Patent Publication Number: US-2023163218-A1

Title: Semiconductor device, power diode, and rectifier

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The technical field of the present invention relates to a semiconductor device using an oxide semiconductor. 
     In this specification, the semiconductor device refers to all the devices that operate by utilizing semiconductor characteristics. In this specification, a transistor is included in a non-linear element, the non-linear element is a semiconductor device, and an electrooptic device, a semiconductor circuit, and an electronic appliance including the non-linear element are all included in semiconductor devices. 
     2. Description of the Related Art 
     Many of transistors included in display devices typified by flat panel displays (e.g., liquid crystal display devices and light-emitting display devices) include silicon semiconductors such as amorphous silicon or polycrystalline silicon and are formed over glass substrates. 
     Attention has been directed to a technique by which, instead of such silicon semiconductors, metal oxides exhibiting semiconductor characteristics are used for transistors. Note that in this specification, a metal oxide exhibiting semiconductor characteristics is referred to as an oxide semiconductor. 
     As the oxide semiconductor, a single-component metal oxide such as tungsten oxide, tin oxide, indium oxide, or zinc oxide and an In—Ga—Zn-based oxide semiconductor which is a homologous compound are given. Techniques by which a transistor formed using the metal oxide is applied to a switching element of a pixel in a display device or the like have been already disclosed in Patent Documents 1 and 2. 
     As a semiconductor device formed using a silicon semiconductor, there is a semiconductor device for high power application, such as a field-effect transistor including metal and an oxide insulating film (a metal-oxide silicon field-effect transistor: MOSFET), a junction field-effect transistor (JFET), and a Schottky barrier diode. 
     In particular, silicon carbide (SiC), which is a silicon-based semiconductor material, is used in a Schottky barrier diode having small reverse saturation current and excellent withstand voltage characteristics (see Patent Document 3). 
     REFERENCE 
     Patent Document 
     [Patent Document 1] Japanese Published Patent Application No. 2007-123861 
     [Patent Document 2] Japanese Published Patent Application No. 2007-096055 
     [Patent Document 3] Japanese Published Patent Application No. 2000-133819 
     SUMMARY OF THE INVENTION 
     A semiconductor device for high power application needs various electrical characteristics such as high withstand voltage, low reverse saturation current, and high on-state current, but so many problems arise when a semiconductor device having such electrical characteristics is actually manufactured. 
     For example, silicon carbide has problems in that it is difficult to obtain a crystal with good quality and a process temperature for manufacturing a semiconductor device is high. For example, an ion implantation method is used to form an impurity region in silicon carbide; in that case, heat treatment at 1500° C. or higher is necessary in order to repair crystal defects caused by ion implantation. 
     In addition, since carbon is contained, there is a problem in that an insulating layer with good quality cannot be formed by thermal oxidation. Furthermore, silicon carbide is chemically very stable and is not easily etched by normal wet etching. 
     In addition, since the semiconductor device for high power application generates heat when a large current flows therethrough, a structure that allows thermal dissipation is needed for the semiconductor device for high power application. 
     Thus, in view of the above problems, an object of one embodiment of the present invention is to provide a semiconductor device having electrical characteristics such as high withstand voltage, low reverse saturation current, and high on-state current. In particular, an object of an embodiment of the present invention is to provide a power diode and a rectifier which include non-linear elements. 
     An embodiment of the present invention is a semiconductor device including a first electrode, a gate insulating layer covering the first electrode, an oxide semiconductor layer overlapping with the first electrode and in contact with the gate insulating layer, a pair of second electrodes covering end portions of the oxide semiconductor layer, an insulating layer covering the pair of second electrodes and the oxide semiconductor layer, and a third electrode in contact with the insulating layer and between the pair of second electrodes. The pair of second electrodes are in contact with end surfaces of the oxide semiconductor layer. 
     Note that in this specification, an end surface of the oxide semiconductor includes a top surface and a side surface in the case where a surface of the oxide semiconductor layer on the gate insulating layer side is referred to as a bottom surface. That is, a pair of second electrodes are in contact with the oxide semiconductor layer in a region except a channel formation region and a region in contact with a gate insulating layer. Therefore, the pair of second electrodes serve as a heat sink, and when heat is generated due to current flowing in the oxide semiconductor layer including the channel formation region, the pair of second electrodes can dissipate the heat to the outside. 
     Another embodiment of the present invention is in the above semiconductor device in which n +  layers are provided between the gate insulating layer and the end portions of the oxide semiconductor layer, and the pair of second electrodes, in order to reduce contact resistance between the pair of second electrodes and the oxide semiconductor layer. 
     A depletion layer in the oxide semiconductor has a large thickness; therefore, in the semiconductor device, high on-state current can be obtained by increasing the thickness of the oxide semiconductor layer. In other words, another embodiment of the present invention is the semiconductor device in which the thickness of the oxide semiconductor layer is greater than or equal to 0.1 μm and less than or equal to 50 μm, preferably greater than or equal to 0.5 μm and less than or equal to 20 μm. 
     The oxide semiconductor layer may be a crystalline oxide semiconductor layer. With such a structure, a highly reliable semiconductor device in which variation in the electrical characteristics due to irradiation with visible light or ultraviolet light is suppressed can be achieved. The crystalline oxide semiconductor layer includes an oxide including a crystal with c-axis alignment (also referred to as C-Axis Aligned Crystal (CAAC)), which has neither a single crystal structure nor an amorphous structure. Note that part of the crystalline oxide semiconductor layer includes crystal grains. In other words, another embodiment of the present invention is the above-described semiconductor device in which the oxide semiconductor layer is a crystalline oxide semiconductor layer, and the crystalline oxide semiconductor layer has an a-b plane parallel to a surface of the crystalline oxide semiconductor layer and has c-axis alignment in a direction perpendicular to the surface. 
     Another embodiment of the present invention is the above-described semiconductor device in which the crystalline oxide semiconductor layer includes one or both of zinc and indium. 
     Another embodiment of the present invention is the above-described semiconductor device in which the first electrode functions as a gate electrode, the pair of second electrodes function as a source electrode and a drain electrode, and the third electrode functions as a back gate electrode. 
     As the above-described semiconductor device, a power diode in which a plurality of non-linear elements are connected in series in the forward direction is given. In other words, another embodiment of the present invention is a power diode including a plurality of non-linear elements. The non-linear element includes a first electrode, a gate insulating layer covering the first electrode, an oxide semiconductor layer overlapping with the first electrode and in contact with the gate insulating layer, a pair of second electrodes in contact with the oxide semiconductor layer, an insulating layer covering the pair of second electrodes and the oxide semiconductor layer, and a third electrode in contact with the insulating layer and between the pair of second electrodes. The pair of second electrodes are in contact with end surfaces of the oxide semiconductor layer, one of the pair of second electrodes is in contact with the first electrode with the gate insulating layer provided therebetween, and the non-linear elements are connected in series in a forward direction. 
     As the above-described semiconductor device, a rectifier including two non-linear elements having the above-described structure is given. In other words, another embodiment of the present invention is a rectifier including a first non-linear element and a second non-linear element. The first non-linear element and the second non-linear element each include a first electrode, a gate insulating layer covering the first electrode, an oxide semiconductor layer overlapping with the first electrode and in contact with the gate insulating layer, a pair of second electrodes in contact with the oxide semiconductor layer, an insulating layer covering the pair of second electrodes and the oxide semiconductor layer, and third electrode in contact with the insulating layer and between the pair of second electrodes. The pair of second electrodes are in contact with end surfaces of the oxide semiconductor layer, one of the pair of second electrodes is in contact with the first electrode with the gate insulating layer provided therebetween, an anode of the first non-linear element is connected to a reference potential on a low potential side, a cathode of the first non-linear element is connected to an input portion and an anode of the second non-linear element, and a cathode of the second non-linear element is connected to an output portion. 
     Further, as the above-described semiconductor device, a rectifier including four non-linear elements having the above-described structure is given. In other words, another embodiment of the present invention is a rectifier including first to fourth non-linear elements. The first to fourth non-linear elements each includes first electrode, a gate insulating layer covering the first electrode, an oxide semiconductor layer overlapping with the first electrode and in contact with the gate insulating layer, a pair of second electrodes in contact with the oxide semiconductor layer, an insulating layer covering the pair of second electrodes and the oxide semiconductor layer, and a third electrode in contact with the insulating layer and between the pair of second electrodes. The pair of second electrodes are in contact with end surfaces of the oxide semiconductor layer, one of the pair of second electrodes is in contact with the first electrode with the gate insulating layer provided therebetween. An anode of the first non-linear element is connected to a reference potential on a low potential side and a cathode thereof is connected to a first input portion. An anode of the second non-linear element is connected to the first input portion and a cathode thereof is connected to an output portion. An anode of the third non-linear element is connected to a second input portion and a cathode thereof is connected to the output portion. An anode of the fourth non-linear element is connected to the reference potential on the low potential side and a cathode thereof is connected to the second input portion. 
     A semiconductor which has characteristics such as higher withstand voltage and lower reverse saturation current and can have higher on-state current as compared to a conventional semiconductor device can be provided. In addition, a semiconductor device in which degradation due to heat generation can be suppressed can be provided. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the accompanying drawings: 
         FIGS.  1 A to  1 C  are a plan view and cross-sectional views illustrating a non-linear element that is one embodiment of the present invention; 
         FIGS.  2 A to  2 D  are plan views illustrating a method for manufacturing a non-linear element that is one embodiment of the present invention; 
         FIGS.  3 A to  3 D  are cross-sectional views illustrating a method for manufacturing a non-linear element that is one embodiment of the present invention; 
         FIGS.  4 A to  4 C  are a plan view and cross-sectional views illustrating a non-linear element that is one embodiment of the present invention; 
         FIGS.  5 A to  5 C  are cross-sectional views illustrating a method for manufacturing a non-linear element that is one embodiment of the present invention; 
         FIGS.  6 A and  6 B  are cross-sectional views illustrating a non-linear element that is one embodiment of the present invention; 
         FIGS.  7 A and  7 B  illustrate a two-dimensional crystal; 
         FIGS.  8 A to  8 D  are cross-sectional views illustrating a method for manufacturing a non-linear element that is one embodiment of the present invention; 
         FIGS.  9 A and  9 B  are cross-sectional views illustrating a non-linear element that is one embodiment of the present invention; 
       FIGS.  10 A 1  and  10 A 2 , FIGS.  10 B 1  and  10 B 2 , and FIGS.  10 C 1  to  10 C 2  illustrate power diodes and rectifiers that are embodiments of the present invention; 
         FIGS.  11 A and  11 B  are a plan view and a cross-sectional view illustrating a diode that is one embodiment of the present invention; 
         FIG.  12    is a top view of a manufacturing apparatus for manufacturing one embodiment of the present invention; 
         FIGS.  13 A to  13 C  are cross-sectional views illustrating a structure of a non-linear element used for calculation; 
         FIG.  14    is a graph showing a calculation result of a drain current in the non-linear element illustrated in  FIGS.  13 A to  13 C ; 
         FIGS.  15 A to  15 C  are cross-sectional views illustrating a structure of a non-linear element used for calculation as a comparative example; 
         FIGS.  16 A to  16 C  are cross-sectional views illustrating a structure of a non-linear element used for calculation as a comparative example; 
         FIGS.  17 A to  17 C  are cross-sectional views illustrating a structure of a non-linear element used for calculation as a comparative example; 
         FIG.  18    is a graph showing calculation results of drain currents in the non-linear elements of  FIGS.  13 A to  13 C  and  FIGS.  15 A to  17 C . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Hereinafter, embodiments of the present invention will be described in detail with reference to the accompanying drawings. However, the present invention is not limited to the following description and it is easily understood by those skilled in the art that the mode and details can be variously changed without departing from the scope and spirit of the present invention. Accordingly, the invention should not be construed as being limited to the description of the embodiments below. In describing structures of the present invention with reference to the drawings, the same reference numerals are used in common for the same portions in different drawings. The same hatching pattern is applied to similar parts, and the similar parts are not especially denoted by reference numerals in some cases. Note that the size, the layer thickness, or the region of each structure illustrated in each drawing is exaggerated for clarity in some cases. Therefore, the present invention is not necessarily limited to such scales illustrated in the drawings. 
     In an illustration of a stack of layers (or electrodes) included in a transistor, an end portion of a lower layer which protrudes from an end portion of an upper layer is not illustrated in some cases for convenience in a plan view of the transistor. 
     Further, when it is described that “A and B are connected to each other”, the case where A and B are electrically connected to each other and the case where A and B are directly connected to each other are included. Here, each of A and B corresponds to an object (e.g., a device, an element, a circuit, a wiring, an electrode, a terminal, a conductive film, or a layer). 
