JP2003238768A - Epoxy resin composition and semiconductor device - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To obtain an epoxy resin composition which is useful for the operation simplification and process stabilization in the semiconductor manufacturing process without lowering the characteristics of heat resistance, reliability, and the like, which are required for semiconductor device and which can meet a variety of recent requirements for chromatic color, and to provide a semiconductor device sealed with this epoxy resin composition. <P>SOLUTION: The epoxy resin composition is the one which contains an epoxy resin (A), a curing agent (B), a filler (C), a curing accelerator (D) and a colorant (E) and whose color is applicable to the following range, i.e., a*≤-5 or a*≥5 or b*≤-5 or b*≥5, when the color is indicated by the coordinates (a*, b*) of the L*a*b* color specification system. <P>COPYRIGHT: (C)2003,JPO

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an epoxy resin composition which is useful for simplifying work and stabilizing a process in a semiconductor manufacturing process and has an excellent color tone.

[0002]

2. Description of the Related Art Conventionally, as a method of manufacturing a semiconductor device, a thermoplastic resin, a phenol resin, a silicone resin,
A method in which a semiconductor element is set in a mold using an epoxy resin or the like and sealed by a transfer molding method or the like is generally performed. The resin used for such sealing is generally called a sealing material resin. Among them, economy,
From the viewpoint of the balance between productivity and physical properties, resin encapsulation with epoxy resin is most actively performed. The epoxy resin encapsulation method uses a composition obtained by adding a curing agent, a filler, and the like to an epoxy resin, but currently available semiconductor encapsulation epoxy resin compositions are black or achromatic.

Various types of semiconductor devices are manufactured, but in many cases semiconductor devices having the same size but different types of semiconductor elements are manufactured. The semiconductor element sealed with the sealing resin composition is indistinguishable from the outside, and in such a case, a symbol or the like is written on the semiconductor device (particularly the sealing resin portion) by laser marking or ink printing (printing). ) And read it to identify the internal semiconductor element, that is, the semiconductor device. But,
In such a method, there are many printing failures, and since it is necessary to read fine symbols and the like on the surface of the semiconductor device, misrecognition of operators occurs, which complicates the work and lowers work efficiency. . In addition, the semiconductor device which cannot be identified is discarded and the yield is lowered.

Further, as described above, most of the epoxy resin compositions for semiconductor encapsulation are black or achromatic, and therefore, the application such as the design of the semiconductor device itself cannot be applied. Conventionally, an opaque outer box in which the contents cannot be seen has been selected as the outer box for storing semiconductor devices such as home electric appliances and personal computers, but recently, products using a transparent outer box in which the contents can be seen have frequently appeared. There is. However, in an all black or achromatic semiconductor device, even if the semiconductor device housed inside is visible (so-called skeleton), the visual vividness cannot be obtained. That is, the current semiconductor devices and epoxy resin compositions for semiconductor encapsulation also have the problem that they have not been able to meet various demands for chromatic colors that have recently appeared, such as designs for semiconductor devices in products. there were.

[0005]

SUMMARY OF THE INVENTION The object of the present invention has been made in view of the above circumstances, and semiconductors can be manufactured without lowering the characteristics such as heat resistance and reliability required for semiconductor devices. Epoxy resin composition which is useful for simplifying the work in the manufacturing process and stabilizing the process, and can meet a variety of recent demands for chromatic colors, and a semiconductor device encapsulated with the epoxy resin composition The purpose is to provide.

[0006]

The present inventors have arrived at the present invention as a result of extensive studies to achieve the above object.

The present invention mainly has the following constitutions. That is, "an epoxy resin composition containing an epoxy resin (A), a curing agent (B), a filler (C), a curing accelerator (D) and a colorant (E), the epoxy resin composition An epoxy resin composition characterized by having the following range when the product is represented by coordinates (a ^* , b ^* ) in the L ^* a ^* b ^* color system: (Range 1) a ^* ≦ − 5 or a ^* ≧ 5 or b ^* ≦ −5 or b ^* ≧ 5 ”.

[0008]

BEST MODE FOR CARRYING OUT THE INVENTION The present invention will be described in detail below.

