Well Anand, that may or may not be a good idea.
It depends on the circuit and application. First, the battery fully charged is 4.2V so if you have 3.3V parts that can't go over 3.6V (most of them) you will destroy your components. Then the battery voltage drops pretty linearly over time to about 2.75V when you should recharge it. If you are using the ADC, this change in voltage may mess up your measurements. There are other voltage dependent parameters that may or may not be critical to the application. Finally, these self discharge about 15% per month sitting on the shelf, so if the low power requirement comes from trying to make a battery last as long as possible, there are far better types of batteries - Lithium/Iron Disulfide for instance that have a 20 year shelf life and a much flatter discharge voltage. Without the actual application we can't be completely sure what's important.
But you are correct to consider going away from the 9V battery. Nearly half the power is currently being wasted as heat. 4 AAs (Li/FeS2) in conjunction with a low drop out and low idle current regulator would be a great generalized solution.
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