     Note that a voltage refers to a difference between potentials of two points, and a potential refers to electrostatic energy (electric potential energy) of a unit charge at a given point in an electrostatic field. Note that in general, a difference between a potential of one point and a reference potential is merely called a potential or a voltage, and a potential and a voltage are used as synonymous words in many cases. Thus, in this specification, a potential may be rephrased as a voltage and a voltage may be rephrased as a potential unless otherwise specified. 
     Note that, functions of “source” and “drain” may become switched in the case that a direction of a current flow is changed during circuit operation, for example. Therefore, the terms “source” and “drain” can be used to denote the drain and the source, respectively, in this specification. 
     On-state current refers to a current (a drain current) which flows between a source electrode and a drain electrode when a transistor is on. For example, in the case of an n-channel transistor, on-state current refers to a drain current when a gate voltage of the transistor is higher than a threshold voltage of the transistor. Off-state current refers to a current (a drain current) which flows between a source electrode and a drain electrode when a transistor is off For example, in an n-channel transistor, off-state current is a drain current when a gate voltage is lower than a threshold voltage of the transistor. Further, an on/off ratio refers to the ratio of on-state current to off-state current. 
     In this specification, an n-channel transistor whose the threshold voltage is positive is defined as a normally-off transistor, while a p-channel transistor whose threshold voltage is negative is defined as a normally-off transistor. Further, an n-channel transistor whose threshold voltage is negative is defined as a normally-on transistor, while a p-channel transistor whose threshold voltage is positive is defined as a normally-on transistor. 
     Embodiment 1 
     In this embodiment, a structure of a non-linear element that is one embodiment of the present invention and a method for manufacturing the non-linear element will be described with reference to  FIGS.  1 A to  1 C ,  FIGS.  2 A to  2 D , and  FIGS.  3 A to  3 D . Note that a transistor is described as an example in this embodiment. 
       FIG.  1 A  is a plan view of a transistor  100 , and  FIG.  1 B  is a cross-sectional view taken along line A-B in the transistor  100 .  FIG.  1 C  is a cross-sectional view taken along line C-D in the transistor  100 . Note that a base insulating layer  102 , a gate insulating layer  105 , and an insulating layer  111  are not illustrated in  FIG.  1 A  for convenience.  FIG.  1 A  illustrates a wiring  104  including a first electrode  103  functioning as a gate electrode; an oxide semiconductor layer  107  including a channel formation region; a wiring  110  including a pair of second electrodes  109   a  and  109   b  functioning as a source electrode and a drain electrode; and a wiring  114  provided between the pair of second electrodes  109   a  and  109   b,  overlapping with the oxide semiconductor layer  107  with the insulating layer  111  provided therebetween, and including a third electrode  113  serving as a back gate electrode. Although the first electrode  103 , the pair of second electrodes  109   a  and  109   b,  and the third electrode  113  are integrated with the wiring  104 , the wiring  110 , and the wiring  114 , respectively here, the electrodes and the wirings may be formed separately and electrically connected to each other. 
     In this embodiment, although the first electrode  103 , the pair of second electrodes  109   a  and  109   b,  and the third electrode  113  function as a gate electrode, a source and a drain electrode, and a back gate electrode, respectively, one embodiment of the present invention is not limited thereto. The first electrode  103 , the pair of second electrodes  109   a  and  109   b,  and the third electrode  113  each can function as any of a gate electrode, a source electrode, a drain electrode, and a back gate electrode. 
     As illustrated in  FIG.  1 B , the transistor  100  is a dual-gate transistor including a gate electrode and a back gate electrode and includes the base insulating layer  102 , the first electrode  103 , the gate insulating layer  105 , the oxide semiconductor layer  107 , the pair of second electrodes  109   a  and  109   b,  the insulating layer  111 , and the third electrode  113 , which are provided over a substrate  101 . 
     The first electrode  103  is provided in contact with the base insulating layer  102 . The gate insulating layer  105  is provided to cover the first electrode  103 . The oxide semiconductor layer  107  is provided in contact with the gate insulating layer  105  to overlap with the first electrode  103 . The pair of second electrodes  109   a  and  109   b  cover end portions of the oxide semiconductor layer  107 . The insulating layer  111  covers part of the oxide semiconductor layer  107  and the pair of second electrodes  109   a  and  109   b.  The third electrode  113  is provided on and in contact with the insulating layer  111  and between the pair of second electrodes  109   a  and  109   b.    
     Further, since the pair of second electrodes  109   a  and  109   b  cover the end portions of the oxide semiconductor layer  107 , the pair of second electrodes  109   a  and  109   b  are in contact with end surfaces of the oxide semiconductor layer  107 . Therefore, in a region where the pair of second electrodes  109   a  and  109   b  are in contact with the oxide semiconductor layer  107 , the width of each of the pair of second electrodes  109   a  and  109   b  is larger than the width of a channel formed in the oxide semiconductor layer  107  (see  FIG.  1 A ). 
     Further, as illustrated in  FIG.  1 C , since the pair of second electrodes  109   a  and  109   b  are in contact with the end surfaces of the oxide semiconductor layer  107 , the pair of second electrodes  109   a  and  109   b  serve as a heat sink, and when heat is generated due to high on-state current flowing in the oxide semiconductor layer  107 , the pair of second electrodes  109   a  and  109   b  can dissipate the heat to the outside. As a result, degradation of the transistor  100  due to heat generation can be suppressed. 
     In addition, in the transistor  100 , the width of the first electrode  103  is larger than the width of the oxide semiconductor layer  107  in the channel length direction, the third electrode  113  overlaps with the oxide semiconductor layer  107  with the insulating layer  111  provided therebetween, and the width of the third electrode  113  is at least larger than or equal to the channel length. 
     Accordingly, end portions of the oxide semiconductor layer  107  which are not covered with the pair of the second electrodes  109   a  and  109   b  are covered with the first electrode  103  and the third electrode  113  with the gate insulating layer  105  and the insulating layer  111  provided therebetween. In other words, all of the end portions of the oxide semiconductor layer  107  are covered with the first electrode  103 , the pair of second electrodes  109   a  and  109   b,  and the third electrode  113 . 
     When the end portions of the oxide semiconductor layer  107  are covered even in the case where the gate insulating layer  105  and the insulating layer  111  are provided between the first electrode  103  and the oxide semiconductor layer  107  and between the third electrode  113  and the oxide semiconductor layer  107 , the pair of second electrodes  109   a  and  109   b  can serve as a heat sink. Therefore, in the transistor  100 , heat which is generated when high on-state current flows in the oxide semiconductor layer  107  can be effectively dissipated to the outside, so that degradation of the transistor  100  due to heat generation can be suppressed. 
     As the substrate  101 , an alkali-free glass substrate formed with a fusion method or a float method, a plastic substrate having heat resistance sufficient to withstand heat treatment performed later, or the like can be used. In addition, a substrate where an insulating film is provided on a surface of a metal substrate such as a stainless steel substrate, or a substrate where an insulating film is provided on a surface of a semiconductor substrate may be used. 
     As a glass substrate, if the temperature of the heat treatment to be performed later is high, a glass substrate whose strain point is 730° C. or higher is preferably used. As a glass substrate, a glass material such as aluminosilicate glass, aluminoborosilicate glass, or barium borosilicate glass is used, for example. By containing a larger amount of barium oxide (BaO) than boric oxide, a more practical heat-resistant glass substrate is obtained. Therefore, a glass substrate containing BaO and B 2 O 3  so that the amount of BaO is larger than that of B 2 O 3  is preferably used. 
     Note that a substrate formed of an insulator, such as a ceramic substrate, a quartz substrate, or a sapphire substrate, may be used instead of the glass substrate. Alternatively, crystallized glass or the like may be used. 
     The base insulating layer  102  provided between the substrate  101  and the first electrode  103  can prevent not only diffusion of an impurity element from the substrate  101  but also etching of the substrate  101  during an etching step included in the steps for manufacturing the transistor. Therefore, the thickness of the base insulating layer  102  is preferably, but not limited to, 50 nm or more. Note that the base insulating layer  102  is formed with a single-layer structure or a stacked structure using an oxide insulator and/or a nitride insulator such as silicon oxide, gallium oxide, aluminum oxide, silicon nitride, silicon oxynitride, aluminum oxynitride, and silicon nitride oxide. In particular, aluminum nitride, aluminum nitride oxide, and silicon nitride which have a high thermal conductivity are effective in improving thermal dissipation when used for the base insulating layer  102 . In addition, alkali metal such as Li and Na is an impurity for the oxide semiconductor layer  107  described later. Therefore, it is preferable to reduce the content of alkali metal. In the case where a glass substrate including an impurity such as alkali metal is used as the substrate  101 , the base insulating layer  102  is preferably formed using a nitride insulator such as silicon nitride or aluminum nitride in order to prevent the entry of the alkali metal. 
     The first electrode  103  serving as a gate electrode can be formed with a single-layer structure or a stacked structure using a metal material such as molybdenum, titanium, tantalum, tungsten, aluminum, copper, chromium, neodymium, or scandium or an alloy material which includes any of these materials as its main component. In addition, the first electrode  103  can have a single-layer structure or a stacked structure including two or more layers. For example, a single-layer structure of an aluminum film containing silicon, a two-layer structure of an aluminum film and a titanium film stacked thereover, a two-layer structure of a tungsten film and a titanium film stacked thereover, a three-layer structure in which a titanium film, an aluminum film, and a titanium film are stacked in this order, and the like can be given. 
     There is no limitation on the thickness of the first electrode  103 , and the thickness of the first electrode  103  can be determined as appropriate considering the electrical resistance of a conductive film formed using any of the above-described materials and a time necessary for forming the conductive film. 
     The gate insulating layer  105  is in contact with the oxide semiconductor layer  107  and thus needs to have high quality. This is because the oxide semiconductor layer  107  which is an i-type or a substantially i-type oxide semiconductor layer obtained by removal of impurities (an oxide semiconductor layer whose hydrogen concentration is reduced and to which oxygen is supplied) is extremely sensitive to an interface state and interface electric charge, and thus an interface between the oxide semiconductor layer  107  the gate insulating layer  105  is important. 
     The gate insulating layer  105  can be formed using any of the above-described oxide insulators. A portion of the gate insulating layer  105  which is in contact with the oxide semiconductor layer  107  preferably contains oxygen, and, in particular, the gate insulating layer  105  preferably contains oxygen whose amount exceeds the stoichiometric proportion. For example, the gate insulating layer  105  may be formed using silicon oxide (SiO 2+α  (note that α&gt;0)) which contains oxygen whose amount exceeds the stoichiometric proportion. By forming the gate insulating layer  105  with the use of the silicon oxide, part of oxygen contained in the gate insulating layer  105  can be supplied to the oxide semiconductor layer  107  in the heat treatment performed in the manufacturing steps of the transistor  100 , so that the transistor  100  can have favorable electrical characteristics. 
     The gate insulating layer  105  may be formed with either a single-layer structure or a stacked structure. When the thickness of the gate insulating layer  105  is increased, gate leakage current can be reduced. When the gate insulating layer  105  is formed using, for example, a high-k material such as hafnium oxide, yttrium oxide, hafnium silicate (HfSi x O y  (x&gt;0, y&gt;0)), hafnium silicate to which nitrogen is added (HfSiO x N y  (x&gt;0, y&gt;0)), or hafnium aluminate (HfAl x O y  (x&gt;0, y&gt;0)), gate leakage current can be reduced. Note that the thickness of the gate insulating layer is preferably greater than or equal to 50 nm and less than or equal to 500 nm. 
     The oxide semiconductor layer  107  can be formed using any of the following: a four-component metal oxide such as an In—Sn—Ga—Zn-based metal oxide; a three-component metal oxide such as an In—Ga—Zn-based metal oxide, an In—Sn—Zn-based metal oxide, an In—Al—Zn-based metal oxide, a Sn—Ga—Zn-based metal oxide, an Al—Ga—Zn-based metal oxide, a Sn—Al—Zn-based metal oxide, an In—Hf—Zn-based metal oxide, an In—La—Zn-based metal oxide, an In—Ce—Zn-based metal oxide, an In—Pr—Zn-based metal oxide, an In—Nd—Zn-based metal oxide, an In—Sm—Zn-based metal oxide, an In—Eu—Zn-based metal oxide, an In—Gd—Zn-based metal oxide, an In—Tb—Zn-based metal oxide, an In—Dy—Zn-based metal oxide, an In—Ho—Zn-based metal oxide, an In—Er—Zn-based metal oxide, an In—Tm—Zn-based metal oxide, an In—Yb—Zn-based metal oxide, or an In—Lu—Zn-based metal oxide; a two-component metal oxide such as an In—Zn-based metal oxide, a Sn—Zn-based metal oxide, an Al—Zn-based metal oxide, a Zn—Mg-based metal oxide, a Sn—Mg-based metal oxide, an In—Mg-based metal oxide, or an In—Ga-based metal oxide; or a single-component metal oxide containing indium, tin, zinc or the like. The oxide semiconductor layer  107  including the channel region is preferably formed using the metal oxide containing zinc or the metal oxide containing zinc and indium, considering the manufacture of a crystalline oxide semiconductor described later. Here, for example, an In—Ga—Zn-based metal oxide means an oxide containing indium (In), gallium (Ga), and zinc (Zn), and there is no particular limitation on the composition ratio thereof Further, the In—Ga—Zn-based metal oxide may contain an element other than In, Ga, and Zn. 