The epoxy resin composition of the present invention comprises, as essential components, an epoxy resin (A), a curing agent (B), a filler (C), a curing accelerator (D) and a colorant (E),
When the color of the epoxy resin composition is represented by L ^* a ^* b ^* color coordinate system (a ^* , b ^* ), it is essential that it is within the following range. (Range 1) a ^* ≦ −5 or a ^* ≧ 5 or b ^* ≦ −5 or b ^* ≧ 5 (that is, a range outside the square of one side 10 with the origin at the center) More preferably (range) 2) −20 ≦ a ^* ≦ 20 and b ^* ≧ a ^* and b ^* ≧ −a ^* and 70 ≧ b ^* ≧ 5 (range 3) −20 ≦ a ^* ≦ 20 and b ^* ≦ a ^* and b ^* ≦ −a ^* and −70 ≦ b ^* ≦ −5 (range 4) −20 ≦ b ^* ≦ 20 and b ^* ≦ a ^* and b ^* ≧ −a ^* and 70 ≧ a ^* ≧ 5 (range 5) −20 ≦ It is preferable that b * ≦ 20 and b ^* ≧ a ^* and b ^* ≦ −a ^* and −70 ≦ a ^* ≦ −5.

Where L^*a^*b^*Color in color coordinate system
The measuring method is described in JIS Z8722.
Use a commercially available color measurement device that conforms to the
L ^*Value, a^*Value, b^*The value can be measured.

Since the epoxy resin composition of the present invention exhibits a vivid color, it can meet the recent demand for chromatic colors and can be applied to a semiconductor device or the like installed on a mother board. If such a motherboard is applied to the skeleton outer box, it is not only colorful and gorgeous, but if it is color-coded for each device, it will have the advantage of being able to see the configuration (design) of the electronic circuit from the outside. To be In addition, by color-coding according to the type of the semiconductor device, there is an advantage in the manufacturing process such that the type can be easily distinguished in the manufacturing process.

The L ^* a ^* b ^* color system is JIS Z872.
As defined in No. 9, a three-dimensional color space consisting of a chromaticity diagram based on Cartesian coordinates on the horizontal axis a ^* and vertical axis b ^* , and an L ^* axis that is an index representing the lightness in the direction perpendicular to these axes. It consists of a ^* axis, the origin of the b ^* axis is achromatic, the so-called gray, white by going forward in this state the L ^* axis, a black by going negative direction of the L ^* axis. On the other hand, the positive direction of the a ^* axis is the red region, the negative direction is the green region, the positive direction of the b ^* axis is the yellow region, and the negative direction is the blue region. Saturation increases with distance from the achromatic origin. It is a color system in which the area of each hue is increased and the arrangement is easily understood. Normally, when drawing a chromaticity diagram on a plane, an L ^* axis cannot be seen because it is an orthogonal graph with the a ^* axis as the horizontal axis and the b ^* axis as the vertical axis.

When the a ^* value and the b ^* value of the epoxy resin composition or its cured product do not fall within the range 1, the epoxy resin composition is white, black or achromatic, and the above merit cannot be obtained. However, when it falls within the range 1, the epoxy resin composition becomes chromatic and the above-mentioned merits are obtained. Then, it is preferable to correspond to any one of the range 2, the range 3, the range 4 and the range 5.

The colorant (E) in the present invention has an a ^* value and ab ^* value in the L ^* a ^* b ^* color coordinate system in which the epoxy resin composition or the cured product thereof after the colorant is added. There is no limitation as long as it falls within the above range, and it is possible to use colorants in general, but pigments are particularly preferable. For example, inorganic pigments such as ultramarine blue, cobalt blue, red iron oxide, and titanium oxide, insoluble azo pigments that do not have hydrophilic groups such as carboxyl groups and sulfoxyl groups in their molecules, and azo lakes that are insoluble metal salts that have hydrophilic groups. And the like, phthalocyanine pigments such as copper phthalocyanine containing a tetraazaporphine ring, low chlorinated copper phthalocyanine, highly halogenated phthalocyanine and metal-free phthalocyanine, and anthraquinone pigments such as quinacridone red. Of these, low-chlorinated copper phthalocyanine, quinacridone red, and benzimidazolone are preferably used. The addition amount of these colorants is preferably 0.01 to 1% in the total epoxy resin composition. As a preferable reason, if it is less than 0.01, the coloring effect is weak and the sufficient effect cannot be exhibited.
On the other hand, if it is more than 1%, there is a possibility that characteristics such as fluidity of the epoxy resin composition may be changed. In addition, a preferable form is a hexagonal plate crystal or a crystal in a metastable state such as an orthorhombic system, and these are dispersed into primary particles having an appropriate particle size (0.01 to 0.5 μm). The state is preferred.

