generated if the ~g coefficient for PZT used is 35 mV m/N. Assume that a compressive force of 10 kN is applied on the circular face. Definition of the g~ coefficient Stress σ = F/A = 5.09 MN/m2
Therefore, the electric field is E = g~σ =1.78·105 V/m U = E d = 3565 V.
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8.4 Ferroelectrics
Material with a finite polarization even in the absence of any external electric field or applied stress Name misleading: nothing to do with iron; properties resembles those of ferromagnetic solids Hysteresis of polarization
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Hysteresis P
Ps
Pr Ec
E
Hysteresis curve as the polarization versus the electric field. Characteristic values are the saturation polarization Ps, the remanent polarization Pr, and the coercive field Ec.
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Microscopic properties of ferroelectrics
Neighboring dipoles are aligned in mutual interaction. Arranged dipoles form domains (compare the magnetic domains in ferromagnets). Characteristic transition temperature, where the alignment of the dipoles is destroyed (ferroelectric Curie temperature) Typically, material with complicated crystal structure of low symmetry Finite polarization even without external electric field All ferroelectrics are also piezoelectric materials Ferroelectric materials have a very high dielectric constant (as large as several thousand). Permanent polarization can be used to store non-volatile digital information.
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Ferroelectric domains
Ferroelectric domains in BaTiO3 shown by imaging with crossed polarizers hsl 2006 – Physics of materials 9 – Dielectric properties
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Alignment of dipoles
The ferroelectric hysteresis loop and the alignment of the dipoles hsl 2006 – Physics of materials 9 – Dielectric properties
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[Askeland 1996]
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Hysteresis loop of barium titanate P P
E
E
Ferroelectric hysteresis loop of single-crystalline BaTiO3 and a BTO polycrystal (right)
[Askeland 1996]
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Dielectric constant vs. temperature
The effect of temperature and grain size on the dielectric constant in barium titanate. Above the Curie temperature, the spontaneous polarization is lost due to a change in crystal structure and barium titanate is in the paraelectric state. The grain size dependence shows that the dielectric constant is a microstructure sensitive property. [Askeland 1996/Moulson, Herbert 1990] hsl 2006 – Physics of materials 9 – Dielectric properties
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Barium titanate structure Ti4+ O2− Ba2+
d
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Compared to the ideal cubic arrangement the positive and negative ions are displaced by a distance d ≈ 0.01 nm.
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Ferroelectric perovskite phase Tetragonal perovskite phase: each titanium atom is bonded to six nearest-neighbor oxygen atoms, but is not exactly in the centre of the octahedron. This offset means that tetragonal BaTiO3 is ferroelectric. The electrical polarization may be reversed by applying an external electric field. Single crystals of BaTiO3 generally contain many domains, corresponding to different directions of Ti off-centering. The net effect of the different domain orientations is to cancel out any macroscopic polarisation. hsl 2006 – Physics of materials 9 – Dielectric properties
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Polarization Example:
Determine the magnitude of the remanent polarization in BaTiO3. The molar volume is 3.8·10−5 m3.
Solution:
1 mol of BaTiO3 consists of NA ions Ba2+, NA ions Ti4+, 3NA ions O2− The total charge associated with 1 mol is ±6NAe The dipole moment for 1 mol is: Pm = 6NAed Polarization is dipole moment per unit volume
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BaTiO3 phase transitions
Different phases of barium titanate as a function of the temperature [Moulson, Herbert 1990] hsl 2006 – Physics of materials 9 – Dielectric properties
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BTO polymorphs
Ps
Ps
Cubic
130 °C
Tetragonal
0 °C
Orthorhombic
Ps
–90 °C
Rhombohedral
The polymorphs of barium titanate at different temperatures
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Ps
Magnitude of polarization
Tc
Magnitude of the polarization of the different polymorphs of barium titanate at different temperatures. Tc is the Curie temperature. [Moulson, Herbert 1990] hsl 2006 – Physics of materials 9 – Dielectric properties
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Dielectric constant
Dielectric constant of barium titanate as a function of the temperature [Moulson, Herbert 1990] hsl 2006 – Physics of materials 9 – Dielectric properties
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Design of a multilayer capacitor Example:
A multi-layer capacitor is to be designed using a BaTiO3-based formulation containing SrTiO3. The dielectric constant of the material is 3000. Calculate the capacitance of a multi-layer capacitor consisting of 100 layers connected in parallel using Ni electrodes. The sides of the capacitor are 10 mm × 5 mm and the thickness of each layer is 10 µm. What is the role of SrTiO3? What processing technique will be used to make these?
Solution:
Capacitance Capacitance of one layer: 13.27 µF Total capacitance of 100 layers connected in parallel: 1327 µF
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8.5 Dielectric breakdown
At a critical (high) field, the dielectric material becomes a conductor. Increasing number of electrons by an avalanche process
Conductivity
e− e− e− e− e− e− e−
Eb
e− e−
e− e−
Scheme of the avalanche process. The accelerated electrons are able to excite other electrons into the conduction band. Each of these electrons then excites another electron, and so on.
Electrical field
Conductivity against the applied electric field. If the field is below the breakdown field then no current flows. hsl 2006 – Physics of materials 9 – Dielectric properties
e− e−
e− e−
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8.6 High and low-k dielectrics
Materials with certain dielectrical properties are important in microelectronics Gate dielectrics with shrinking dimensions require high-k materials (i. e. high dielectric constant εr) Static power consumption of a transistor depends on the leakage current International Technology Roadmap for Semiconductors: effective oxide thickness < 1 nm, gate leakage current < 103 A/cm2 at 100 °C (2007) in MPUs Standard SiO2 cannot support these requirements (high tunneling probability through thin layer)
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Band gap (eV)
Variation of the DK with the band gap
Dielectric constant Variation of the dielectric constant with the band gap in binary oxides [Bersuker:2004]
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Low-k dielectric materials
Increase in the speed of integrated circuits by reducing the size (feature size 1 µm to 90 nm in the last decades) Not all IC components work faster when decreased in size: Interconnections work slower Figure of merit is the product (resistance × capacitance, in units of time) Shrinking cross section of wire → increase in the resistance, reduction in wire distance → increase in the capacitance As a result delay increases
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Interconnections Embedded Cu lines with interline capacitance [Shamiryan:2004]
Cross section of interconnections in damascence technology [www.tecchannel.de] hsl 2006 – Physics of materials 9 – Dielectric properties
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Dual damascence technology 1. Etch of trenches for contacts and interconnections 2. Deposition of insulating film (barrier layer) to prevent the in-diffusion of copper in the semiconductor 3. Evaporation of a thin metal layer where the contacts and the interconnects grow galvanically (dual damascence process) 4. When the trenches are filled the excess copper has to be removed by chemo–mechanical polishing (similar to the intarsia in damascence swords). 5. Deposition of a insulating oxide layer 6. Deposition of up to six additional interconnect layers
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Reduction of the dielectric constant
Decreased polarizability with less polar bonds Decreased density by using of porous material
Low-k materials
Non-Si
Si-based
Polymers Amorphous carbon
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Silica-based
SSQ-based
SiOF SiOCH
HSSQ MSSQ
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Summary
A dielectric placed in a electric field becomes polarized. 3 factors which can contribute to polarization: induced polarization, ionic, and permanent dipole contributions Specific frequency dependence of these contributions Requirement to control the dielectric constant in modern devices
Read about this lecture: R. Turton: The physics of solids. Oxford University Press 2000, chapter 10. References to pictures and data used are given in the file References.pdf
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