     In addition, it is preferable that impurities such as hydrogen be sufficiently removed from the oxide semiconductor layer  107  and oxygen be sufficiently supplied thereto. Specifically, the hydrogen concentration of the oxide semiconductor layer  107  is 5×10 19  atoms/cm 3  or lower, preferably 5×10 18  atoms/cm 3  or lower, more preferably 5×10 17  atoms/cm 3  or lower. Note that the hydrogen concentration of the oxide semiconductor layer  107  is measured by secondary ion mass spectroscopy (SIMS). By supply of sufficient oxygen, a defect level due to oxygen deficiency in the energy gap of the oxide semiconductor layer  107  is reduced. Accordingly, a carrier density of the oxide semiconductor layer  107  due to a donor such as hydrogen is higher than or equal to 1×10 10  cm −3  and lower than or equal to 1×10 13  cm −3 . In this manner, the transistor  100  with extremely favorable off-state current characteristics can be obtained with the use of an i-type (intrinsic) or substantially i-type oxide semiconductor for the oxide semiconductor layer  107 . For example, the off-state current (per unit channel width (1 μm) here) at room temperature (25° C.) is 100 zA (1 zA (zeptoampere) is 1×10 −21  A) or less, preferably 10 zA or less. 
     In addition, the content of alkali metal such as Li and Na is preferably low, and the concentration of alkali metal in the oxide semiconductor layer  107  is preferably 2×10 16  cm −3  or lower, preferably 1×10 15  cm −3  or lower. Further the content of alkaline earth metal is preferably low because alkaline earth metal is also an impurity. The reason is described below. Note that it has been pointed out that an oxide semiconductor is insensitive to impurities, there is no problem when a considerable amount of metal impurities is contained in the film, and therefore, soda-lime glass which contains a large amount of alkali metal such as sodium (Na) and is inexpensive can also be used (Kamiya, Nomura, and Hosono, “Carrier Transport Properties and Electronic Structures of Amorphous Oxide Semiconductors: The present status”,  KOTAI BUTSURI  ( SOLID STATE PHYSICS ), 2009, Vol. 44, pp. 621-633). But such consideration is not appropriate. Alkali metal is not an element included in an oxide semiconductor, and therefore, is an impurity. Also, alkaline earth metal is impurity in the case where alkaline earth metal is not included in an oxide semiconductor. Alkali metal, in particular, Na becomes Na +  when an insulating film in contact with the oxide semiconductor layer is an oxide and Na diffuses into the insulating film. Further, in the oxide semiconductor layer, Na cuts or enters a bond between metal and oxygen which are included in an oxide semiconductor. As a result, degradation of the electrical characteristics occurs; for example, the field-effect mobility is reduced or the transistor becomes a normally-on transistor in which a drain current flows even in the state where no voltage is applied to the gate electrode (Vg=0), which is caused by the shift of the threshold voltage in the negative direction. In addition, variation in the electrical characteristics also occurs. Such degradation of electrical characteristics of the transistor and variation in the electrical characteristics due to the impurities remarkably appear when the hydrogen concentration in the oxide semiconductor layer is sufficiently low. 
     An In—Ga—Zn-based metal oxide has a sufficiently high resistance when no electric field is applied thereto and thus can sufficiently reduce off-state current. Further, an In—Ga—Zn-based metal oxide has high field-effect mobility and is therefore a preferable semiconductor material for the transistor of one embodiment of the present invention. 
     A depletion layer is thicker in the case of using an oxide semiconductor for a channel formation region like the transistor  100 , as compared to the case of using a silicon semiconductor for a channel formation region. Accordingly, the channel region is thicker because the channel region is also formed in a depth direction of the oxide semiconductor. Further, when the thickness of the oxide semiconductor is large, a large number of carriers can flow. As a result, high on-state current can be obtained. 
     The drain withstand voltage of the transistor  100  depends on the thickness of the oxide semiconductor layer  107 . Therefore, in order to increase the drain withstand voltage, the thickness of the oxide semiconductor layer  107  is preferably large and may be selected in accordance with the desired drain withstand voltage. 
     Accordingly, the thickness of the oxide semiconductor layer  107  is preferably greater than or equal to 0.1 μm and less than or equal to 50 μm, preferably greater than or equal to 0.5 μm and less than or equal to 20 μm in consideration of the electrical characteristics such as the on-state current and the drain withstand voltage. 
     The drain withstand voltage of a transistor using an oxide semiconductor is now described. 
     When the electric field in the semiconductor reaches a certain threshold value, impact ionization occurs, carriers accelerated by the high electric field impact crystal lattices in a depletion layer, thereby generating pairs of electrons and holes. When the electric field becomes even higher, the pairs of electrons and holes generated by the impact ionization are further accelerated by the electric field, and the impact ionization is repeated, resulting in an avalanche breakdown in which current is increased exponentially. The impact ionization occurs because carriers (electrons and holes) have kinetic energy that is larger than or equal to the band gap of the semiconductor. It is known that the impact ionization coefficient that shows probability of impact ionization has correlation with the band gap and that the impact ionization is unlikely to occur as the band gap is increased. 
     Since the band gap of the oxide semiconductor is about 3.15 eV, which is larger than the band gap of silicon, i.e., about 1.12 eV, the avalanche breakdown is unlikely to occur. Therefore, a transistor using the oxide semiconductor has a high drain withstand voltage, and an exponential sudden increase of on-state current is expected to be unlikely to occur when a high electric field is applied. 
     Next, hot-carrier degradation of a transistor using an oxide semiconductor is described. 
     The hot-carrier degradation means deterioration of transistor characteristics, e.g., shift in the threshold voltage or gate leakage current, which is caused as follows: electrons that are accelerated to be rapid are injected in the vicinity of a drain in a channel into a gate insulating film and become fixed electric charge or form trap levels at the interface between the gate insulating film and the oxide semiconductor. The factors of the hot-carrier degradation are, for example, channel-hot-electron injection (CHE injection) and drain-avalanche-hot-carrier injection (DAHC injection). 
     Since the band gap of a silicon semiconductor is narrow, electrons are likely to be generated like an avalanche owing to an avalanche breakdown, and electrons that are accelerated to be so rapid as to go over a barrier to the gate insulating film are increased in number. However, the oxide semiconductor described in this embodiment has a wide band gap; therefore, the avalanche breakdown is unlikely to occur and resistance to the hot-carrier degradation is higher than that of a silicon semiconductor. In this manner, the transistor including an oxide semiconductor has high drain withstand voltage, and the transistor including an oxide semiconductor is suitable for a semiconductor device for high power application such as an insulated-gate field effect transistor (IGFET), a junction field-effect transistor, and a Schottky barrier diode. 
     The pair of second electrodes  109   a  and  109   b  can be formed using any of the materials given in the description of the first electrode  103 . The thickness and structure of the electrodes are appropriately selected based on the description of the first electrode  103 . Note that since the pair of second electrodes  109   a  and  109   b  function as a heat sink which dissipates heat generated when the on-state current flows in the oxide semiconductor layer  107  to the outside, the pair of second electrodes  109   a  and  109   b  are preferably formed using a metal material or an alloy material which easily conducts heat. 
     The insulating layer  111  can be formed using any of the oxide insulators given in the description of the gate insulating layer  105 . Since the insulating layer  111  is also in contact with the oxide semiconductor layer  107 , a portion of the insulating layer  111  which is in contact with the oxide semiconductor layer  107  preferably includes oxygen, and the insulating layer  111  is preferably formed using silicon oxide (SiO 2+α  (note that α&gt;0)) which contains oxygen whose amount exceeds the stoichiometric proportion, in particular. By forming the insulating layer  111  with the use of the silicon oxide, part of oxygen contained in the insulating layer  111  can be supplied to the oxide semiconductor layer  107  in the heat treatment performed in the manufacturing steps of the transistor  100 , so that the transistor  100  can have favorable electrical characteristics. In addition, the insulating layer  111  may be formed using the high-k material given in the description of the gate insulating layer  105 . Further the insulating layer  111  may be formed with either a single-layer structure or a stacked structure. When the thickness of the insulating layer  111  is increased, gate leakage current on the back gate side can be reduced. The thickness of the insulating layer  111  is preferably greater than or equal to 50 nm and less than or equal to 500 nm. 
     The third electrode  113  serving as a back gate electrode can be formed using any of the materials given in the description of the first electrode  103 , and the thickness and structure of the third electrode  113  may be appropriately selected based on the description of the first electrode  103 . 
     In view of reliability, the electrical characteristics of the transistor including an oxide semiconductor are changed by irradiation with visible light or ultraviolet light or application of heat or an electric field. As an example of a change of the electrical characteristics, the transistor becomes a normally-on transistor in which the drain current flows even in the state where no voltage is applied to the gate electrode (Vg=0). In the case of an n-channel transistor, in which an electron is a majority carrier, an electron in the drain current flows in a region where a depletion layer is formed. Therefore, in the transistor, the region where an electron flows includes a region in the vicinity of a top surface of the oxide semiconductor layer  107 , where the pair of second electrodes  109   a  and  109   b  and the insulating layer  111  are provided. Therefore, a hole is induced in the insulating layer  111  in contact with the oxide semiconductor layer  107  (in particular, in a region in the vicinity of a bottom surface of the insulating layer  111  in contact with the oxide semiconductor layer  107 ) and the transistor becomes normally-on as time passes. Thus, since the transistor described in this embodiment is a dual-gate transistor including the third electrode  113 , voltage can be freely applied to the third electrode  113 , and the threshold voltage (Vth) can be controlled, so that the transistor can be prevented from becoming normally-on. 
     In addition, since the transistor described in this embodiment is a dual-gate transistor, voltage can be applied to the third electrode  113 , and a channel can be efficiently formed even when the thickness of the oxide semiconductor layer  107  is large, so that high on-state current can be obtained. 
     Here, the shape of the third electrode  113  is described with reference to  FIGS.  2 A to  2 D . 
     The third electrode  113  illustrated in  FIG.  2 A  has the same shape as the third electrode  113  illustrated in  FIG.  1 A . The third electrode  113  is provided to be parallel to the first electrode  103  and overlaps with the pair of second electrodes  109   a  and  109   b  with the insulating layer  111  provided therebetween. In that case, voltage applied to the third electrode  113  and voltage applied to the first electrode  103  can be controlled independently. 
     The third electrode  113  illustrated in  FIG.  2 B  is parallel to the first electrode  103  but does not overlap with the pair of second electrodes  109   a  and  109   b.  Also in this structure, the voltage applied to the third electrode  113  and the voltage applied to the first electrode  103  can be controlled independently. 
     The third electrode  113  illustrated in  FIG.  2 C  can be connected to the first electrode  103 . In other words, the first electrode  103  and the third electrode  113  are connected to each other in an opening portion  150  formed in the gate insulating layer  105  and the insulating layer  111 . In this structure, the voltage applied to the third electrode  113  is equal to the voltage applied to the first electrode  103 . 
     Further, a structure illustrated in  FIG.  2 D  may be employed, in which the third electrode  113  is not connected to the first electrode  103  but is in a floating state. 
     In addition, in the structures illustrated in  FIGS.  2 C and  2 D , the third electrode  113  may overlap with the pair of second electrodes  109   a  and  109   b  with the insulating layer  111  provided therebetween. 
     Although not illustrated in  FIGS.  1 A to  1 C , a protective insulating layer may be provided over the insulating layer  111  and the third electrode  113  in the transistor  100 . 
     Next, a method for manufacturing the transistor  100  is described with reference to  FIGS.  3 A to  3 D . 
     The base insulating layer  102  is formed over the substrate  101 . Through this process, impurities in a glass substrate can be prevented from entering the transistor which is to be formed. 
     The base insulating layer  102  can be formed by a sputtering method, a CVD method, a coating method, or the like. In this embodiment, a silicon oxide film is formed by a sputtering method with the use of a silicon target. In order to remove moisture and hydrogen in the base insulating layer  102 , the substrate  101  may be subjected to heat treatment after the base insulating layer  102  is formed. 