Next, other constituent components of the epoxy resin composition of the present invention will be described.

The epoxy resin used for the epoxy resin (A) is not particularly limited as long as it is a compound having two or more epoxy groups in one molecule, and may be a monomer, an oligomer or a polymer in general. For example, bisphenol F type epoxy resin having no alkyl substituent, cresol novolac type epoxy resin, phenol novolac type epoxy resin, 4,4′-bis (2,3-epoxypropoxy) biphenyl, 4,4′-bis (2, 3-epoxypropoxy) -3,3 ', 5,5'-tetramethylbiphenyl, 4,4'-bis (2,3-epoxypropoxy) -3,3', 5,5'-tetraethylbiphenyl,
4,4'-bis (2,3-epoxypropoxy) -3,
Biphenyl type epoxy resin such as 3 ′, 5,5′-tetrabutylbiphenyl, phenol aralkyl type epoxy resin, biphenyl skeleton-containing phenol aralkyl type epoxy resin, naphthalene type epoxy resin, bisphenol A type epoxy resin, triphenol type epoxy resin, Examples thereof include dicyclopentadiene skeleton-containing epoxy resin, triphenylmethane type epoxy resin, and halogenated epoxy resin. Among them, preferred is, for example, 4,4'-bis (2,3-epoxypropoxy)-
Examples thereof include 3,3 ′, 5,5′-tetramethylbiphenyl and bisphenol F type epoxy resin having no alkyl substituent. Also, two or more of these epoxy resins may be used in combination. From the viewpoint of compatibility with the colorant (E), various bisphenol F type epoxy resins and various biphenyl type epoxy resins are particularly preferable.

The curing agent (B) in the present invention is not particularly limited as long as it can be cured by reacting with an epoxy resin, and specific examples thereof include novolac resins such as phenol novolac, cresol novolac and naphthol novolac. , Phenol aralkyl resins, biphenyl skeleton-containing phenol aralkyl resins, dicyclopentadiene skeleton-containing phenol resins, naphthol aralkyl resins, bisphenol compounds such as bisphenol A, maleic anhydride, phthalic anhydride, pyromellitic anhydride and other acid anhydrides and meta Examples thereof include aromatic amines such as phenylenediamine, diaminodiphenylmethane and diaminodiphenyl sulfone. Among them, preferable examples include phenol aralkyl resin and biphenyl skeleton-containing phenol aralkyl resin. These may be used alone or in combination of two or more kinds of curing agents.

The filler (C) in the present invention is preferably an inorganic filler, specifically, amorphous silica, crystalline silica, calcium carbonate, magnesium carbonate, alumina, magnesia, clay, talc, calcium silicate. ,
Examples thereof include metal oxides such as titanium oxide and antimony oxide, asbestos, glass fibers and glass spheres.

The curing accelerator (D) in the present invention is not particularly limited as long as it accelerates the curing reaction between the epoxy resin (A) and the curing agent (B), and for example, 2-methylimidazole, 2,4. -Dimethylimidazole, 2-
Ethyl-4-methylimidazole, 2-phenylimidazole, 2-phenyl-4-methylimidazole, 2-
An imidazole compound such as heptadecyl imidazole,
Triethylamine, benzyldimethylamine, α-methylbenzylmethylamine, 2- (dimethylaminomethyl) phenol, 2,4,6-tris (dimethylaminomethyl) phenol, 1,8-diazabicyclo (5.
4,0) tertiary amine compounds such as undecene-7, organometallic compounds such as zirconium tetramethoxide, zirconium tetrapropoxide, tetrakis (acetylacetonato) zirconium, tri (acetylacetonato) aluminum, and triphenylphosphine, trimethyl Examples thereof include organic phosphine compounds such as phosphine, triethylphosphine, tributylphosphine, tri (p-methylphenyl) phosphine, tri (nonylphenyl) phosphine, and tetraphenylphosphine / tetraphenylborate. These curing accelerators may be used alone or in combination of two or more.

