For example, I want to express that the company is in CPU industry within semiconductor sector. I would like to let other know the specific indsutry of the company's business and the broad sector which the industry belong s to. Is the epxression "in CPU industry within semiconductor sector" correct or is there some better way to say this?
Both terms are used much more strictly in the financial world than in general use. You'll find businesses are listed on the stock market along with their industry and sector which are clearly defined.
In general use, 'industry' can refer to all the people and companies involved in a particular kind of business, for example, the manufacturing industry could refer to any business that produces something. It can also be used more specifically, for example, the car industry for those businesses who produce cars. 'Industry' predominantly refers to businesses and areas of business that manufacture or produce things, although computer software produced by the software industry, as well as the internet and e-commerce have perhaps changed that somewhat, and businesses dealing in less tangible assets may still be termed as industries.
Similarly, in general use, 'sector' has a much broader meaning and, much like the related word 'section', it just refers to a specific area of a greater whole. So, depending on the context, it can mean different things. 'Sector' is often used to refer to a particular part of a country's economy, for example, 'the banking sector'. One key difference to 'industry' is that we also refer to all money-making businesses collectively as 'the private sector', while government services such as policing, welfare, healthcare (depending on the country you live in) are collectively called the public sector. You could not refer to these as 'industry' because public services are not supposed to make many like businesses.
In the past, 'trade' and 'industry' were considered very different things, ie a business that manufactured something might be an 'industry', but those that sold the product on to consumers were not. However, as the size and complexity of retail businesses has grown, it is now common to hear 'the retail sector' and possibly even 'the retail industry'.
With reference to your specific example, I think 'semiconductors' are far too specific an industry to be termed a 'sector'. For example, the Intel Corporation are listed on the stock market as being in the 'semiconductor industry', and the 'technology sector'. If a business manufactured only CPUs, I would probably say that they were in the semiconductor industry but specialising in CPUs.