     Next, the first electrode  103  is formed over the base insulating layer  102 . Note that the step for forming the first electrode  103  is combined with a step for forming the wiring  104  (see  FIG.  1 A ). The first electrode  103  can be formed in such a manner that a conductive film is formed over the substrate  101  by a sputtering method, a vacuum evaporation method, or a CVD method, a resist mask is formed over the conductive film by a first photolithography step, and the conductive film is etched using the resist mask. Alternatively, the resist mask is formed by a printing method or an ink-jet method instead of the photolithography step, so that the number of steps for forming the first electrode  103  can be reduced. Note that end portions of the first electrode  103  preferably have a tapered shape because the coverage with the gate insulating layer  105  formed later can be improved. Note that the taper shape can be obtained by etching while the resist mask is made to recede. 
     In this embodiment, a conductive film (e.g., a tungsten film) with a thickness of 150 nm is formed by a sputtering method and etched with the use of a resist mask formed through the first photolithography step, so that the first electrode  103  is formed. Note that the etching step with the use of the resist mask includes a step for removing the resist mask even when its description is not expressly made in this specification. 
     Next, the gate insulating layer  105  covering the first electrode  103  is formed. The gate insulating layer  105  is in contact with the oxide semiconductor layer  107  which is to be formed later, and thus, needs to have high quality. When the oxide semiconductor layer  107  is in contact with the gate insulating layer  105  having high quality, the interface state density between the oxide semiconductor layer  107  and the gate insulating layer  105  is reduced and the interface characteristics become favorable. As a result, the transistor  100  completed can have favorable electrical characteristics. 
     The gate insulating layer  105  can be formed by any of the methods given in the description of the base insulating layer  102 . In this embodiment, silicon oxide (SiO 2+α  (note that α&gt;0)) containing oxygen whose amount exceeds the stoichiometric proportion is formed as the gate insulating layer  105 . Note that the silicon oxide formed has a thickness of 200 nm. 
     When the silicon oxide is formed by a sputtering method, a silicon target or a quartz target is used as a target and oxygen or a mixed gas of oxygen and argon is used as a sputtering gas. At that time, the gate insulating layer  105  is preferably formed while hydrogen, moisture, hydroxyl groups, hydride, or the like remaining in a process chamber are removed. An entrapment vacuum pump is preferably used in order to remove hydrogen, water, hydroxyl groups, hydride, or the like remaining in the process chamber. As the entrapment vacuum pump, for example, a cryopump, an ion pump, or a titanium sublimation pump is preferably used. The evacuation unit may be a turbo pump provided with a cold trap. Hydrogen, water, hydroxyl groups, hydride, or the like is removed from the process chamber which is evacuated with a cryopump; thus, when the gate insulating layer  105  is formed in the process chamber, the concentration of hydrogen, water, hydroxyl groups, or hydride contained in the gate insulating layer  105  can be reduced. 
     It is preferable to use a high-purity gas from which an impurity such as hydrogen, water, a hydroxyl group, or hydride is removed to a concentration of several ppm or several ppb, as a sputtering gas used when the gate insulating layer  105  is formed. 
     In this embodiment, the substrate  101  is transferred to the process chamber, a sputtering gas containing high-purity oxygen, from which hydrogen, water, hydroxyl groups, hydride, or the like is removed, is introduced, and the silicon oxide is formed as the gate insulating layer  105  over the substrate  101  using a silicon target. Note that the gate insulating layer  105  may be formed while the substrate  101  is heated. 
     In addition, in the case where the gate insulating layer  105  is formed with a stacked structure, for example, silicon nitride may be formed between the silicon oxide and the substrate  101 . The silicon nitride is formed using a silicon target and sputtering gas containing high-purity nitrogen from which hydrogen, water, hydroxyl groups, hydride, or the like is removed. Further it is preferable that the silicon nitride be formed while hydrogen, water, hydroxyl groups, hydride, or the like remaining in the process chamber is removed in a manner similar to the case of the silicon oxide. 
     In the case where the silicon nitride and the silicon oxide are stacked to form the gate insulating layer  105 , the silicon nitride and the silicon oxide can be formed in the same process chamber with the same silicon target. In that case, a sputtering gas containing nitrogen is introduced and silicon nitride is formed using a silicon target mounted in the process chamber, and then, the sputtering gas is switched to a sputtering gas containing oxygen and the same silicon target is used to form silicon oxide. The silicon nitride and the silicon oxide can be successively formed by this method without exposure to the air, whereby hydrogen, water, a hydroxyl group, hydride, or the like can be prevented from being adsorbed on a surface of the silicon nitride. 
     Further, preheat treatment is preferably performed before the gate insulating layer  105  is formed, in order to remove hydrogen, water, hydroxyl groups, or hydride which remains on the inner wall of the process chamber, on a surface of the target, or inside the target material. After the preheat treatment, the substrate  101  or the process chamber is cooled and then the gate insulating layer  105  is formed without exposure to the air. In that case, not water but oil or the like is preferably used as a coolant for the target. 
     Further, in the case where the gate insulating layer  105  is formed by a CVD method, for example, by a high-density plasma CVD with the use of a microwave (e.g., its frequency is 2.45 GHz), the gate insulating layer which is dense and has high withstand voltage and high quality can be formed. In addition, since the gate insulating layer formed by high-density plasma CVD can have a uniform thickness, the gate insulating layer has excellent step coverage. Further, as for the gate insulating layer formed using high-density plasma CVD, the thickness can be controlled precisely. 
     Next, an oxide semiconductor film  106  is formed to overlap with the first electrode  103  with the gate insulating layer  105  provided therebetween. The oxide semiconductor film  106  can be formed over the gate insulating layer  105  by a sputtering method, a molecular beam epitaxy method, an atomic layer deposition method, a pulsed laser deposition method, a coating method, or a printing method. 
     In this embodiment, the oxide semiconductor film  106  is formed by a sputtering method. The oxide semiconductor film  106  is formed over the gate insulating layer  105  in such a manner that a sputtering gas from which hydrogen, water, hydroxyl groups, hydride, or the like is removed is introduced into the process chamber and a metal oxide is used as a target while the substrate is held in the process chamber held in a reduced-pressure state and moisture remaining in the process chamber is removed. Hydrogen, water, hydroxyl groups, hydride, or the like remaining in the process chamber may be removed in a manner similar to the gate insulating layer  105 . Consequently, impurities such as hydrogen, water, hydroxyl groups, or hydride, (and preferably a compound containing a carbon atom) are exhausted from the process chamber, so that the concentration of the impurities contained in the oxide semiconductor film  106  can be reduced. Further, the oxide semiconductor film  106  may be formed while the substrate  101  is heated. 
     As the target used for forming the oxide semiconductor film  106  by a sputtering method, a metal oxide target containing at least zinc or a metal oxide target containing at least zinc and indium can be used. In this embodiment, the oxide semiconductor film  106  with a thickness of 500 nm is deposited using an In—Ga—Zn-based metal oxide target (In 2 O 3 :Ga 2 O 3 :ZnO=1:1:2 [molar ratio]). As other examples of the metal oxide target, a target having a composition of In 2 O 3 :Ga 2 O 3 :ZnO=1:1:1 [molar ratio], a target having a composition of In:Ga:Zn=1:1:0.5 [atomic ratio], a target having a composition of In:Ga:Zn=1:1:1 [atomic ratio], and the like are given. Any of the above-described metal oxide targets may contain SiO 2  at 2 wt % or more and 10 wt % or less. Note that the filling rate of the metal oxide target is greater than or equal to 90% and less than or equal to 100%, preferably greater than or equal to 95% and less than or equal to 99.9%. By using the metal oxide target with high filling rate, a dense oxide semiconductor film is formed. 
     The oxide semiconductor film  106  is formed in a rare gas (typically argon) atmosphere, an oxygen atmosphere, or an atmosphere including a rare gas (typically argon) and oxygen. It is preferable to use a high-purity gas from which an impurity such as hydrogen, water, a hydroxyl group, or hydride is removed to a concentration of several ppm or several ppb, as a sputtering gas used when the oxide semiconductor film  106  is formed. 
     For example, formation conditions are set as follows: the distance between the substrate  101  and the target is 170 mm, the substrate temperature is 250° C., the pressure is 0.4 Pa, and the direct current (DC) power is 0.5 kW. 
     As pretreatment, it is preferable that the substrate  101  subjected to the steps up to the formation of the gate insulating layer  105  be preheated and impurities such as hydrogen, water, hydroxyl groups, or hydride adsorbed on the substrate  101  be eliminated and exhausted so that hydrogen is contained in the oxide semiconductor film  106  as little as possible. Note that exhaust is preferably performed using a cryopump at the time of the preheating. Note that this preheating may be omitted. Alternatively, the preheating may be performed on the substrate  101  before the first electrode  103  is formed or after the oxide semiconductor layer  107  is formed later. 
     Note that before the oxide semiconductor film  106  is formed by sputtering, reverse sputtering in which plasma is generated by introduction of an argon gas is preferably performed so that dust or an oxide film which is attached to a surface of the gate insulating layer  105  is removed, in which case the resistance at an interface between the gate insulating layer  105  and the oxide semiconductor film  106  can be reduced. The reverse sputtering refers to a method of modifying a surface of a substrate by applying a voltage to the substrate using an RF power source in an argon atmosphere to form plasma in a vicinity of the substrate. Note that instead of an argon atmosphere, a nitrogen atmosphere, a helium atmosphere, or the like may be used. Alternatively, an argon atmosphere to which oxygen, nitrous oxide, or the like is added may be used. Further alternatively, an argon atmosphere to which chlorine, carbon tetrafluoride, or the like is added may be used. 
     The structure obtained through the steps up to here is illustrated in  FIG.  3 A . 
     Next, the oxide semiconductor film  106  is etched using a resist mask formed through a second photolithography step, so that a first island-shaped oxide semiconductor layer is formed. 
     The step for forming the first island-shaped oxide semiconductor layer is described. The first island-shaped oxide semiconductor layer is formed by etching with the use of the resist mask formed through the second photolithography step. The second photolithography step is similar to the first photolithography step. 
     For the etching of the oxide semiconductor film  106 , either wet etching or dry etching may be employed. In addition, these may be combined. As an etchant used for wet etching, a mixed solution of phosphoric acid, acetic acid, and nitric acid, an ammonia hydrogen peroxide mixture (hydrogen peroxide water at 31 wt %:ammonia water at 28 wt %:water=5:2:2), or the like can be used. In addition, ITO07N (produced by KANTO CHEMICAL CO., INC.) may also be used. 
     The etchant after the wet etching is removed together with the etched material by cleaning. The waste liquid including the etchant and the etched material may be purified and the material may be reused. When a material such as indium is collected from the waste liquid after the etching and reused, the resources can be efficiently used and the cost can be reduced. 
     As the etching gas for dry etching, a gas containing chlorine (chlorine-based gas such as chlorine (Cl 2 ), boron trichloride (BCl 3 ), silicon tetrachloride (SiCl 4 ), or carbon tetrachloride (CCl 4 )) is preferably used. 
     Alternatively, a gas containing fluorine (fluorine-based gas such as carbon tetrafluoride (CF 4 ), sulfur hexafluoride (SF 6 ), nitrogen trifluoride (NF 3 ), or trifluoromethane (CHF 3 )); hydrogen bromide (HBr); oxygen (O 2 ); any of these gases to which a rare gas such as helium (He) or argon (Ar) is added; or the like can be used. 
     As the dry etching method, a parallel plate RIE (reactive ion etching) method or an ICP (inductively coupled plasma) etching method can be used. In order to etch the film into a desired shape, the etching conditions (the amount of electric power applied to a coil-shaped electrode, the amount of electric power applied to an electrode on a substrate side, the temperature of the electrode on the substrate side, or the like) is adjusted as appropriate. 
     Next, first heat treatment is performed on the first island-shaped oxide semiconductor layer to form an oxide semiconductor layer  126 . 
     The temperature of the first heat treatment is higher than or equal to 400° C. and lower than or equal to 750° C., preferably higher than or equal to 400° C. and lower than the strain point of the substrate  101 . Here, the substrate is introduced into an electric furnace that is a kind of heat treatment apparatus and heat treatment is performed on the first island-shaped oxide semiconductor layer at 450° C. in an atmosphere of an inert gas such as nitrogen or a rare gas for one hour. After that, the oxide semiconductor layer  126  is not exposed to the air; consequently, hydrogen, water, a hydroxyl group, hydride, or the like can be prevented from entering the oxide semiconductor layer  126 . As a result, the oxide semiconductor layer  126  whose hydrogen concentration is reduced can be obtained. That is, at least one of dehydration and dehydrogenation of the first island-shaped oxide semiconductor layer can be performed by this first heat treatment. 