In the present invention, a coupling agent such as a silane coupling agent or a titanium coupling agent can be added. More preferably, the filler is treated with these coupling agents prior to blending with the other components. A silane coupling agent is preferably used as the coupling agent, and as the silane coupling agent,
Generally used are those in which a hydrolyzable group such as an alkoxy group, a halogen atom and an amino group, and an organic group are directly bonded to a silicon atom, and a partial hydrolyzed condensate thereof. As the organic group in the silane coupling agent, a hydrocarbon group substituted with a nitrogen atom, an oxygen atom, a halogen atom, a sulfur atom or the like is used. Specific examples of the silane coupling agent include γ-glycidoxypropyltrimethoxysilane, γ-glycidoxypropylmethyldimethoxysisilane, γ- (2,3-epoxycyclohexyl) propyltrimethoxysilane, γ- (N-phenylamino) propyltrimethoxysilane, γ- (N-phenylamino) propylmethyldimethoxysilane, γ-
(N-methylamino) propyltrimethoxysilane, γ
-(N-methylaminopropyl) methyldimethoxysilane, γ- (N-ethylamino) propyltrimethoxysilane, γ- (N-ethylamino) propylmethyldimethoxysilane, γ-glycidoxypropyltrimethoxysilane, γ- (N-ethylamino) propylmethyltrimethoxysilane, γ-methacryloxypropyltrimethoxysilane, γ-methacryloxypropylmethyldimethoxysilane, γ-mercaptopropyltrimethoxysilane, γ-mercaptopropylmethyldimethoxysilane,
N-β- (aminoethyl) -γ-aminopropyltrimethoxysilane, N-β- (aminoethyl) -γ-aminopropylmethyldimethoxysilane, N-β- (aminoethyl) -γ-aminopropyltriethylsilane, γ-aminopropyltriethoxysilane, γ-aminopropylmethyldiethoxysilane, γ-aminopropyltrimethoxysilane, γ-aminopropylmethyldimethoxysilane, γ- (N, N-dimethylamino) propyltrimethoxysilane and the like. To be

The coupling agent is preferably added in an amount of 0.1 to 2% by weight based on the total amount of the epoxy resin composition from the viewpoint of fluidity and filling property.

In the epoxy resin composition of the present invention, a bromine compound can be blended for the purpose of improving flame retardancy, though it is not an essential component. The bromine compound is not particularly limited as long as it is usually added to the epoxy resin composition as a flame retardant. Specific preferred examples of the bromine compound include:
Examples include brominated epoxy resins such as brominated bisphenol A type epoxy resins and brominated phenol novolac type epoxy resins, brominated polycarbonate resins, brominated polystyrene resins, brominated polyphenylene oxide resins, tetrabromobisphenol A, decabromodiphenyl ether. Among them, brominated epoxy resins such as brominated bisphenol A type epoxy resin and brominated phenol novolac type epoxy resin are
It is particularly preferable in terms of moldability.

Similarly, in the epoxy resin composition of the present invention, an antimony compound can be blended although it is not an essential component. This is usually added as a flame retardant aid to the epoxy resin composition for semiconductor encapsulation, and is not particularly limited, and known ones can be used. Preferable specific examples of the antimony compound include antimony trioxide, antimony tetroxide and antimony pentoxide.

When these flame retardants and flame retardant aids are added,
From the viewpoint of easy disposal of unnecessary substances generated from the epoxy resin composition and reliability of the semiconductor device, it is preferable that the halogen atom and the antimony atom are each 0.2 wt% or less with respect to the epoxy resin composition.

The epoxy resin composition of the present invention may further contain various additives listed below.
Various elastomers such as silicone rubber, olefin copolymer, modified nitrile rubber and modified polybutadiene rubber, various thermoplastic resins such as silicone oil and polyethylene, fluorine-based and silicone-based surfactants, long-chain fatty acids, long-chain fatty acids Various release agents such as metal salts of, long-chain fatty acid esters, long-chain fatty acid amides and paraffin wax, ion trapping agents such as hydrotalcites, and cross-linking agents such as organic peroxides.

The epoxy resin composition of the present invention is preferably produced by melt-kneading the above components. For example, after mixing various raw materials by a known method such as a mixer,
It is produced by melt-kneading using a known kneading method such as a Banbury mixer, a kneader, a roll, a single or twin screw extruder and a cokneader. The resin temperature during melt kneading is usually in the range of 70 to 150 ° C.