     Further, in addition to the dehydration and dehydrogenation, the first heat treatment is also combined with treatment by which, in the case where the gate insulating layer  105  contains oxygen, part of the oxygen is diffused into an interface between the gate insulating layer  105  and the first island-shaped oxide semiconductor layer or the vicinity thereof. As a result of the treatment, the oxide semiconductor layer  126  whose oxygen deficiency is reduced can be obtained. 
     Note that it is preferable that in the first heat treatment, hydrogen, water, a hydroxyl group, hydride, or the like be not contained in nitrogen or a rare gas such as helium, neon, or argon. Alternatively, the purity of nitrogen or a rare gas such as helium, neon, or argon which is introduced into the heat treatment apparatus is greater than or equal to 6 N (99.9999%), preferably greater than or equal to 7 N (99.99999%) (i.e., the impurity concentration is lower than or equal to 1 ppm, preferably lower than or equal to 0.1 ppm). 
     The heat treatment apparatus used for the first heat treatment is not limited to a particular apparatus, and the apparatus may be provided with a device for heating an object to be processed by heat radiation or heat conduction from a heating element such as a resistance heating element. For example, an electric furnace, or a rapid thermal annealing (RTA) apparatus such as a gas rapid thermal annealing (GRTA) apparatus or a lamp rapid thermal annealing (LRTA) apparatus can be used. An LRTA apparatus is an apparatus for heating an object to be processed by radiation of light (an electromagnetic wave) emitted from a lamp such as a halogen lamp, a metal halide lamp, a xenon arc lamp, a carbon arc lamp, a high pressure sodium lamp, or a high pressure mercury lamp. A GRTA apparatus is an apparatus for heat treatment using a high-temperature gas. 
     Further, the first heat treatment may be performed before the first island-shaped oxide semiconductor layer is formed. In other words, the first heat treatment may be performed on the oxide semiconductor film  106 . In that case, the substrate is taken out of the heat treatment apparatus after the first heat treatment, and then the second photolithography step and the etching step are performed. 
     The structure obtained through the steps up to here is illustrated in  FIG.  3 B . 
     Then, a conductive film which is to be processed into the pair of second electrodes  109   a  and  109   b  is formed over the gate insulating layer  105  and the oxide semiconductor layer  126 . After that, the conductive film is etched using a resist mask formed through a third photolithography step, so that the pair of second electrodes  109   a  and  109   b  are formed. The method for forming the pair of second electrodes  109   a  and  109   b  may be similar to the method for forming the first electrode  103 . In this embodiment, the conductive film is formed to a thickness of 150 nm using titanium by a sputtering method. Note that the step for forming the pair of second electrodes  109   a  and  109   b  is combined with a step for forming the wiring  110  (see  FIG.  1 A ). 
     The structure obtained through the steps up to here is illustrated in  FIG.  3 C . 
     Next, the insulating layer  111  is formed in contact with the pair of second electrodes  109   a  and  109   b  and part of the oxide semiconductor layer  126 . The method for forming the insulating layer  111  may be similar to the method for forming the gate insulating layer  105 . In this embodiment, the insulating layer  111  is formed by a sputtering method with the use of silicon oxide. Note that the thickness of the insulating layer  111  is 200 nm. After that, second heat treatment whose heat temperature is different from that of the first heat treatment is performed. By the second heat treatment, part of oxygen included in the gate insulating layer  105  and the insulating layer  111  is supplied to the oxide semiconductor layer  126 , so that the oxide semiconductor layer  107  is formed. As the heat temperature of the second heat treatment gets higher, the amount of change in the threshold voltage which is caused by light irradiation or application of a BT stress becomes small. However, when the heat temperature is higher than 320° C., the on-state characteristics are degraded. Thus, the second heat treatment is performed under the conditions that the atmosphere is an inert atmosphere, an oxygen atmosphere, or a mixed atmosphere of oxygen and nitrogen, and the heat temperature is higher than or equal to 200° C. and lower than or equal to 400° C., preferably higher than or equal to 250° C. and lower than or equal to 320° C. In addition, heating time of the heat treatment is longer than or equal to 1 minute and shorter than or equal to 24 hours. Note that the second heat treatment may be performed after the formation of the third electrode  113  to be formed later. In addition, silicon nitride may be formed over the insulating layer  111  to prevent the intrusion of moisture or alkali metal. Since alkali metal such as Li or Na is an impurity, the content of alkali metal is preferably reduced. The concentration of the alkali metal in the oxide semiconductor layer  107  is lower than or equal to 2×10 16  cm −3 , preferably, lower than or equal to 1×10 15  cm −3 . Further the content of alkaline earth metal is preferably low because alkaline earth metal is also an impurity. Note that after the third electrode  113  described later is formed, silicon nitride may be formed as a protective insulating layer. In that case, the following steps and the like are appropriately performed: a step for forming opening portions in the protective insulating layer; a step for forming a conductive film to be electrically connected to the first electrode  103 , the pair of second electrodes  109   a  and  109   b,  and the third electrode  113 . 
     Next, the third electrode  113  is formed in contact with the insulating layer  111  so as to overlap with a channel formation region of the oxide semiconductor layer  107 . The third electrode  113  is formed in such a manner that a conductive film is formed on the insulating layer  111  and then the conductive film is etched using a resist mask formed through a fourth photolithography step. The method for forming the third electrode  113  may be similar to the method for forming the first electrode  103 . In this embodiment, the conductive film is formed to a thickness of 150 nm by a sputtering method with the use of molybdenum. Note that the step for forming the third electrode  113  is combined with a step for forming the wiring  114  (see  FIG.  1 A ). 
     The structure obtained through the steps up to here is illustrated in  FIG.  3 D . 
     Through the above steps, a non-linear element with high withstand voltage, low reverse saturation current, and high on-state current can be obtained. Note that this embodiment can be implemented in combination with any of the structures described in other embodiments, as appropriate. 
     Embodiment 2 
     In this embodiment, a non-linear element whose structure is partly different from that of the non-linear element described in Embodiment 1 will be described. Note that a transistor is used as an example also in this embodiment. 
       FIG.  4 A  is a plan view of a transistor  200 , and  FIG.  4 B  is a cross-sectional view taken along line E-F in the transistor  200 .  FIG.  4 C  is a cross-sectional view taken along line G-H in the transistor  200 . Note that the transistor  200  is a modified example of the transistor  100 ; therefore, in  FIGS.  4 A to  4 C , the same reference numerals are used for the same parts as those in  FIGS.  1 A to  1 C , and detailed description of the same reference numerals is omitted. 
     As illustrated in  FIG.  4 B , the transistor  200  is a dual-gate transistor and includes a base insulating layer  102 , a first electrode  103 , a gate insulating layer  105 , an oxide semiconductor layer  107 , n +  layers  117   a  and  117   b,  a pair of second electrodes  109   a  and  109   b,  an insulating layer  111 , and a third electrode  113 , which are provided over a substrate  101 . 
     The first electrode  103  is provided in contact with the base insulating layer  102 . The gate insulating layer  105  is provided to cover the first electrode  103 . The oxide semiconductor layer  107  is provided in contact with the gate insulating layer  105  to overlap with the first electrode  103 . The n +  layers  117   a  and  117   b  are formed to cover the gate insulating layer  105  and end portions of the oxide semiconductor layer  107 . The pair of second electrodes  109   a  and  109   b  are provided over the n +  layers  117   a  and  117   b  to cover the end portions of the oxide semiconductor layer  107 . The insulating layer  111  covers part of the oxide semiconductor layer  107  and the pair of second electrodes  109   a  and  109   b.  The third electrode  113  is provided on and in contact with the insulating layer  111  and between the pair of second electrodes  109   a  and  109   b.    
     The n +  layers  117   a  and  117   b  are formed between the oxide semiconductor layer  107  and the pair of second electrodes  109   a  and  109   b,  whereby contact resistance between the oxide semiconductor layer  107  and the pair of second electrodes  109   a  and  109   b  can be reduced. As a result, high on-state current can be obtained. In addition, when the n +  layers  117   a  and  117   b  are formed, parasitic capacitance can be reduced and the amount of change in on-state current (Ion degradation) between before and after application of a negative gate stress in a BT test can be suppressed. 
     Although only one of the pair of second electrodes  109   a  and  109   b  is illustrated in  FIG.  1 C , an etching step is performed using the pair of second electrodes  109   a  and  109   b  as masks to form the n +  layers  117   a  and  117   b,  and thus the n +  layers  117   a  and  117   b  are processed so that the end portions of the n +  layers  117   a  and  117   b  protrude from the pair of second electrodes  109   a  and  109   b.  Thus, the channel length of the transistor  200  is determined by the distance between the n +  layer  117   a  and the n +  layer  117   b.  Although the pair of second electrodes  109   a  and  109   b  face end surfaces of the oxide semiconductor layer  107  with the n +  layers  117   a  and  117   b  provided therebetween, the pair of second electrodes  109   a  and  109   b  serve as a heat sink in a manner similar to Embodiment 1 because the n +  layers  117   a  and  117   b  are not formed to be extremely thick and do not block the conduction of heat generated in the oxide semiconductor layer  107 . As a result, heat which is generated when high on-state current flows in the oxide semiconductor layer  107  can be dissipated to the outside, so that degradation of the transistor  200  due to heat generation can be suppressed. 
     Steps for manufacturing the transistor  200  are described below. The steps for manufacturing the transistor  200  are the same as those for manufacturing the transistor  100  except a step for forming the n +  layers  117   a  and  117   b.  Therefore, description is made with reference to  FIGS.  3 A and  3 B  and  FIGS.  5 A to  5 C . 
     In accordance with the manufacturing steps described in Embodiment 1 (after the structure of  FIG.  3 A  is obtained), the structure of  FIG.  3 B  is obtained. 
     Next, a film  115  to be the n +  layers  117   a  and  117   b  is formed to a thickness greater than or equal to 1 nm and less than or equal to 200 nm over the gate insulating layer  105  and the oxide semiconductor layer  107  with the use of an In—Zn-based metal oxide, an In—Sn-based metal oxide, or a single-component metal oxide containing indium or tin. The method for forming the film  115  may be the same as the method for forming the oxide semiconductor layer  107 . In addition, SiO 2  may be contained in the above material for the n +  layer. In this embodiment, an In—Sn-based metal oxide film containing SiO 2  is formed to a thickness of 100 nm. 
     Next, a conductive film  108  for forming the pair of second electrodes  109   a  and  109   b  is formed over the film  115  which is to be the n +  layers (see  FIG.  5 A ). The conductive film is processed to form the pair of second electrodes  109   a  and  109   b.  Note that the step for forming the pair of second electrodes  109   a  and  109   b  is combined with a step for forming a wiring  110  (see  FIG.  4 A ). 
     Then, the film  115  is processed using the pair of second electrodes  109   a  and  109   b  as masks to form the n +  layers  117   a  and  117   b.  Through this processing, the n +  layers  117   a  and  117   b  are formed so that the end portions thereof protrude from the pair of second electrodes  109   a  and  109   b  (see  FIG.  5 B ). Thus, the channel length of the transistor  200  is determined by the distance between the n +  layer  117   a  and the n +  layer  117   b.  On the other hand, the channel length of the transistor  100  described in Embodiment 1 is determined by the distance between the pair of second electrodes  109   a  and  109   b.  It is preferable that the taper angle of each of the end portions of the n +  layers  117   a  and  117   b  (an angle formed by the side surface of either the n +  layer  117   a  or the n +  layers  117   b  and a planar surface of the substrate  101 ) be less than or equal to 30°. 
     The subsequent steps are the same as those of Embodiment 1. The insulating layer  111  covering part of the oxide semiconductor layer  107  and the pair of second electrodes  109   a  and  109   b  is formed. Further, the third electrode  113  is formed in contact with the insulating layer  111  so as to overlap with a channel formation region of the oxide semiconductor layer  107  (see  FIG.  5 C ). Details of the steps are the same as those of Embodiment 1. 
     Through the above steps, a non-linear element with high withstand voltage, low reverse saturation current, and high on-state current can be obtained. Note that this embodiment can be implemented in combination with any of the structures described in other embodiments, as appropriate. 
     Embodiment 3 
     In this embodiment, a non-linear element whose structure is partly different from that of the non-linear element described in Embodiment 1 will be described. Note that a transistor is used as an example also in this embodiment. 
     A transistor  300  described in this embodiment is a transistor obtained in such a manner that the oxide semiconductor layer  107  of the transistor  100  described in Embodiment 1 is replaced with a crystalline oxide semiconductor stack  120  including a first crystalline oxide semiconductor layer  107   a  and a second crystalline oxide semiconductor layer  107   b.  In short, a plan structure of the transistor  300  is similar to that of the transistor  100 ; therefore,  FIG.  1 A  can be referred to for the plan view of the transistor  300 .  FIG.  6 A  is a cross-sectional view taken along line A-B in the transistor  300 .  FIG.  6 B  is a cross-sectional view taken along line C-D in the transistor  300 . Note that the transistor  300  is a modified example of the transistor  100 ; therefore, in  FIGS.  6 A and  6 B , the same reference numerals are used for the same parts as those in  FIGS.  1 A to  1 C , and detailed description of the same reference numerals is omitted. 
     As illustrated in  FIG.  6 A , the transistor  300  is a dual-gate transistor and includes a base insulating layer  102 , a first electrode  103 , a gate insulating layer  105 , the crystalline oxide semiconductor stack  120 , a pair of second electrodes  109   a  and  109   b,  an insulating layer  111 , and a third electrode  113 , which are provided over a substrate  101 . 
     The first electrode  103  is provided in contact with the base insulating layer  102 . The gate insulating layer  105  is provided to cover the first electrode  103 . The crystalline oxide semiconductor stack  120  is provided in contact with the gate insulating layer  105  to overlap with the first electrode  103 . The pair of second electrodes  109   a  and  109   b  cover end portions of the crystalline oxide semiconductor stack  120 . The insulating layer  111  covers part of the crystalline oxide semiconductor stack  120  and the pair of second electrodes  109   a  and  109   b.  The third electrode  113  is provided on and in contact with the insulating layer  111  and between the pair of second electrodes  109   a  and  109   b.    
     In addition, since the pair of second electrodes  109   a  and  109   b  are provided to cover the end portions of the crystalline oxide semiconductor stack  120 , the pair of second electrodes  109   a  and  109   b  are in contact with the end surfaces of the crystalline oxide semiconductor stack  120 . Therefore, at least in a region where the pair of second electrodes  109   a  and  109   b  are in contact with the crystalline oxide semiconductor stack  120 , the width of each of the pair of second electrodes  109   a  and  109   b  is larger than the width of a channel formed in the crystalline oxide semiconductor stack  120  (see  FIG.  1 A ). 
     Further, only one of the pair of second electrodes  109   a  and  109   b  is illustrated in  FIG.  6 B . Since the pair of second electrodes  109   a  and  109   b  are in contact with the end surfaces of the crystalline oxide semiconductor stack  120 , the pair of second electrodes  109   a  and  109   b  serve as a heat sink in a manner similar to other embodiments, and when heat is generated due to high on-state current flowing in the crystalline oxide semiconductor stack  120 , the pair of second electrodes  109   a  and  109   b  can dissipate the heat to the outside. As a result, degradation of the transistor  300  due to heat generation can be suppressed. 
     Here, the crystalline oxide semiconductor stack  120  is described. The crystalline oxide semiconductor stack  120  has a stacked structure of the first crystalline oxide semiconductor layer  107   a  and the second crystalline oxide semiconductor layer  107   b.    
     The first crystalline oxide semiconductor layer  107   a  has c-axis alignment. In addition, the second crystalline oxide semiconductor layer  107   b  also has c-axis alignment. Note that the first crystalline oxide semiconductor layer  107   a  and the second crystalline oxide semiconductor layer  107   b  include an oxide including a crystal with c-axis alignment (also referred to as c-axis aligned crystal (CAAC)), which has neither a single crystal structure nor an amorphous structure. 
     An oxide including CAAC refers to an oxide including a crystal with c-axis alignment, which has a triangular or hexagonal atomic arrangement when seen from the direction of an a-b plane, a surface, or an interface. In the crystal, metal atoms are arranged in a layered manner, or metal atoms and oxygen atoms are arranged in a layered manner along the c-axis, and the direction of the a-axis or the b-axis is varied in the a-b plane (the crystal rotates around the c-axis). 
     In a broad sense, an oxide including CAAC means a non-single-crystal oxide including a phase which has a triangular, hexagonal, regular triangular, or regular hexagonal atomic when seen from the direction perpendicular to the a-b plane and in which metal atoms are arranged in a layered manner or metal atoms and oxygen atoms are arranged in a layered manner when seen from the direction perpendicular to the c-axis. 
     The CAAC includes a crystalline region (a crystal region) but a boundary between one crystal region and another crystal region is not necessarily clear. That is, the first crystalline oxide semiconductor layer  107   a  and the second crystalline oxide semiconductor layer  107   b  partly include a crystal grain boundary. 
     In the case where oxygen is included in the CAAC, nitrogen may be substituted for part of oxygen included in the CAAC. The c-axes of individual crystalline portions included in the CAAC may be aligned in one direction (e.g., a direction perpendicular to a surface of a substrate over which the CAAC is formed, a surface, a film surface, or an interface of the CAAC). In addition, normals of the a-b planes of individual crystal regions included in the CAAC may be aligned in one direction (e.g., a direction perpendicular to the surface of the substrate over which the CAAC is formed, or the surface, the film surface, or the interface of the CAAC). 
     The CAAC may be a conductor, a semiconductor, or an insulator depending on its composition or the like. Further, The CAAC may transmit or not transmit visible light depending on its composition or the like. 
     As an example of such a CAAC, there is a crystal which is formed into a film shape and has a triangular or hexagonal atomic arrangement when observed from the direction perpendicular to a surface of the film or, a surface of a substrate, or an interface and in which metal atoms are arranged in a layered manner or metal atoms and oxygen atoms (or nitrogen atoms) are arranged in a layered manner when a cross section of the film is observed. 
     The first crystalline oxide semiconductor layer  107   a  and the second crystalline oxide semiconductor layer  107   b  are preferably formed using metal oxide containing at least zinc or metal oxide containing at least zinc and indium. For example, among the metal oxides described in Embodiment 1, an In—Sn—Ga—Zn-based metal oxide which is a four-component oxide; an In—Ga—Zn-based metal oxide, an In—Sn—Zn-based metal oxide, an In—Al—Zn-based metal oxide, a Sn—Ga—Zn-based metal oxide, an Al—Ga—Zn-based metal oxide, or a Sn—Al—Zn-based metal oxide, which is a three-component metal oxide; an In—Zn-based metal oxide, a Sn—Zn-based metal oxide, or an Al—Zn-based metal oxide which is a two-component metal oxide; a single-component metal oxide containing Zn; or the like can be used. 
     The first crystalline oxide semiconductor layer  107   a  is formed by a sputtering method in which a substrate temperature is higher than or equal to 200° C. and lower than or equal to 400° C., and after the formation, first heat treatment (at a temperature higher than or equal to 400° C. and lower than or equal to 750° C.) is performed. 
     Here, the crystal structure of the CAAC is described. Although depending on the temperature of the first heat treatment, the first heat treatment causes crystallization from a film surface and crystal grows from the film surface toward the inside of the film; thus, a c-axis aligned crystal is obtained. By the first heat treatment, a large amount of zinc and oxygen gather to the film surface, and one or more layers of graphene-type two-dimensional crystal including zinc and oxygen and having a hexagonal upper plane (a schematic plan view thereof is shown in  FIG.  7 A ) are formed at the outermost surface; the layers of crystal at the outermost surface grow in the thickness direction to form a stack of layers. In  FIG.  7 A , a white circle indicates a zinc atom, and a black circuit indicates an oxygen atom. By increasing the temperature of the heat treatment, crystal growth proceeds from the surface to the inside and further from the inside to the bottom. Further,  FIG.  7 B  schematically shows a stack formed of six layers of two-dimensional crystal as an example of a stack in which two-dimensional crystal has grown. 
     In the case where oxygen is contained in the gate insulating layer  105 , part of oxygen is diffused into an interface between the gate insulating layer  105  and the first crystalline oxide semiconductor layer  107   a  or the vicinity thereof by the first heat treatment, so that oxygen deficiency of the first crystalline oxide semiconductor layer  107   a  is reduced. Accordingly, at least oxygen whose amount exceeds the stoichiometric proportion is preferably contained in the film (in the bulk) of the gate insulating layer  105  or the interface between the first crystalline oxide semiconductor layer  107   a  and the gate insulating layer  105 . 
     The second crystalline oxide semiconductor layer  107   b  is formed by a sputtering method in which a substrate temperature in deposition is higher than or equal to 200° C. and lower than or equal to 400° C. By setting the substrate temperature in the deposition to be higher than or equal to 200° C. and lower than or equal to 400° C., precursors can be arranged in the oxide semiconductor layer formed on and in contact with the surface of the first crystalline oxide semiconductor layer  107   a  and so-called orderliness can be obtained. Then, second heat treatment is preferably performed at a temperature higher than or equal to 400° C. and lower than or equal to 750° C. after the formation. The second heat treatment is performed in a nitrogen atmosphere, an oxygen atmosphere, or a mixed atmosphere of nitrogen and oxygen, whereby the density of the second crystalline oxide semiconductor layer  107   b  can be increased and oxygen deficiency can be reduced. By the second heat treatment, crystal growth proceeds in the thickness direction with the use of the first crystalline oxide semiconductor layer  107   a  as a nucleus, that is, crystal growth proceeds from the bottom to the inside; thus, the second crystalline oxide semiconductor layer  107   b  is formed. 
     In a manner similar to the transistor  100 , when the thickness of the crystalline oxide semiconductor stack  120  is large, large current between the source electrode and the drain electrode can be ensured in the transistor  300 . 
     The drain withstand voltage of the transistor  300  depends on the thickness of the crystalline oxide semiconductor stack  120 . Therefore, in order to increase the drain withstand voltage, the thickness of the crystalline oxide semiconductor stack  120  is preferably large and may be selected in accordance with the desired drain withstand voltage. 
     Therefore, considering the amount of the on-state current and the drain withstand voltage, the thickness of the crystalline oxide semiconductor stack  120  is preferably greater than or equal to 0.1 μm and less than or equal to 50 μm, more preferably greater than or equal to 0.5 μm and less than or equal to 20 μm. 
     Further, the transistor  300  in which the crystalline oxide semiconductor stack  120  includes the channel region has orderliness in a direction along the interface. Therefore, in the transistor  300 , in the case where carriers flow along the interface of the crystalline oxide semiconductor stack  120 , that is, in the case where carriers flow in a direction substantially parallel to the a-b plane, the crystalline oxide semiconductor stack  120  does not block the flow. Therefore, degradation of the electrical characteristics of the transistor  300  is suppressed even with light irradiation or application of a BT stress. 
     Without limitation to the two-layer structure in which the second crystalline oxide semiconductor layer  107   b  is formed over the first crystalline oxide semiconductor layer  107   a,  a stacked structure including three or more layers may be formed by repeatedly performing a process of deposition and heat treatment for forming a third crystalline oxide semiconductor layer after the second crystalline oxide semiconductor layer  107   b  is formed. 
     Steps for manufacturing the transistor  300  are described below. The steps for manufacturing the transistor  300  are the same as those for manufacturing the transistor  100  except a step for manufacturing the crystalline oxide semiconductor stack  120 . Therefore, the steps are described with reference to  FIGS.  8 A to  8 D . 
     In accordance with the manufacturing steps described in Embodiment 1, steps up to the formation of the gate insulating layer  105  are performed, whereby a structure illustrated in  FIG.  8 A  is obtained. 
     Next, a first oxide semiconductor film is formed over the gate insulating layer  105 . The thickness of the oxide semiconductor film formed is smaller than the thickness of a second oxide semiconductor film which is to be the second crystalline oxide semiconductor layer  107   b.    
     In this embodiment, the first oxide semiconductor film is formed to a thickness of 100 nm in an oxygen atmosphere, an argon atmosphere, or an atmosphere including argon and oxygen under conditions where an In—Ga—Zn-based metal oxide target (In 2 O 3 :Ga 2 O 3 :ZnO=1:1:2 [molar ratio]) is used, the distance between the substrate and the target is 170 mm, the substrate temperature is 250° C., the pressure is 0.4 Pa, and the direct current (DC) power is 0.5 kW. 
     Next, the atmosphere in a chamber in which the substrate is put is set to a nitrogen atmosphere or dry air, and first heat treatment is performed. The temperature of the first heat treatment is higher than or equal to 400° C. and lower than or equal to 750° C. In addition, heating time of the first heat treatment is longer than or equal to 1 minute and shorter than or equal to 24 hours. By the first heat treatment, a first crystalline oxide semiconductor film is formed (see  FIG.  8 B ). Details of the first heat treatment are described in Embodiment 1 and thus, are omitted here. 
     Next, a second oxide semiconductor film which has a large thickness than the first crystalline oxide semiconductor film is formed over the first crystalline oxide semiconductor film. 