The epoxy resin composition of the present invention is melted by heating and kneading, cooled and further pulverized into the shape of powder, the shape of the tablet obtained by tableting the powder, melted by heating and kneading and cooled and solidified in the mold. It can be used in the form of tablets, pellets melted by heating and kneading, extruded, and then cut.

From these shapes, the semiconductor element is sealed and the semiconductor device is manufactured. The epoxy resin composition of the present invention is applied to a member having a semiconductor fixed on a substrate, for example, at 120 to 250 ° C, preferably 150 to 20 ° C.
At a temperature of 0 ° C., molding is performed by a method such as transfer molding, injection molding, or casting method to manufacture a semiconductor device sealed with a cured product of an epoxy resin composition. If necessary, additional heat treatment (for example, 150 to 2
00 ° C., 2 to 16 hours).

Further, the epoxy resin composition of the present invention can be applied to those conventionally molded with an epoxy resin composition such as a semiconductor mounting substrate and a connector.

[0030]

EXAMPLES The present invention will be described in detail below with reference to examples, but the present invention is not limited to the examples given here. In addition,% in an Example shows weight%. [Examples 1 to 15 and Comparative Examples 1 to 4] The components shown in Table 1 were dry-blended with a mixer at the composition ratios (weight ratios) shown in Tables 2 to 3, and then a mixing roll having a roll surface temperature of 90 ° C was used. The mixture was heated and kneaded for 5 minutes, cooled and pulverized to obtain an epoxy resin composition for semiconductor encapsulation.

[0031]

[Table 1]

[0032]

[Chemical 1]

[0033]

[Chemical 2]

[0034]

[Chemical 3]

[0035]

[Chemical 4] <Measurement of L ^* a ^* b ^* colorimetric system value> About the obtained resin composition, 160 pin QFP (outer shape: 28 mm × 28 mm ×
Using a mold for 3.4 mm, frame material: 42 alloy), a package was molded by a low-pressure transfer molding machine at a mold temperature of 175 ° C., a cure time of 1 minute, and a pressure of 130 kgf. As the evaluation chip, a chip having a chip size of 10.4 mm × 10.4 mm × 0.5 mm on which a simulated element having a surface coated with a silicon nitride film was mounted was used.

160 pin QF obtained by the above molding
The a ^* value and b ^* value of P were measured by an SM color computer "SM-5-IS-2B" manufactured by Suga Test Instruments. <Evaluation of Occurrence Rate of Operation Error> 160 obtained by the above molding
The pinQFP was visually observed under a white lamp to examine the degree of color tone. Also, the obtained 160pinQFP was mixed with the black 160pinQFP to obtain the obtained 160pinQFP.
160pinQFP selected by selecting nQFP
The number of cells that could be distinguished from the black ones was checked to determine whether they could be distinguished. <Evaluation of crack resistance (reflow resistance reliability)> 10 pieces of 160 pinQFP obtained by the above molding were subjected to 180 ° C. and 6
After post-curing under the condition of time, this is 85 ℃ / 8
IR of maximum temperature 260 ℃ after humidified at 5% RH for 168 hours
Heat treatment was performed in a reflow furnace. The temperature profile of the reflow furnace is in the range of 150 ° C to 200 ° C for 60 seconds to 1
00 seconds, heating rate from 200 ℃ to 260 ℃ 1.5 ~
The temperature was maintained at 2.5 ° C / sec and the maximum temperature of 255 ° C to 265 ° C for 10 to 20 seconds, and the temperature decreasing rate from 260 ° C to 200 ° C was 1.5 to 2.5 ° C / sec. After the heat treatment, the number of packages obtained favorably, excluding defective packages in which external cracks were generated, was examined. <Evaluation of Void (Moldability)> 16 obtained by the above molding
After molding 10 pieces of 0 pinQFP and after visually observing and cutting the cross section, the pieces were observed with a 20 × microscope to examine the presence or absence of voids. The number of packages obtained well, excluding defective packages in which voids were generated, was determined. The results of evaluation as described above are shown in Tables 2 and 3.

[0037]

[Table 2]

[0038]

[Table 3] Tables 2 to 3 show the evaluation results. As can be seen from the table, in the case of the semiconductor devices outside the range 1, the occurrence rate of work error is high (Comparative Example 1). Further, as seen in all of the examples, the solder heat resistance is not deteriorated by adding the coloring agent.