     In this embodiment, the second oxide semiconductor film is formed to a thickness of 400 nm in an oxygen atmosphere, an argon atmosphere, or an atmosphere including argon and oxygen under conditions where an In—Ga—Zn-based metal oxide target (In 2 O 3 :Ga 2 O 3 :ZnO=1:1:2 [molar ratio]) is used, the distance between the substrate and the target is 170 mm, the substrate temperature is 400° C., the pressure is 0.4 Pa, and the direct current (DC) power is 0.5 kW. 
     Next, the atmosphere in a chamber in which the substrate is put is set to a nitrogen atmosphere or dry air, and second heat treatment is performed. The temperature of the second heat treatment is higher than or equal to 400° C. and lower than or equal to 750° C. 
     In addition, heating time of the second heat treatment is longer than or equal to 1 minute and shorter than or equal to 24 hours. By the second heat treatment, the second crystalline oxide semiconductor film is formed (see  FIG.  8 C ). Details of the second heat treatment for forming the second crystalline oxide semiconductor film are similar to those of Embodiment 1. Note that in the drawings, an interface between the first crystalline oxide semiconductor film and the second crystalline oxide semiconductor film is denoted by a dashed line for description of the oxide semiconductor stack; however, the interface is actually not distinct and is illustrated for easy understanding. 
     When the first heat treatment and the second heat treatment are performed at a temperature higher than 750° C., a crack (a crack extended in the thickness direction) is easily generated in the formed oxide semiconductor film due to shrink of the glass substrate. Accordingly, in the case where the temperatures of the first heat treatment and the second heat treatment and the substrate temperature at the time of forming the oxide semiconductor film by a sputtering method are set to temperatures lower than or equal to 750° C., preferably lower than or equal to 450° C., a highly reliable transistor can be manufactured over a large area glass substrate. 
     It is preferable that steps from the formation of the gate insulating layer  105  to the second heat treatment be successively performed without exposure to the air. For example, a manufacturing apparatus whose top view is illustrated in  FIG.  12    may be used. The manufacturing apparatus illustrated in  FIG.  12    is the single wafer multi-chamber equipment, which includes three sputtering devices  10   a,    10   b,  and  10   c,  a substrate supply chamber  11  provided with three cassette ports  14  for holding a process substrate, load lock chambers  12   a  and  12   b,  a transfer chamber  13 , a substrate heating chamber  15 , and the like. Note that a transfer robot for transferring a substrate to be treated is provided in each of the substrate supply chamber  11  and the transfer chamber  13 . The atmospheres of the sputtering devices  10   a,    10   b,  and  10   c,  the transfer chamber  13 , and the substrate heating chamber  15  are preferably controlled so as to hardly contain hydrogen and moisture (i.e., as an inert atmosphere, a reduced pressure atmosphere, or a dry air atmosphere). For example, a preferable atmosphere is a dry nitrogen atmosphere in which the dew point of moisture is −40° C. or lower, preferably −50° C. or lower. An example of a procedure of the manufacturing steps with use of the manufacturing apparatus illustrated in  FIG.  12    is as follows. A process substrate is transferred from the substrate supply chamber  11  to the substrate heating chamber  15  through the load lock chamber  12   a  and the transfer chamber  13 ; moisture attached to the process substrate is removed by vacuum baking in the substrate heating chamber  15 ; the process substrate is transferred to the sputtering device  10   c  through the transfer chamber  13 ; and the gate insulating layer  105  is deposited in the sputtering device  10   c.  Then, the process substrate is transferred to the sputtering device  10   a  through the transfer chamber  13  without exposure to air, and the first oxide semiconductor film is formed in the sputtering device  10   a.  Then, the process substrate is transferred to the substrate heating chamber  15  though the transfer chamber  13  without exposure to air and the first heat treatment is performed. Then, the process temperature is transferred to the sputtering device  10   b  through the transfer chamber  13 , and the second oxide semiconductor film is formed in the sputtering device  10   b.  Then, the process substrate is transferred to the substrate heating chamber  15  through the transfer chamber  13 , and the second heat treatment is performed. As described above, with use of the manufacturing apparatus illustrated in  FIG.  12   , the steps for manufacturing a transistor can proceed without exposure to air. Further, the sputtering device in the manufacturing apparatus in  FIG.  12    can achieve a manufacturing process without exposure to air by changing a sputtering target. For example, the following process can be performed. The substrate over which the gate insulating layer  105  is formed in advance is placed in the cassette port  14 , and the steps from the formation of the first oxide semiconductor film to the second heat treatment are performed without exposure to air, so that the first crystalline oxide semiconductor film and the second crystalline oxide semiconductor film are formed. After that, in the sputtering device  10   c,  a conductive film to be the pair of second electrodes  109   a  and  109   b  can be formed with use of a metal target over the second crystalline oxide semiconductor film, without exposure to air. 
     Next, a crystalline oxide semiconductor stack including the first crystalline oxide semiconductor film and the second crystalline oxide semiconductor film is processed, so that the crystalline oxide semiconductor stack  120  in which the first crystalline oxide semiconductor layer  107   a  and the second crystalline oxide semiconductor layer  107   b  are stacked is formed (see  FIG.  8 D ). 
     A mask having a desired shape is formed over the crystalline oxide semiconductor stack and then the crystalline oxide semiconductor stack is etched with the use of the mask to perform the processing of the crystalline oxide semiconductor stack. The mask may be formed by a method such as photolithography or an ink-jet method. 
     For the etching of the crystalline oxide semiconductor stack, either dry etching or wet etching may be employed. It is needless to say that both of them may be employed in combination. Details of the dry etching and the wet etching are similar to those of Embodiment 1. 
     The subsequent steps are the same as those of Embodiment 1. The pair of second electrodes  109   a  and  109   b  are formed, and the insulating layer  111  covering part of the crystalline oxide semiconductor stack  120  and the pair of second electrodes  109   a  and  109   b  is formed. Further, the third electrode  113  is formed in contact with the insulating layer  111  so as to overlap with a channel formation region of the crystalline oxide semiconductor stack  120  (see  FIG.  6 A ). Details of the steps are similar to those of Embodiment 1. Note that the step for forming the pair of second electrodes  109   a  and  109   b  is combined with a step for forming a wiring  110  (see  FIG.  1 A ), and the step for forming the third electrode  113  is combined with a step for forming a wiring  114  (see  FIG.  1 A ). 
     Through the above steps, a non-linear element with high withstand voltage, low reverse saturation current, and high on-state current can be obtained. Note that this embodiment can be implemented in combination with any of the structures described in other embodiments, as appropriate. 
     Embodiment 4 
     In this embodiment, a non-linear element whose structure is partly different from that of the non-linear element described the above embodiment will be described. Note that a transistor is used as an example also in this embodiment. 
     A transistor  400  described in this embodiment is a transistor obtained in such a manner that the oxide semiconductor layer  107  of the transistor  200  described in Embodiment 2 is replaced with a crystalline oxide semiconductor stack  120  including a first crystalline oxide semiconductor layer  107   a  and a second crystalline oxide semiconductor layer  107   b.  In short, a plan structure of the transistor  400  is similar to that of the transistor  200 ; therefore,  FIG.  4 A  can be referred to for the plan view of the transistor  400 .  FIG.  9 A  is a cross-sectional view taken along line E-F in the transistor  400  (see  FIG.  4 A ).  FIG.  9 B  is a cross-sectional view taken along line G-H in the transistor  400  (see  FIG.  4 A ). Note that in  FIGS.  9 A and  9 B , the same reference numerals are used for the same parts as those in  FIGS.  1 A to  1 C , and details of the same reference numerals are omitted. 
     As illustrated in  FIG.  9 B , the transistor  400  is a dual-gate transistor and includes a base insulating layer  102 , a first electrode  103 , a gate insulating layer  105 , a crystalline oxide semiconductor stack  120 , n +  layers  117   a  and  117   b,  a pair of second electrodes  109   a  and  109   b,  an insulating layer  111 , and a third electrode  113 , which are provided over a substrate  101 . 
     The first electrode  103  is provided in contact with the base insulating layer  102 . The gate insulating layer  105  is provided to cover the first electrode  103 . The crystalline oxide semiconductor stack  120  is provided in contact with the gate insulating layer  105  to overlap with the first electrode  103 . The n +  layers  117   a  and  117   b  cover the gate insulating layer  105  and end portions of the crystalline oxide semiconductor stack  120 . The pair of second electrodes  109   a  and  109   b  cover the end portions of the crystalline oxide semiconductor stack  120  and end portions of the n +  layers  117   a  and  117   b.  The insulating layer  111  covers part of the crystalline oxide semiconductor stack  120  and the pair of second electrodes  109   a  and  109   b.  The third electrode  113  is provided on and in contact with the insulating layer  111  and between the pair of second electrodes  109   a  and  109   b.    
     The n +  layers  117   a  and  117   b  are formed between the crystalline oxide semiconductor stack  120  and the pair of second electrodes  109   a  and  109   b,  whereby contact resistance between the crystalline oxide semiconductor stack  120  and the pair of second electrodes  109   a  and  109   b  can be reduced. As a result, high on-state current can be obtained. In addition, when the n +  layers  117   a  and  117   b  are formed, parasitic capacitance can be reduced and the amount of change in on-state current (Ion degradation) between before and after application of a negative gate stress in a BT test can be suppressed. 
     Although only one of the pair of second electrodes  109   a  and  109   b  is illustrated in  FIG.  1 C , an etching step is performed using the pair of second electrodes  109   a  and  109   b  as masks to form the n +  layers  117   a  and  117   b,  and thus the n +  layers  117   a  and  117   b  are processed so that the end portions of the n +  layers  117   a  and  117   b  protrude from the pair of second electrodes  109   a  and  109   b.  Thus, the channel length of the transistor  400  is determined by the distance between the n +  layer  117   a  and the n +  layer  117   b.  Although the pair of second electrodes  109   a  and  109   b  face the crystalline oxide semiconductor stack  120  with the n +  layers  117   a  and  117   b  provided therebetween, the pair of second electrodes  109   a  and  109   b  serve as a heat sink in a manner similar to other embodiments because the n +  layers  117   a  and  117   b  are not formed to be extremely thick and do not block the conduction of heat generated in the crystalline oxide semiconductor stack  120 . As a result, heat which is generated when high on-state current flows in the crystalline oxide semiconductor stack  120  can be dissipated to the outside, so that degradation of the transistor  400  due to heat generation can be suppressed. 
     The crystalline oxide semiconductor stack  120  has a stacked structure of the first crystalline oxide semiconductor layer  107   a  and the second crystalline oxide semiconductor layer  107   b.  Details of the first crystalline oxide semiconductor layer  107   a  and the second crystalline oxide semiconductor layer  107   b  are similar to those of Embodiment 3. In the transistor  400  described in this embodiment, at least part of the first crystalline oxide semiconductor layer  107   a  and part of the second crystalline oxide semiconductor layer  107   b  are crystallized and have c-axis alignment, and the crystalline oxide semiconductor stack  120  has orderliness in a direction along the interface between the crystalline oxide semiconductor stack  120  and the gate insulating layer. Therefore, in the case where carriers flow along the interface, the crystalline oxide semiconductor stack  120  does not block the flow. Therefore, degradation of the electrical characteristics of the transistor  400  is suppressed even with light irradiation or application of a BT stress. 
     Steps for manufacturing the transistor  400  are described below. The steps for manufacturing the transistor  400  are the same as those for manufacturing the transistor  200  except a step for manufacturing the crystalline oxide semiconductor stack  120 . Therefore, the steps are described with reference to  FIGS.  5 A to  5 C  and  FIGS.  8 A to  8 D . 
     In accordance with the manufacturing steps described in Embodiment 1 and Embodiment 3, steps up to the formation of the crystalline oxide semiconductor stack  120  are performed, whereby a structure illustrated in  FIG.  8 D  is obtained. 
     Next, the film  115  to be the n +  layers  117   a  and  117   b  is formed to a thickness greater than or equal to 1 nm and less than or equal to 200 nm over the gate insulating layer  105  and the crystalline oxide semiconductor stack  120  with the use of an In—Zn-based metal oxide, an In—Sn-based metal oxide, or a single-component metal oxide containing indium or tin. The method for forming the film  115  is similar to that of Embodiment 2. In addition, SiO 2  may be contained in the above material for the n +  layers. In this embodiment, an In—Sn-based metal oxide film containing SiO 2  is formed to a thickness of 100 nm. 
     Next, a conductive film for forming the pair of second electrodes  109   a  and  109   b  is formed over the film  115  which is to be the n +  layers (see  FIG.  5 A ). The conductive film is processed to form the pair of second electrodes  109   a  and  109   b.  Note that the step for forming the pair of second electrodes  109   a  and  109   b  is combined with a step for forming a wiring  110  (see  FIG.  4 A ). 