[0039]

As described above, according to the present invention, it is possible to prevent the mixing of different kinds of semiconductor elements in the semiconductor device manufacturing process and to meet the demand for the chromatic color in the semiconductor device mounted on the mother board or the like. It is possible to obtain an excellent epoxy resin composition for semiconductor encapsulation and a semiconductor device encapsulated with the epoxy resin composition.

According to the epoxy resin composition of the present invention, in addition to the above effects, the characteristics conventionally required for semiconductor devices such as peeling resistance and moldability are not deteriorated.

[Brief description of drawings]

FIG. 1 is a schematic diagram showing a rough relationship between coordinates and colors in an L ^* a ^* b ^* color system.

FIG. 2 is a schematic diagram showing (range 1).

FIG. 3 is a schematic view showing (range 2), (range 3), (range 4) and (range 5).

─────────────────────────────────────────────────── ─── Continuation of front page (51) Int.Cl. ⁷ Identification code FI theme code (reference) H01L 23/31 F term (reference) 4J002 CC032 CD041 CD061 DE076 DE126 DE136 DE146 DE236 DJ016 DJ036 DJ046 EU027 EU057 EU107 FD016 FD097 FD142 GQ05 4J036 AA01 DA01 FA02 FA05 FA12 FA14 JA07 4M109 AA01 BA01 CA21 EA02 EB02 EB04 EB08 EB11 EC20

Claims (6)

[Claims] 1. An epoxy resin composition containing an epoxy resin (A), a curing agent (B), a filler (C), a curing accelerator (D) and a colorant (E), said epoxy resin composition. An epoxy resin composition, wherein the color of the resin composition corresponds to the following range when expressed by coordinates (a ^* , b ^* ) of L ^* a ^* b ^* color system. (Range 1) a ^* ≦ −5 or a ^* ≧ 5 or b ^* ≦ −5 or b ^* ≧ 5 2. The color of the epoxy resin composition, when represented by coordinates (a ^* , b ^* ) in the L ^* a ^* b ^* color system, falls within one of the following ranges: The epoxy resin composition according to claim 1. (Range 2) −20 ≦ a ^* ≦ 20 and b ^* ≧ a ^* and b ^* ≧ −a ^* and 70 ≧ b ^* ≧ 5 (Range 3) −20 ≦ a ^* ≦ 20 and b ^* ≦ a ^* and b ^* ≦ −a ^* and −70 ≦ b ^* ≦ −5 (range 4) −20 ≦ b ^* ≦ 20 and b ^* ≦ a ^* and b ^* ≧ −a ^* and 70 ≧ a ^* ≧ 5 (range 5) − 20 ≦ b * ≦ 20 and b ^* ≧ a ^* and b ^* ≦ −a ^* and −70 ≦ a ^* ≦ −5 3. The epoxy resin composition according to claim 1, wherein the colorant (E) contains a pigment selected from phthalocyanine pigments, azo pigments and anthraquinone pigments. . 4. A semiconductor device, which is encapsulated with the cured product of the epoxy resin composition according to claim 1. 5. A cured product of an epoxy resin composition containing an epoxy resin (A), a curing agent (B), a filler (C), a curing accelerator (D) and a coloring agent (E). A semiconductor device, wherein a cured product of the epoxy resin composition corresponds to the following range when the coordinates (a ^* , b ^* ) of the L ^* a ^* b ^* color system are displayed. Semiconductor device. (Range 1) a ^* ≦ −5 or a ^* ≧ 5 or b ^* ≦ −5 or b ^* ≧ 5 6. The semiconductor device according to claim 5, wherein when displayed by coordinates (a ^* , b ^* ) of the L ^* a ^* b ^* color system, the semiconductor device satisfies any one of the following ranges. (Range 2) −20 ≦ a ^* ≦ 20 and b ^* ≧ a ^* and b ^* ≧ −a ^* and 70 ≧ b ^* ≧ 5 (Range 3) −20 ≦ a ^* ≦ 20 and b ^* ≦ a ^* and b ^* ≦ −a ^* and −70 ≦ b ^* ≦ −5 (range 4) −20 ≦ b ^* ≦ 20 and b ^* ≦ a ^* and b ^* ≧ −a ^* and 70 ≧ a ^* ≧ 5 (range 5) − 20 ≦ b * ≦ 20 and b ^* ≧ a ^* and b ^* ≦ −a ^* and −70 ≦ a ^* ≦ −5