     Then, the film  115  is processed using the pair of second electrodes  109   a  and  109   b  as masks to form the n +  layers  117   a  and  117   b.  Through this processing, the n +  layers  117   a  and  117   b  are formed so that the end portions thereof protrude from the pair of second electrodes  109   a  and  109   b  (see  FIG.  5 B ). Thus, the channel length of the transistor  400  is determined by the distance between the n +  layer  117   a  and the n +  layer  117   b.  On the other hand, the channel length of the transistor  300  described in Embodiment 3 is determined by the distance between the pair of second electrodes  109   a  and  109   b.  It is preferable that the taper angle of each of the end portions of the n +  layers  117   a  and  117   b  (an angle formed by the side surface of either the n +  layer  117   a  or the n +  layers  117   b  and a planar surface of the substrate  101 ) be less than or equal to 30°. 
     The subsequent steps are the same as those of Embodiment 2. The insulating layer  111  covering part of the crystalline oxide semiconductor stack  120  and the pair of second electrodes  109   a  and  109   b  is formed. Further, the third electrode  113  is formed in contact with the insulating layer  111  so as to overlap with a channel formation region of the crystalline oxide semiconductor stack  120  (see  FIG.  9 A ). Note that the step for forming the third electrode  113  is combined with a step for forming a wiring  114  (see  FIG.  4 A ), and details of the steps are the same as those of Embodiment 2. 
     Through the above steps, a non-linear element with high withstand voltage, low reverse saturation current, and high on-state current can be obtained. Note that this embodiment can be implemented in combination with any of the structures described in other embodiments, as appropriate. 
     Embodiment 5 
     In this embodiment, calculation results of on-state currents of non-linear elements will be described. Note that calculation is performed on simplified structures of the non-linear elements. Further, Sentaurus Device manufactured by Synopsys Inc. is used in the calculation. 
     First, a calculation result of a change in a drain current with respect to a change in a gate voltage is described. 
       FIG.  13 A  is a simplified view of a cross-sectional structure (Structure 1) along the channel length direction in the transistor  200  described in Embodiment 2 (a cross-sectional structure taken along line E-F in  FIG.  4 A ).  FIG.  13 B  is a simplified view of a cross-sectional structure taken along line G-H in  FIG.  4 A .  FIG.  13 C  is a simplified view of a cross-sectional structure along a direction perpendicular to the line E-F in a channel formation region of the transistor  200 . Note that components of  FIGS.  13 A to  13 C  corresponding to those of  FIGS.  4 A to  4 C  are denoted by the same numerals as those of  FIGS.  4 A to  4 C . 
     Parameters reflected on the calculation result of on-state current in the structure of  FIGS.  13 A to  13 C  are as follows: 
     1. Channel length L 1 : 10 μm 
     2. Length L 2  of the pair of second electrodes  109   a  and  109   b:  5 μm 
     3. Thickness T os  of the oxide semiconductor layer  107 : 10 μm 
     4. Thickness T G  of the gate insulating layer  105  and the thickness T BG  of the insulating layer  111 : 0.2 μm 
     5. Channel width W 1 : 100 μm 
     6. Width W 2  of the pair of second electrodes  109   a  and  109   b:  5 μm 
     7. Work function ϕM of tungsten used for the first electrode  103 : 4.9 eV 
     8. Work function ϕM of titanium used for the pair of second electrodes  109   a  and  109   b:  4.0 eV 
     9. Work function ϕM of molybdenum used for the third electrode  113 : 4.8 eV 
     10. Band gap Eg, electron affinity χ, relative permittivity, and electron mobility of In—Ga—Zn—O-based metal oxide used for the oxide semiconductor layer  107 : 3.15 eV, 4.3 eV, 15, and 10 cm 2 /Vs 
     11. Relative permittivity of silicon oxynitride used for the gate insulating layer  105 : 4.1 
     12. Relative permittivity of silicon oxide used for the insulating layer  111 : 3.8 
     Note that the calculation is performed on the assumption that the first electrode  103 , the pair of second electrodes  109   a  and  109   b,  the third electrode  113 , and the n +  layers  117   a  and  117   b  have the same potential regardless of their thicknesses, and thus, the calculation results do not reflect the thicknesses. 
       FIG.  14    shows the calculation result of the drain current (Id) where the drain voltage is 15 V and the gate voltage varies from 0 V to 20 V. As found from  FIG.  14   , the non-linear element in which the end portions of the oxide semiconductor layer  107  are covered with the n +  layers  117   a  and  117   b  and the pair of second electrodes  109   a  and  109   b  can have high on-state current. 
     Next, a calculation result of a change in the drain current with respect to a change in the drain voltage in Structure 1 is described. Comparative examples are Structure 2 to Structure 4, which are described below. 
     Structure 2 is a structure of the transistor  200  (see  FIGS.  15 A to  15 C ), in which the pair of second electrodes  109   a  and  109   b,  the n +  layers  117   a  and  117   b,  and the third electrode  113  are not in contact with the side surfaces of the oxide semiconductor layer  107 . As for Structure 2,  FIG.  15 A  is a view simplified in the same manner as  FIG.  13 A .  FIG.  15 B  is a view simplified in the same manner as  FIG.  13 B .  FIG.  15 C  corresponds to the structure of  FIG.  13 C . 
     Structure 3 is a structure (see  FIGS.  16 A to  16 C ) in which the pair of second electrodes  109   a  and  109   b  and the n +  layers  117   a  and  117   b  are in contact with the side surfaces of the oxide semiconductor layer  107 , and the third electrode  113  is not in contact with the side surfaces of the oxide semiconductor layer  107 . As for Structure 3,  FIG.  16 A  is a view simplified in the same manner as  FIG.  13 A .  FIG.  16 B  is a view simplified in the same manner as  FIG.  13 B .  FIG.  16 C  corresponds to the structure of  FIG.  13 C . 
     Structure 4 is a structure (see  FIGS.  17 A to  17 C ) in which the pair of second electrodes  109   a  and  109   b  and the n +  layers  117   a  and  117   b  are not in contact with the side surfaces of the oxide semiconductor layer  107 , and the third electrode  113  is in contact with the side surfaces of the oxide semiconductor layer  107 . As for Structure 4,  FIG.  17 A  is a view simplified in the same manner as  FIG.  13 A .  FIG.  17 B  is a view simplified in the same manner as  FIG.  13 B .  FIG.  17 C  is a view simplified in the same manner as  FIG.  13 C . 
     Parameters of Structure 2 to Structure 4 which are reflected in calculation results are the same as those of Structure 1. Note that the calculation is performed on the assumption that the first electrode  103 , the pair of second electrodes  109   a  and  109   b,  the third electrode  113 , and the n +  layers  117   a  and  117   b  have the same potential regardless of their thicknesses, and thus, the calculation results do not reflect the thicknesses. 
     The calculation results of the on-state currents of Structure 1 to Structure 4 are shown in  FIG.  18   .  FIG.  18    shows the calculation results of drain currents (Id) corresponding to drain voltages, where the gate voltage (Vg) is 10 V and the drain voltages (Vd) vary from 0 V to 20 V. 
     As shown in  FIG.  18   , the drain current of Structure 1 is higher than any of the drain currents of Structure 2 to Structure 4. In other words, a structure like Structure 1 in which the pair of second electrodes  109   a  and  109   b,  the n +  layers  117   a  and  117   b,  and the third electrode  113  are in contact with the side surfaces of the oxide semiconductor layer  107  is employed, carriers can be efficiently injected into the oxide semiconductor layer  107  that includes the channel formation region and high on-state current can be obtained, which is favorable to a non-linear element for large current application. 
     This embodiment can be implemented in appropriate combination with any of the structures described in the other embodiments. 
     Embodiment 6 
     In this embodiment, examples of a power diode and a rectifier with the use of a non-linear element that is one embodiment of the present invention will be described with reference to FIG.  10 A 1  to  10 C 2  and  FIGS.  11 A and  11 B . 
     FIG.  10 A 1  illustrates an example of a structure of a power diode that is one embodiment of the present invention. In a power diode illustrated in FIG.  10 A 1 , a plurality of diodes are connected in series. 
     FIG.  10 B 1  illustrates an example of a structure of a rectifier that is one embodiment of the present invention. The rectifier illustrated in FIG.  10 B 1  is a half-wave rectifier including two diodes. An anode of a first diode is connected to a lower potential side reference potential (preferably, a ground potential). A cathode of the first diode is connected to an input portion and an anode of a second diode. A cathode of the second diode is connected to an output portion. 
     FIG.  10 C 1  illustrates an example of a structure of a rectifier that is one embodiment of the present invention. The rectifier illustrated in FIG.  10 C 1  is a full-wave rectifier including four diodes. The four diodes are referred to as first to fourth diodes clockwise from the diode on the upper left. Anodes of the first diode and the fourth diode are connected to a reference potential (preferably a ground potential) on the lower potential side. A cathode of the first diode and an anode of the second diode are connected to a first input portion. An anode of the third diode and a cathode of the fourth diode are connected to a second input portion. A cathode of the second diode and a cathode of the third diode are connected to an output portion. 
     As the diodes used in the power diode, the half-wave rectifier, and the full-wave rectifier, non-linear elements described in the above embodiment, in each of which one of the pair of second electrodes  109   a  and  109   b  which functions as the source electrode or the drain electrode is electrically connected (diode-connected) to the first electrode  103  functioning as the gate electrode can be used (see  FIGS.  11 A and  11 B ). In the case where the non-linear element described in the above embodiment is an n-type non-linear element, electrodes with diode connection are an anode and an electrode without diode connection is a cathode. 
       FIG.  11 A  is a plan view of the diode-connected non-linear element.  FIG.  11 B  is a cross-sectional view taken along line I-J in  FIG.  11 A . As illustrated in  FIG.  11 B , a wiring  110  including the pair of second electrodes  109   a  and  109   b  is electrically connected to a wiring  104  including the first electrode  103  through an opening portion  150 . Note that although the transistor  100  described in Embodiment 1 is used in this embodiment, the transistor of any of Embodiment 2 to Embodiment 4 can also have a diode connection. 
     The power diode of FIG.  10 A 1  can have a structure illustrated in FIG.  10 A 2  with the use of diode-connected transistors. The half-wave rectifier of FIG.  10 B 1  can have a structure illustrated in FIG.  10 B 2  with the use of diode-connected transistors. The full-wave rectifier of FIG.  10 C 1  can have a structure illustrated in FIG.  10 C 2  with the use of diode-connected transistors. 
     In FIG.  10 A 2 , the transistors included in the power diode, which are dual-gate transistors, each include a third electrode  113  (including a wiring  114 ) (see  FIGS.  11 A and  11 B ). The third electrodes  113  to which control signals G 1  to G 5  are supplied control threshold voltages of the respective diode-connected transistors. Also in FIGS.  10 B 2  and  10 C 2 , since every transistor includes the third electrode  113 , the threshold voltages of the diode-connected transistors can be controlled by the control signals G 1  to G 4 . For example, in view of reliability as described in the above embodiment, the electrical characteristics of a transistor including an oxide semiconductor vary by irradiation with visible light and ultraviolet light or application of heat or an electric field. For example, a transistor becomes normally-on. Further in the case where the half-wave rectifier and the full-wave rectifier are formed using n-channel transistors, when the n-channel transistors become normally-on, current flows in the half-wave rectifier and the full-wave rectifier even when a reverse bias is applied thereto, whereby a normal rectification action cannot be obtained. Thus, negative potential is applied to the third electrodes  113  functioning as back gate electrodes of the transistors included in the half-wave rectifier and the full-wave rectifier, which prevents the transistors from becoming normally-on and reduces reverse current, so that a favorable rectification action can be obtained. 
     Note that in this embodiment, the third electrodes  113  of the transistors included in the power diode and the rectifier are supplied with respective control signals; however, the third electrodes  113  may be connected to each other and the transistors included in the power diode and the rectifier may be supplied with the same signals. Further, in the drawings, “OS” written in the vicinity of a circuit symbol of the transistor included in the power diode and the rectifier indicates that the transistor includes an oxide semiconductor layer. 
     Further, an oxide semiconductor can be used in the transistors included in the power diode and the rectifier described in this embodiment. Therefore, according to any of the above embodiments, a power diode and a rectifier having excellent drain withstand voltage and high drain current can be obtained. 
     Through the above steps, a non-linear element with characteristics such as high withstand voltage and low reverse saturation current, which can have high on-state current can be obtained. Note that this embodiment can be implemented in combination with any of the structures described in other embodiments, as appropriate. 
     This application is based on Japanese Patent Application serial no. 2010-204693 filed with Japan Patent Office on Sep. 13, 2010, the entire contents of which are hereby incorporated by reference.